mirror of https://github.com/CGAL/cgal
3335 lines
110 KiB
TeX
3335 lines
110 KiB
TeX
\documentclass[a4paper]{article}
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\usepackage{html}
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\usepackage[dvips]{graphics,color,epsfig}
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\usepackage{path}
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\usepackage{amssymb}
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\usepackage{amsfonts}
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\usepackage{amsmath}
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\usepackage{amsthm}
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\usepackage{psfrag}
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\usepackage{algorithm}
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\usepackage{algpseudocode}
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\newcommand{\N}{\ensuremath{\mathbb{N}}}
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\newcommand{\F}{\ensuremath{\mathbb{F}}}
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\newcommand{\Z}{\ensuremath{\mathbb{Z}}}
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\newcommand{\R}{\ensuremath{\mathbb{R}}}
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\newcommand{\Q}{\ensuremath{\mathbb{Q}}}
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\newcommand{\C}{\ensuremath{\mathbb{C}}}
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\newcommand{\MBI}[1]{\ensuremath{M_{#1}^{-1}}}
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\newcommand{\RMBI}[1]{\ensuremath{\check{M}_{#1}^{-1}}}
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\newcommand{\pmu}[2]{\ensuremath{p_{\mu_{j}}^{(#1)}(\varepsilon, #2)}}
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\newcommand{\pmuz}[2]{\ensuremath{\dot{p}_{\mu_{j}}^{(#1)}(\varepsilon, #2)}}
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\newcommand{\px}[3]{\ensuremath{p_{x_{#1}}^{(#2)}(\varepsilon, #3)}}
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\newcommand{\pxz}[3]{\ensuremath{\dot{p}_{x_{#1}}^{(#2)}(\varepsilon, #3)}}
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\newcommand{\xe}[1]{\ensuremath{x_{#1}(\varepsilon)}}
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\newcommand{\cab}[3]{\ensuremath{\mathcal{A}_{>}(#1, #2, #3)}}
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\newcommand{\can}[3]{\ensuremath{\mathcal{A}_{<}(#1, #2, #3)}}
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\newtheorem{lemma}{Lemma}
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\newtheorem{assumption}{Assumption}
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\newtheorem{definition}{Definition}
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\title{Degeneracy}
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\author{Frans J.\ Wessendorp}
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%
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\begin{document}
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\maketitle
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\section{Row rank and column rank assumption}
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The QP-solver so far assumes nondegeneracy of the quadratic program to
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solve~\cite{Sven}, page 19. For ease of reference we will restate the
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nondegeneracy assumptions, that is given the QP,
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%\begin{equation}
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\begin{eqnarray}
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\label{def:QP}
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(QP)\quad minimize& c^{T}x + x^{T} D x & \nonumber \\
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s.t. & \sum_{j=0}^{n-1}a_{ij}x_{j} = b_{i} & i \in E \nonumber \\
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& \sum_{j=0}^{n-1}a_{ij}x_{j} \leq b_{i} & i \in I^{\leq} \\
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& \sum_{j=0}^{n-1}a_{ij}x_{j} \geq b_{i} & i \in I^{\geq} \nonumber \\
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& x_{j} \geq 0 & j \in \{0 \ldots n-1 \}
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\nonumber
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\end{eqnarray}
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%\end{equation}
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with $D$ positive semi definite, where $I:= I^{\leq} \cup I^{\geq}$ and
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$\left| E \right| + \left| I \right| = m$,
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the following conditions
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\begin{assumption} \label{ass:nondegeneracy}
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Nondegeneracy
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\begin{enumerate}
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\item $Rank\left( A \right) = m$
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\item The subsystem $A_{G}x_{G} = b$ has only solutions for sets $G \subseteq
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\{0 \ldots n-1\}$ with $\left|G \right| \geq m$.
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\end{enumerate}
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\end{assumption}
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must be met. In the following we will show how these
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assumptions can be dropped. To this end we will first describe how the auxiliary
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problem is set up.
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\section{The auxiliary problem}
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The auxiliary problem is constructed by augmenting the constraint matrix $A$
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with columns corresponding to slack and artificial variables. The slack columns
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are added in the standard way. For each of the equality constraints
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$\sum_{j=0}^{n-1} {a_{i j}x_{j}} = b_{i}, i \in E$ the original matrix $A$
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is augmented by an artificial column
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\begin{equation} \label{def:art_col}
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\tilde{a} = \left\{
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\begin{array}{ll}
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-e_{i} & \mbox{if $b_{i} < 0$} \\
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e_{i} & \mbox{otherwise,}
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\end{array}
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\right.
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\end{equation}
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where $e_{i}$ denotes the $i$-th column of the identity matrix.
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If the set of inequality constraints with infeasible origin
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$I_{inf}:=\{i \in I^{\leq} \left| \right. b_{i} < 0 \} \cup
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\{i \in I^{\geq} \left| \right. b_{i} > 0 \}$
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is nonempty the original matrix $A$
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is augmented by a special artificial column $\tilde{a}^{s}$, defined as
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\begin{equation}\label{def:spec_art_col}
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\tilde{a}^{s}_{i} = \left\{
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\begin{array}{ll}
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-1 & \mbox{if $i \in I^{\leq}, b_{i} < 0$} \\
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1 & \mbox{if $i \in I^{\geq}, b_{i} > 0$} \\
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0 & \mbox{otherwise}
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\end{array}
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\right.
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\end{equation}
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If we denote by $O$ the index set of original variables, by $S$ the index set
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of slack variables, by $art$ the index set of artificial variables and by
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$\tilde{A}$ the original constraint matrix $A$ augmented in the above way
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the auxiliary problem may be expressed as
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\begin{eqnarray*}
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\mbox{minimize} & \tilde{c}^{T}x & \\
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s.t. & \tilde{A}x = b_{i} & i \in \{0 \ldots m-1 \} \\
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& x_{j} \geq 0 & j \in O \cup S \cup art
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\end{eqnarray*}
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where
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\begin{equation} \label{def:aux_c}
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\tilde{c}_{j} = \left\{
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\begin{array}{ll}
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0 & \mbox{if $j \in O \cup S$} \\
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> 0 & \mbox{if $j \in art$}
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\end{array}
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\right.
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\end{equation}
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For later use we introduce the bijection $\sigma$ defined as
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\begin{equation}
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\sigma: S \cup art \setminus \{\tilde{a}^{s}\} \rightarrow I \cup E
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\end{equation}
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which maps the index sets of slack and artificial variables to
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the index sets of their inequality and equality constraints.
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\subsection{Initialization of the auxiliary problem}
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Since only constraints which are satisfied with equality determine the values of
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the basic variables with respect to a given basis only the equality constraints
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as well as the currently active inequality constraints are
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considered (see also ~\cite{Sven}, Section 2.4).
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To this end the set of basic variables $B$ is partitioned into original and
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slack variables, that is
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$B=B_{O} \cup B_{S}$, where $B_O \subseteq O \cup art$,
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and the set of inequality constraints $I$ is partitioned as
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$I=S_{B} \cup S_{N}$, where $S_{B}:=\sigma(B_{S})$
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and
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$S_{N}:=\sigma(S \setminus B_{S})$, if $\sigma$ denotes the bijection
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$S \rightarrow I$.
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The set of active constraints
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$C=E \cup S_{N}$ is
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introduced, such that a `reduced' basis matrix $\check{A}_{B}$
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with respect to $B$ is defined as
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\begin{equation}
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\label{def:red_basis_phaseI}
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\check{A}_{B}:=\tilde{A}_{C, B_{O}}
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\end{equation}
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Note that for degenerate bases active constraints do not neccessarily occur in
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the index set $C$ only.
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For later use, we introduce $A_{B}$ for the unreduced basis,
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\begin{equation}
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\label{def:basis_phaseI}
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A_{B}:= \tilde{A}_{C \cup S_{B}, B_{O} \cup B_{S}}
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\end{equation}
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Let $i_{0} \in I_{inf}$ be the index of a constraint that has a most infeasible
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origin, that is
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\[
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\left| b_{i_{0}} \right| \geq \left|b_{i}\right|, \quad i \in I_{inf}
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\]
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then $B_{O}$, $B_{S}$ and the initial set of basic and nonbasic constraints
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$S_{B}$ and $S_{N}$are initialized as
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\begin{equation}
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\begin{array}{ccccccc}
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\label{def:headings_init_io}
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B_{O}^{(0)} &:=& art && B_{S}^{(0)} &:=& S \setminus
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\{\sigma^{-1}\left(i_{0}\right)\} \\
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S_{B}^{(0)} &:=& I \setminus \{i_{0}\} && S_{N}^{(0)} & := & \{ i_{0} \}
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\end{array}
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\end{equation}
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If on the other hand, $I_{inf}=\emptyset$ then $B_{O}$, $B_{S}$, $S_{B}$ and
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$S_{N}$ are initialized as
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\begin{equation}
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\begin{array}{ccccccc}
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\label{def:headings_init_fo}
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B_{O}^{(0)} &:=& art && B_{S}^{(0)} &:=& S \\
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S_{B}^{(0)} &:=&I && S_{N}^{(0)}&:=& \emptyset
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\end{array}
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\end{equation}
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where $art$ does not contain a special artificial variable.
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\subsection{Expelling artificial variables from the Basis}
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At the end of phase I some artificial variables may remain in the basis.
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In that case, if the original problem is to be feasible, the basis has to be
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degenerate.
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Chv\'{a}tal~\cite{Chvatal}, Chapter 8, describes a procedure that,
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given a system $Ax = b$ and an optimal
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basis $B$, computes a subsystem $A'x = b'$ of constraints with $A'$ having full
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row rank and a basis $B'$ such that the set of feasible solutions for both
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systems is the same. The procedure tries to pivot the artificial variables out of
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the basis, the constraints corresponding to the artificial variables that can
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not be driven out of the basis in this manner can be removed without changing
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the solution set of the system.
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Since the proof for the equality of the solution sets of $Ax = b$ and
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$A'x = b'$ only works if the special artificial variable can be pivoted out of
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the basis we show first that this can always be achieved.
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For the sake of notational convenience we assume here that the index set of the
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artificial variables $art\setminus \{\tilde{a}^{s}\}$ is defined as
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$art\setminus \{\tilde{a}^{s}\}=\{l+1 \ldots l+\left|E\right| \}$,
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$l \geq n-1$ and that the basic artificial variable $x_{i+l}$ appears in the
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basis heading $B_{O}$ at position $i$.
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\subsubsection{The special artificial variable}
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\label{sec:spec_art_unpert}
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Suppose the special artificial variable is the $k$-th entry in the basis
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heading $B$. Then
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\[
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r = e_{k}^{T} \check{A}_{B}^{-1}
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\]
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denotes the corresponding row of the basis inverse and
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the nonbasic variable $x_{j}$ can be pivoted into the basis
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with the special artificial variable leaving iff
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\begin{equation} \label{eq:piv_precond}
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e_{k}^{T}\check{A}_{B}^{-1}\tilde{A}_{C,j} \neq 0
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\end{equation}
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This can easily be verified by considering the corresponding eta matrix
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whose $k$-th column is $\check{A}_{B}^{-1}\tilde{A}_{C,j}$ and whose determinant
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is nonzero iff condition (\ref{eq:piv_precond}) holds.
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Since $r \check{A}_{B} = e_{k}^{T}$, there exists by definition of
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the special artificial column $\tilde{a}^{s}$ and the fact that $B$ is a basis
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at least one
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$i \in I_{inf} \cap S_{N} \supset \emptyset$ such that
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$r_{i} \neq 0$, which in turn implies that condition
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(\ref{eq:piv_precond}) holds for some nonbasic
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$j = \sigma^{-1} \left( i \right)$.
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\subsubsection{Pivoting the artificial variables out of the basis}
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The procedure described in~\cite{Chvatal} is outlined in
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algorithm~(\ref{alg:expel_art_var}). We avoid iterating
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over sets that change during the computation, we only use member tests on such
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sets. The primitive $update(j,i)$
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updates the reduced basis inverse $\check{A}_{B}^{-1}$ with the entering
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variable $x_{j}$ and leaving variable $x_{i}$ and updates the basis heading
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accordingly. The procedure claims that every solution of
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\begin{equation}
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\sum_{j=0}^{n-1}a_{ij}x_{j}=b_{i} \quad i \in E
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\end{equation}
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is also a solution of
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\begin{equation}
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\sum_{j=0}^{n-1}a_{ij}x_{j}=b_{i} \quad i \in E \setminus J
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\end{equation}
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the proof of which we omit here.
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\begin{algorithm}
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\caption{Expel basic artificial variables from basis}
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\label{alg:expel_art_var}
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\begin{algorithmic}[0]
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\ForAll{$i \in art$}
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\If{$i \in B_{O}$}
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\State $r \gets e_{i}^{T}\check{A}_{B}^{-1}$
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\ForAll{$j \in O \cup S$}
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\If{$j \in N$}
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\If{$r\tilde{A}_{C,j}\neq 0$}
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\State $update(j,i)$
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\EndIf
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\EndIf
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\EndFor
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\EndIf
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\EndFor
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\State $J \gets \sigma(B_{O} \cap art)$
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\end{algorithmic}
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\end{algorithm}
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\subsubsection{Removing the remaining artificial variables and their
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constraints}
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Let $B^{(k)}$ and $N^{(k)}$ denote the set of indices of basic and nonbasic
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variables at the end of Algorithm~(\ref{alg:expel_art_var}).
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Since for the remaining basic artificial variables $x_{i}$,
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$i \in B_{O}^{(k)} \cap art$
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\begin{equation}
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e_{i}^{T}\check{A}_{B^{(k)}}^{-1}\tilde{A}_{N^{(k)}} = 0
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\end{equation}
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holds,
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we may, according to the procedure described in~\cite{Chvatal}, remove
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the artificial variable $x_{i}$ together with its constraint
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$\tilde{A}_{\sigma(i), \bullet}$ without changing the set of feasible
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solutions.
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This can be achieved by applying a slightly modified update
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of type U8, defined in Section~6.3.2 in~\cite{Sven}. An $U8(j,i)$ update is a
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LP-update and replaces an original variable $x_{i}$, $i \in B_{O}$ in the basis
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by a slack variable $x_{j}$, $j \in S \setminus B_{S}$,
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thus $S_{N}$ and $B_{O}$, or more generally, the sets
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$C=E \cup S_{N}$ and $B_{O}$ each decrease by one element. Thus the reduced
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basis matrix $\check{A}_{B}$ is shrunk by the
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row $\left(\check{A}_{B}\right)_{\sigma(j), \bullet}$ and
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the column $\left(\check{A}_{B}\right)_{\bullet, i}$. Provided the
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update mechanism is general enough to handle variables $x_{j}$, $j \in art$ as
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well instead of $x_{j}$, $j \in S \setminus B_{S}$ only, or equivalently,
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the update mechanism is capable of
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removing rows $\left(\check{A}_{B}\right)_{\sigma(j)}$ with $\sigma(j) \in E$
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as well instead of $\sigma(j) \in S_{N}$ only, we can use the update $U8(i,i)$
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for our purposes.
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Of course the update of the basis headings needs appropriate modification
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in this case, that is, $B_{O}^{\prime}:=B_{O} \setminus \{i\}$ and
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$E^{\prime}:=E \setminus \{ \sigma(i)\}$, if $\check{A}_{B^{\prime}}$ denotes
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the basis matrix after the update.
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\subsubsection{Dropping the row rank assumption}
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Due to performance reasons alluded to in the last subsection above the feature
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of removing redundant equalities is provided only if the compile time tag
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\texttt{Has\_no\_inequalities} has type \texttt{Tag\_false}.
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So if the constraint matrix $A$ has
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only equality constraints, $I=\emptyset$ in Definition~(\ref{def:QP}), and the
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constraint matrix is suspected not to have full rank one can define the
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compile time tag \texttt{Has\_no\_inequalities} to be of type
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\texttt{Tag\_false} at the price of
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some small performance penalty. If on the other hand the compile time tag has
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type \texttt{Tag\_true}, in that case the solver aborts in case of $Rank(A)<m$.
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\section{The lexicographic method}
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In order to being able to drop the second condition of
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Assumption~(\ref{ass:nondegeneracy}) we use a variation of the lexicographic
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method. Since the standard lexicographic method enlarges the feasible region
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in the explicit constraints with respect to the original problem by perturbing
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the right hand side, it is only applicable for inequality
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constraints. Even if this obstacle can be overcome for equality constraints we
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may still have artificial variables in the optimal basis of the auxiliary
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problem, pivoting them out of the basis as described above may render the
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perturbed problem infeasible, since the pivots here are no longer degenerate.
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Instead of altering the explicit constraints we enlarge the feasible region by
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perturbing the implicit constraints from $x_{j} \geq 0$ to $x_{j} \geq
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-\varepsilon^{j +1}$ in Definition~(\ref{def:QP}) with $0 < \varepsilon < 1$,
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such that the perturbed problem is defined as
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\begin{eqnarray}
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\label{def:QP_eps}
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(QP_{\varepsilon})\quad minimize& c^{T}x + x^{T} D x & \nonumber \\
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s.t. & \sum_{j=0}^{n-1}a_{ij}x_{j} = b_{i} & i \in E \nonumber \\
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& \sum_{j=0}^{n-1}a_{ij}x_{j} \leq b_{i} & i \in I^{\leq} \\
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& \sum_{j=0}^{n-1}a_{ij}x_{j} \geq b_{i} & i \in I^{\geq} \nonumber \\
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& x \geq \epsilon & \nonumber
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\end{eqnarray}
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if we define $\epsilon$ as
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\begin{equation}
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\label{def:epsilon}
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\epsilon_{j}:= -\varepsilon^{j+1}, \quad j \in \{0 \dots n-1 \}
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\end{equation}
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All the entities whose definitions differ for the perturbed and
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unperturbed problem will be denoted as functions of $\varepsilon$,
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such that setting $\varepsilon =0$ will yield the corresponding definitions
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for the unperturbed problem.
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Values will be considered as polynomials in $\varepsilon$,
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$p(\varepsilon)=\sum_{k=0}^{l}p_{k}\varepsilon^{k}$, and values are compared
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lexicographically in ascending order of the exponents of $\varepsilon$. As
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a consequence degenerate bases, in the sense of basic variables taking on their
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implicit lower bound values,
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are no longer possible and therefore cycling is
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avoided.
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For later use we define the leading coefficient of a polynomial
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$p(\varepsilon)=\sum_{k=0}^{l}p_{k}\varepsilon^{k}$
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\begin{equation}
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\label{def:lead_coeff}
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l_{c}(p(\varepsilon)):=\min_{0 \leq k \leq l:p_{k} \neq 0}k
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\end{equation}
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\subsection{The auxiliary problem for the lexicographic method}
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The slack columns are added in the standard way.
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Similar to Definition~\ref{def:art_col}, for each equality constraint the
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constraint matrix $A$ is augmented by an artificial column
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\[
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\tilde{a} =\left\{
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\begin{array}{ll}
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-e_{i} &
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\mbox{if $l_{c}\left(
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b_{i} + \sum_{j=0}^{n-1}a_{ij}\varepsilon^{j+1}\right)< 0$}\\
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e_{i} & \mbox{otherwise}
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\end{array}
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\right.
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\]
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If the set of inequality constraints with infeasible origin
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\[
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I_{Inf}:=\{i \in I^{\leq}\left|\right. l_{c}\left(b_{i} +
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\sum_{j=0}^{n-1}a_{ij}\varepsilon^{j+1}\right) < 0\} \cup
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\{i \in I^{\geq}\left|\right. l_{c}\left(b_{i} +
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\sum_{j=0}^{n-1}a_{ij}\varepsilon^{j+1}\right) > 0\}
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\]
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is nonempty, the original constraint matrix is likewise augmented by a special
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artificial column $\tilde{a}_{i}^{s}$ defined as
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|
\begin{equation}
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\tilde{a}^{s}_{i} =\left\{
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\begin{array}{ll}
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-1 & \mbox{if $i \in I^{\leq},
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l_{c}\left(b_{i} +
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\sum_{j=0}^{n-1}a_{ij}\varepsilon^{j+1}\right) < 0$}\\
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1 & \mbox{if $i \in I^{\geq},
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l_{c}\left(b_{i} +
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\sum_{j=0}^{n-1}a_{ij}\varepsilon^{j+1}\right)> 0$}\\
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0 & \mbox{otherwise}
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\end{array}
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\right.
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\end{equation}
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The auxiliary problem is then defined as
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\begin{eqnarray}
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|
\label{def:aux_prob}
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\mbox{minimize} & \tilde{c}^{T}x(\varepsilon) & \nonumber \\
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s.t. & \tilde{A}x(\varepsilon) = b_{i} & i \in \{0\ldots m-1\} \\
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& x_{j}(\varepsilon) \geq -\varepsilon^{j+1} &
|
|
j \in O \cup S \cup art \nonumber
|
|
\end{eqnarray}
|
|
with $\tilde{c}$ defined as in Definition~\ref{def:aux_c} and the additional
|
|
requirement that
|
|
\begin{equation}
|
|
\label{req:order_eps}
|
|
\max_{i \in O \cup S}i < \min_{i \in art}i
|
|
\end{equation}
|
|
holds.
|
|
%Furthermore we require that the index $ind(\tilde{a}^{s})$ is
|
|
%the largest one with respect to slack and artificial variables
|
|
%\begin{equation}
|
|
%\label{req:order_eps_spec_art}
|
|
%\max_{i \in S \cup art \setminus \{\tilde{a}^{s} \}}i < ind(\tilde{a}^{s})
|
|
%\end{equation}
|
|
\marginpar{may the mapping $\sigma: S \rightarrow I$ be arbitrary?}
|
|
|
|
\subsection{Initialization of the auxiliary problem for the lexicographic
|
|
method}
|
|
The auxiliary problem is initialized as before, that is $B_{O}^{(0)}$
|
|
$B_{S}^{(0)}$, $S_{B}^{(0)}$ and $S_{N}^{(0)}$ are defined as in
|
|
Definitions~(\ref{def:headings_init_io}) and~(\ref{def:headings_init_fo}),
|
|
depending on feasibility of the origin, the only difference being
|
|
the fact that the most infeasible origin $i_{0} \in I_{Inf}$ defined as
|
|
\begin{equation*}
|
|
\left|b_{i_{0}} + \sum_{j=0}^{n-1}a_{i_{0}j}\varepsilon^{j+1} \right|
|
|
>
|
|
\left|b_{i} + \sum_{j=0}^{n-1}a_{ij}\varepsilon^{j+1} \right|
|
|
\quad i \in I_{inf},
|
|
\end{equation*}
|
|
is now unique.
|
|
|
|
\subsection{Resolving ties in phaseI}
|
|
\label{sec:Res_ties_phaseI}
|
|
\subsubsection{LP-case}
|
|
Let $B$ be the current basis and $j \in N$ be the entering variable
|
|
and define $\hat{N}:= N \setminus\{j\}$ and
|
|
$q:= A_{B}^{-1}\tilde{A}_{\bullet,j}$.
|
|
Let $i_{1}, i_{2} \in B$ be
|
|
involved in a tie in the unperturbed problem, that is
|
|
\begin{equation*}
|
|
\min_{i \in B: q_{x_{i}} > 0}
|
|
\frac{\left(A_{B}^{-1}b\right)_{x_{i}}}{q_{x_{i}}}
|
|
=
|
|
\frac{\left(A_{B}^{-1}b\right)_{x_{i_{1}}}}{q_{x_{i_{1}}}}
|
|
=
|
|
\frac{\left(A_{B}^{-1}b\right)_{x_{i_{2}}}}{q_{x_{i_{2}}}}
|
|
\end{equation*}
|
|
by Definition~(\ref{def:aux_prob}), we then have to compare
|
|
polynomials in $\varepsilon$ of the following form in the perturbed problem
|
|
\begin{eqnarray}
|
|
\tilde{p}_{x_{i}}^{(L)}\left(\varepsilon, B\right) & := &
|
|
\frac{\varepsilon^{i+1}
|
|
- \left(A_{B}^{-1}\tilde{A}_{N}\right)_{x_{i}}
|
|
\epsilon_{N}}{q_{x_{i}}} \nonumber \\
|
|
\label{def:p_x_i_tilde}
|
|
& = &
|
|
\frac{\varepsilon^{i+1}
|
|
- \left(A_{B}^{-1}\tilde{A}_{N \setminus \{j\}}\right)_{x_{i}}
|
|
\epsilon_{N \setminus \{j\}}}{q_{x_{i}}}
|
|
+ \varepsilon^{j+1}
|
|
\end{eqnarray}
|
|
Note, that always either $\tilde{p}_{x_{i_{1}}}^{(L)}
|
|
\left(\varepsilon, B\right) <
|
|
\tilde{p}_{x_{i_{2}}}^{(L)}\left(\varepsilon, B\right)$ or
|
|
$\tilde{p}_{x_{i_{1}}}^{(L)}\left(\varepsilon, B\right) >
|
|
\tilde{p}_{x_{i_{2}}}^{(L)}\left(\varepsilon, B\right)$, even if
|
|
$\xe{i_{1}} = \xe{i_{2}}$,
|
|
since the terms $\frac{\varepsilon^{i_{1}+1}}{q_{x_{i_{1}}}}$ and
|
|
$\frac{\varepsilon^{i_{2}+1}}{q_{x_{i_{1}}}}$ are unique to
|
|
$\tilde{p}_{x_{i_{1}}}^{(L)}\left(\varepsilon, B\right)$ and
|
|
$\tilde{p}_{x_{i_{2}}}^{(L)}\left(\varepsilon, B\right)$.
|
|
|
|
\subsubsection{QP-case}
|
|
In preparation of phaseII the QP-case in phaseI uses the QP-machinery for $D=0$.
|
|
The basis $M_{B}$ in phaseI is, given the basis heading
|
|
$\left[C \cup S_{B}, B_{O} \cup B_{S} \right]$,
|
|
according to \cite{Sven} Section~2.3.1, Equation~2.5, defined as
|
|
\begin{equation}
|
|
\label{def:M_B_phaseI}
|
|
M_{B}
|
|
\left(\begin{array}{c}
|
|
\lambda \\
|
|
\hline
|
|
x_{B}^{*}
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B}
|
|
\end{array}
|
|
\right)
|
|
\quad
|
|
M_{B}:=
|
|
\left(\begin{array}{c|c}
|
|
0 & A_{B} \\
|
|
\hline
|
|
A_{B}^{T} & 0
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
such that $M_{B}^{-1}$ is defined as
|
|
\begin{equation}
|
|
\label{def:M_B_inv_phaseI}
|
|
M_{B}^{-1}:=
|
|
\left(\begin{array}{c|c}
|
|
0 & \left(A_{B}^{-1}\right)^{T} \\
|
|
\hline
|
|
A_{B}^{-1} & 0
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
Assume $B$ to be the current basis and $j \in N$ the entering variable and
|
|
define $\hat{N}$ as above $q$ is according to \cite{Sven}, Section~2.3.2,
|
|
Equation~2.2 defined as
|
|
\begin{equation}
|
|
\label{def:q_phaseI}
|
|
q:= M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
\tilde{A}_{\bullet, j} \\
|
|
\hline
|
|
0
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
0 \\
|
|
\hline
|
|
A_{B}^{-1}\tilde{A}_{\bullet, j}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
Using $\tilde{c}_{B}=0$ in Definition~(\ref{def:M_B_phaseI})
|
|
we can express $i_{1}$, $i_{2}$ involved in a tie in the unperturbed problem
|
|
as
|
|
\begin{equation*}
|
|
\min_{i \in B: q_{x_{i}}>0}
|
|
\frac{
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
0
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
}{q_{x_{i}}}
|
|
=
|
|
\frac{
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
0
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i_{1}}}
|
|
}{q_{x_{i_{1}}}}
|
|
=
|
|
\frac{
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
0
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i_{2}}}
|
|
}{q_{x_{i_{2}}}}
|
|
\end{equation*}
|
|
such that by Definition~(\ref{def:aux_prob}) we have to compare polynomials
|
|
in $\varepsilon$ of the following form in the perturbed problem
|
|
\begin{equation}
|
|
\label{def:p_x_i_tilde_Q_1}
|
|
\tilde{p}_{x_{i}}^{(Q)}\left(\varepsilon, B\right) :=
|
|
\frac{
|
|
\varepsilon^{i+1}
|
|
-\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
\tilde{A}_{N} \\
|
|
\hline
|
|
0
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
\epsilon_{N}
|
|
}{q_{x_{i}}}
|
|
\end{equation}
|
|
which is, using Definitions~(\ref{def:M_B_inv_phaseI}) and
|
|
(\ref{def:q_phaseI}) the same as
|
|
Definition~(\ref{def:p_x_i_tilde}),
|
|
but for some shift in the indexing function.
|
|
|
|
|
|
\subsection{Transition to PhaseII}
|
|
For the sake of notational convenience we assume here that the index set of the
|
|
artificial variables $art\setminus \{\tilde{a}^{s}\}$ is defined as
|
|
$art\setminus \{\tilde{a}^{s}\}=\{l+1 \ldots l+\left|E\right| \}$,
|
|
$l \geq n-1$ and that the basic artificial variable $x_{i+l}$ appears in the
|
|
basis heading $B$ at position $i$.
|
|
|
|
\subsubsection{Expelling artificial variables from the basis}
|
|
Let $N_{-1}:=N \cup \{-1\}$ denote the set of nonbasic variables extended
|
|
by $-1$ with $\tilde{A}_{\bullet, -1}:=b$. Define for $i \in B \cap art$
|
|
\begin{eqnarray*}
|
|
l_{B}\left(i\right):=\min_{j \in R_{i}} j & \text{where}&
|
|
R_{i}:=\{ j \in N_{-1} \left| \right.
|
|
e_{i}^{T}A_{B}^{-1}\tilde{A}_{\bullet, j} \neq 0 \}
|
|
\end{eqnarray*}
|
|
\marginpar{Definition restricted to basic artificials only}
|
|
\begin{lemma}
|
|
\label{lem:art_BxN_zero}
|
|
Let $x_{B}(\varepsilon)$ be the optimal solution of the auxiliary problem and
|
|
$B \cap art \neq \emptyset$. Then for $i \in B \cap art$ either
|
|
$l_{B}\left(i\right) = -1$ or $l_{B}\left(i\right) \geq 0$,
|
|
$e_{i}^{T}A_{B}^{-1}\tilde{A}_{\bullet,j} = 0$
|
|
for $j \in N \setminus art$.
|
|
\end{lemma}
|
|
\begin{proof}
|
|
We only consider $l_{B}\left(i\right) \geq 0$ for all $i \in B \cap art$, for
|
|
$l_{B}\left(i\right) = -1$ for some $i \in B \cap art$ implies infeasibility of
|
|
the unperturbed problem.
|
|
Since $x_{B}(\varepsilon) = A_{B}^{-1}b - A_{B}^{-1}
|
|
\tilde{A}_{N}\epsilon_{N}$
|
|
and the implicit constraints are $x_{j}(\varepsilon) \geq -\varepsilon^{j+1}$
|
|
for variable $x_{j}(\varepsilon)$, feasibility requires
|
|
for some variable $i \in B \cap art$ either $l_{B}\left(i\right) < i$,
|
|
$e_{i}^{T}A_{B}^{-1}\tilde{A}_{\bullet, l_{B}\left(i\right)} > 0$ or
|
|
$l_{B}\left(i\right) > i$. Let
|
|
\begin{equation*}
|
|
\ell:= \min_{i \in B \cap art} l_{B}\left(i\right),
|
|
\end{equation*}
|
|
if $\ell \in N \setminus art$,
|
|
then by definition of $\tilde{c}_{B}$ and $\ell$,
|
|
$\tilde{c}_{B}A_{B}^{-1}\tilde{A}_{\bullet, \ell} > 0$,
|
|
in contradiction to optimality which requires
|
|
$\tilde{c}_{\ell} \geq
|
|
\tilde{c}_{B}A_{B}^{-1}\tilde{A}_{\bullet, \ell}$,
|
|
since $\tilde{c}_{\ell}=0$ for
|
|
$\ell \in N \setminus art$.
|
|
So $\ell \in N \cap art$, which by requirement~(\ref{req:order_eps}) implies
|
|
$e_{i}^{T}A_{B}^{-1}\tilde{A}_{\bullet, j}=0$, for
|
|
$i \in B \cap art$ and $j \in N \setminus art$.
|
|
\end{proof}
|
|
|
|
\subsubsection{The special artificial variable}
|
|
Suppose $ind(\tilde{a}^{s}) \in B_{O}$ in the optimal solution of the auxiliary
|
|
problem.
|
|
For the special artificial variable in the perturbed case the same remarks
|
|
apply as in the unperturbed case, that is, by
|
|
Section~(\ref{sec:spec_art_unpert}) there exists
|
|
$j \in S \setminus B_{S}$ such that
|
|
\begin{equation*}
|
|
e_{k}^{T}A_{B}^{-1}\tilde{A}_{\bullet,j} \neq 0
|
|
\end{equation*}
|
|
holds, if $\tilde{a}^{s}$ appears in $k$-th position in the basis heading
|
|
$B$. By Lemma~(\ref{lem:art_BxN_zero}) on the other hand we have
|
|
\begin{equation*}
|
|
e_{k}^{T}A_{B}^{-1}\tilde{A}_{\bullet,j} = 0,
|
|
\end{equation*}
|
|
so $ind(\tilde{a}^{s}) \in N$.
|
|
|
|
\subsubsection{Removing artificial variables}
|
|
By Lemma~(\ref{lem:art_BxN_zero})
|
|
\begin{equation}
|
|
e_{i}^{T}A_{B}^{-1}\tilde{A}_{N \setminus art}=0, \quad i \in B_{O} \cap art
|
|
\end{equation}
|
|
holds for the optimal solution of the auxiliary problem, such that by the proof
|
|
in \cite{Chvatal}, Chapter~8, the equality constraints
|
|
$A_{i}$, $i \in J=\sigma(B_{O} \cap art)$ are linear combinations of the
|
|
equality constraints $A_{i}$, $i \in E \setminus J$.
|
|
|
|
The values of nonbasic artificial variables $x_{j}$, $j \in N \cap art$ are
|
|
increased from $-\varepsilon^{j+1}$ to zero and then $x_{j}$ is removed. This
|
|
may render the the solution infeasible if
|
|
\begin{equation*}
|
|
e_{i}^{T}A_{B}^{-1}\tilde{A}_{\bullet, j} > 0
|
|
\end{equation*}
|
|
for some $i \in B$ and $j \in N \cap art$ with $j < i$, but does not affect the
|
|
linear dependence of the constraints $A_{i}$, $i \in J$ on the constraints
|
|
$A_{i}$, $i \in E \setminus J$.
|
|
Thus by Requirement~(\ref{req:order_eps}) this may only affect
|
|
the feasibility of the basic artificial variables
|
|
$x_{i}$, $i \in B \cap art$, but since these
|
|
basic artificial variables $i \in B \cap art$ can be removed together with
|
|
their corresponding constraints $A_{\sigma(i)}$ we end up, after removal of
|
|
basic artificial variables $x_{i}$, $i \in B \cap art$ and their corresponding
|
|
constraints, with a basic feasible
|
|
solution to the original perturbed problem. Like in the unperturbed case
|
|
the basic artificial variables
|
|
$i \in B \cap art$ are removed by updates of type $U8$.
|
|
\subsection{The lexicographic method in phaseII}
|
|
Since the nonbasic variables are no longer zero, the objective function as well as
|
|
the values of the basic variables are no longer independent of the nonbasic
|
|
variables; we therefore for the sake of explicitness restate the perturbed
|
|
variants of (UQP($B$)), (QP($B$)) and their KKT conditions for optimality
|
|
as well as the perturbed variant of the definition of QP-basis.
|
|
|
|
\begin{lemma}{KKT conditions for $(QP(B_{\varepsilon}))$}
|
|
\label{lemma:KKT_QP(B)_epsilon}
|
|
A feasible solution $x^{*}(\varepsilon) \in \mathbb{R}^{n}$ to the quadratic
|
|
program
|
|
\begin{eqnarray*}
|
|
\mbox{$(QP(B_{\varepsilon}))$} & minimize & c_{B \cup N}^{T}
|
|
x_{B \cup N}(\varepsilon)
|
|
+ x_{B \cup N}^{T}(\varepsilon)
|
|
D_{B \cup N, B \cup N} x_{B \cup N}(\varepsilon) \\
|
|
& s.t. & A_{B}x_{B}(\varepsilon) = b - A_{N}x_{N}(\varepsilon) \\
|
|
& & I_{N}x_{N}(\varepsilon) = \epsilon_{N} \\
|
|
& & x_{B}(\varepsilon) \geq \epsilon_{B}
|
|
\end{eqnarray*}
|
|
with $D$ symmetric, is optimal iff there exists an $m$-vector
|
|
$\lambda(\varepsilon)$ and an $\left|B\right|$-vector $\mu(\varepsilon) \geq 0$
|
|
such that
|
|
\begin{eqnarray}
|
|
c_{B}^{T} + 2x_{B}^{*^{\scriptstyle{T}}}(\varepsilon)D_{B,B} +
|
|
2\epsilon_{N}^{T}D_{N,B} & = &
|
|
-\lambda^{T}(\varepsilon)A_{B} + \mu_{B}^{T}(\varepsilon)I_{B} \\
|
|
\mu_{B}^{T}(\varepsilon) \left( -I_{B}x_{B}^{*}(\varepsilon) +
|
|
\epsilon_{B} \right) & = & 0
|
|
\end{eqnarray}
|
|
\end{lemma}
|
|
|
|
Likewise, the KKT conditions for $(UQP(B_{\varepsilon}))$ are
|
|
\begin{lemma}{KKT conditions for $(UQP(B_{\varepsilon}))$}
|
|
\label{lemma:KKT_UQP(B)_epsilon}
|
|
A feasible solution $x^{*}(\varepsilon) \in \mathbb{R}^{n}$ to the
|
|
unconstrained quadratic program
|
|
\begin{eqnarray*}
|
|
\mbox{$(UQP(B_{\varepsilon}))$} & minimize & c_{B \cup N}^{T}
|
|
x_{B \cup N}(\varepsilon)
|
|
+ x_{B \cup N}^{T}(\varepsilon) D_{B \cup N, B \cup N}
|
|
x_{B \cup N}(\varepsilon) \\
|
|
& s.t. & A_{B}x_{B}(\varepsilon) = b - A_{N}x_{N}(\varepsilon) \\
|
|
& & I_{N}x_{N}(\varepsilon) = \epsilon_{N}
|
|
\end{eqnarray*}
|
|
with $D$ symmetric, is optimal iff there exists an $m$-vector
|
|
$\lambda(\varepsilon)$ such that
|
|
\begin{eqnarray}
|
|
c_{B}^{T} + 2x_{B}^{*^{\scriptstyle{T}}}(\varepsilon)D_{B,B} +
|
|
2\epsilon_{N}^{T}D_{N,B} & = &
|
|
-\lambda^{T}(\varepsilon)A_{B}
|
|
\end{eqnarray}
|
|
\end{lemma}
|
|
|
|
\begin{lemma}
|
|
\label{lemma:strict}
|
|
Any vector $x_{B \cup N}(\varepsilon)$, $x_{B}(\varepsilon) > \epsilon_{B}$
|
|
satisfying
|
|
\begin{eqnarray}
|
|
A_{B}x_{B}(\varepsilon) & = & b - A_{N}x_{N}(\varepsilon) \\
|
|
x_{N}(\varepsilon) & = & \epsilon_{N}
|
|
\end{eqnarray}
|
|
is an optimal solution to $(QP(B_{\varepsilon}))$, iff it is an optimal
|
|
solution to $(UQP(B_{\varepsilon}))$.
|
|
\end{lemma}
|
|
|
|
\begin{proof}
|
|
For $x_{B}(\varepsilon) > \epsilon_{B}$ the second condition of
|
|
Lemma~\ref{lemma:KKT_QP(B)_epsilon} implies $\mu_{B}^{T}(\varepsilon)=0$.
|
|
Thus, the first condition of Lemma~\ref{lemma:KKT_QP(B)_epsilon} and the
|
|
single condition of Lemma~\ref{lemma:KKT_UQP(B)_epsilon} are equivalent.
|
|
On the other hand any optimal solution to $(UQP(B_{\varepsilon}))$ with
|
|
$x_{B}(\varepsilon) > \epsilon_{B}$ is feasible and optimal to
|
|
$(QP(B_{\varepsilon}))$ too, since the feasible region of the latter is
|
|
completely contained in the feasible region of $(UQP(B_{\varepsilon}))$.
|
|
\end{proof}
|
|
|
|
And last but not least the perturbed variant of the definition of a $QP$-basis:
|
|
\begin{definition}
|
|
A subset $B$ of the variables of a quadratic program in standard form defines
|
|
a $QP_{\varepsilon}$-basis iff
|
|
\begin{enumerate}
|
|
\item the unconstrained subproblem
|
|
\begin{eqnarray}
|
|
\mbox{(UQP($B_{\epsilon}$))} & minimize & c_{B \cup N}^{T}
|
|
x_{B \cup N}(\varepsilon)
|
|
+ x_{B \cup N}^{T}(\varepsilon)D_{B \cup N, B \cup N}
|
|
x_{B \cup N}(\varepsilon) \nonumber\\
|
|
\label{eq:QP_eps_basis_feasibility_B}
|
|
& s.t. &A_{B} x_{B}(\varepsilon) = b - A_{N}x_{N}(\varepsilon) \\
|
|
\label{eq:QP_eps_basis_feasibility_N}
|
|
& &I_{N} x_{N}(\varepsilon) = \epsilon_{N}
|
|
\end{eqnarray}
|
|
has an unique optimal solution $x_{B}^{*}(\varepsilon) > \epsilon_{B}$ and
|
|
\item $A_{B}$ has full row rank, $rank(A_{B})=m$.
|
|
\end{enumerate}
|
|
\end{definition}
|
|
In the following subsections we will mimic the arguments of \cite{Sven},
|
|
Sections (2.3.1), (2.3.2) for the perturbed problem.
|
|
|
|
\subsubsection{Pricing}
|
|
Testing whether a nonbasic variable \xe{j} can improve the
|
|
current solution $x_{B}^{*}(\varepsilon)$ by entering the current
|
|
$QP_{\varepsilon}$-basis $B$ is done as follows.
|
|
Let $\hat{B}:=B \cup \{j\}$ and consider the subproblem
|
|
\begin{eqnarray*}
|
|
\mbox{(QP($\hat{B}_{\varepsilon}$))} &minimize& c_{\hat{B} \cup \hat{N}}^{T}
|
|
x_{\hat{B} \cup \hat{N}}(\varepsilon) +
|
|
x_{\hat{B} \cup \hat{N}}^{T}(\varepsilon)
|
|
D_{\hat{B} \cup \hat{N}}x_{\hat{B} \cup \hat{N}}(\varepsilon)
|
|
\\
|
|
& s.t. & A_{\hat{B}}x_{\hat{B}}(\varepsilon) = b - A_{\hat{N}}
|
|
x_{\hat{N}}(\varepsilon) \\
|
|
& & I_{\hat{N}}x_{\hat{N}}(\varepsilon) = \epsilon_{\hat{N}} \\
|
|
& & x_{\hat{B}}(\varepsilon) \geq \epsilon_{\hat{B}}
|
|
\end{eqnarray*}
|
|
By Lemma~\ref{lemma:KKT_QP(B)_epsilon} for the above,
|
|
$x_{\hat{B}}^{*}(\varepsilon)$ is an optimal solution
|
|
iff there exists vectors $\lambda(\varepsilon)$ and
|
|
$\mu(\varepsilon) \geq 0$ such that
|
|
\begin{eqnarray}
|
|
\label{eq:KKT_lagrange_id}
|
|
c_{\hat{B}}^{T} + 2x_{\hat{B}}^{*^{\scriptstyle{T}}}(\varepsilon)D_{\hat{B}, \hat{B}} +
|
|
2\epsilon_{\hat{N}}^{T}D_{\hat{N}, \hat{B}}& = &
|
|
-\lambda^{T}(\varepsilon)A_{\hat{B}}
|
|
+ \mu_{\hat{B}}^{T}(\varepsilon)I_{\hat{B}} \\
|
|
\label{eq:KKT_compl_slackness}
|
|
\mu_{\hat{B}}^{T}(\varepsilon)
|
|
\left(-I_{\hat{B}}x_{\hat{B}}^{*}(\varepsilon)
|
|
+ \epsilon_{\hat{B}}\right) & = & 0
|
|
\end{eqnarray}
|
|
Since $x_{B}^{*}(\varepsilon)> \epsilon_{B}$, $\mu_{B}(\varepsilon)=0$ holds
|
|
using~(\ref{eq:KKT_compl_slackness}). Isolating $x_{j}^{*}(\varepsilon)$
|
|
in~(\ref{eq:KKT_lagrange_id}) and grouping into $B$ and $j$ components yields
|
|
\begin{eqnarray}
|
|
\label{eq:KKT_lagrange_id_B}
|
|
c_{B}^{T} + {2x_{B}^{*}}^{T}(\varepsilon)D_{B,B}
|
|
+ 2\epsilon_{\hat{N}}^{T}D_{\hat{N},B}
|
|
+ 2x_{j}^{*}(\varepsilon)D_{B,j}^{T} & = & -\lambda^{T}(\varepsilon) A_{B} \\
|
|
\label{eq:KKT_lagrange_id_j}
|
|
c_{j} + 2{x_{B}^{*}}^{T}(\varepsilon)D_{B,j}
|
|
+ 2x_{j}^{*}(\varepsilon)D_{j,j}
|
|
+ 2\epsilon_{\hat{N}}^{T}D_{\hat{N},j} & = &
|
|
-\lambda^{T}(\varepsilon) A_{j} + \mu_{j}(\varepsilon)
|
|
\end{eqnarray}
|
|
Equation~(\ref{eq:KKT_lagrange_id_B}) together with the
|
|
feasibility constraints~(\ref{eq:QP_eps_basis_feasibility_B}),
|
|
(\ref{eq:QP_eps_basis_feasibility_N}) of
|
|
$(UQP(B_{\epsilon}))$ and the fact that $N:=\hat{N} \cup \{j\}$ determine
|
|
$\lambda(\varepsilon)$, given $x_{B}^{*}(\varepsilon)$ and
|
|
$x_{j}^{*}(\varepsilon) = -\varepsilon^{j+1}$, by the linear equation system
|
|
\begin{equation}
|
|
M_{B}\left(
|
|
\begin{array}{c}
|
|
\lambda(\varepsilon) \\
|
|
\hline
|
|
x_{B}^{*}(\varepsilon)
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(
|
|
\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B}
|
|
\end{array}
|
|
\right)
|
|
-
|
|
\left(
|
|
\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B,N}
|
|
\end{array}
|
|
\right)\epsilon_{N}
|
|
\end{equation}
|
|
with $M_{B}$ defined as
|
|
\begin{equation}
|
|
\label{def:M_B}
|
|
M_{B}:=\left(
|
|
\begin{array}{c|c}
|
|
0 & A_{B} \\
|
|
\hline
|
|
A_{B}^{T} & 2D_{B,B}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
By the definition of $QP_{\varepsilon}$-basis, $x_{B}^{*}(\varepsilon)$
|
|
is the unique optimal
|
|
solution to $(UQP(B_{\varepsilon}))$ and $A_{B}$ has full row rank. Thus, also
|
|
$\lambda(\varepsilon)$ is unique and $M_{B}$ is regular,
|
|
therefore $M_{B}^{-1}$ exists. Note,
|
|
that $M_{B}$ is the same as in the unperturbed problem.
|
|
|
|
\subsubsection{Ratio Test Step 1}
|
|
Starting with a $QP_{\varepsilon}$-basis $B$ and an entering variable
|
|
\xe{j},
|
|
we want to find a new basis $B^{\prime} \subseteq B \cup \{j\}$ with
|
|
better objective function value.
|
|
Define $\hat{B}:=B \cup \{j\}$, then $x_{\hat{B}}^{*}(\varepsilon)$ with
|
|
$x_{j}^{*}(\varepsilon)=-\varepsilon^{j+1}$ is the optimal solution to
|
|
\begin{eqnarray*}
|
|
(UQP_{j}^{t}(\hat{B}_{\varepsilon})) & minimize &
|
|
c_{\hat{B} \cup \hat{N}}^{T}x_{\hat{B} \cup \hat{N}}(\varepsilon)
|
|
+ x_{\hat{B} \cup \hat{N}}^{T}(\varepsilon)
|
|
D_{\hat{B} \cup \hat{N},\hat{B} \cup \hat{N}}
|
|
x_{\hat{B} \cup \hat{N}}(\varepsilon) \\
|
|
& s.t & A_{\hat{B}}x_{\hat{B}}(\varepsilon) =
|
|
b - A_{\hat{N}}x_{\hat{N}}(\varepsilon) \\
|
|
& & I_{\hat{N}}x_{\hat{N}}(\varepsilon) = \epsilon_{\hat{N}} \\
|
|
& & \xe{j} = - \varepsilon^{j+1} + t
|
|
\end{eqnarray*}
|
|
for $t=0$. $(UQP_{j}^{t}(\hat{B}_{\varepsilon}))$ has a unique solution
|
|
$x_{\hat{B}}^{*}(\varepsilon, t)$ for each value of t, given by
|
|
\begin{equation}
|
|
\label{eq:UQP_j_t_opt_explicit}
|
|
M_{B}\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, t\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, t\right) \\
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B} \\
|
|
\end{array}
|
|
\right)
|
|
-
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B,N} \\
|
|
\end{array}
|
|
\right) \epsilon_{N}
|
|
-t
|
|
\left(\begin{array}{c}
|
|
A_{j} \\
|
|
\hline
|
|
2D_{B,j} \\
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
and $x_{j}^{*}\left(\varepsilon,t\right)= -\varepsilon^{j+1} + t$. This follows
|
|
from the KKT conditions given by Lemma~\ref{lemma:KKT_UQP(B)_epsilon} for the
|
|
reformulation of $(UQP_{j}^{t}(\hat{B}_{\varepsilon}))$ as
|
|
\begin{eqnarray*}
|
|
(UQP_{j}^{t}(\hat{B}_{\varepsilon}))
|
|
& minimize & c_{B \cup N}^{T} x_{B \cup N}(\varepsilon) +
|
|
x_{B \cup N}^{T}(\varepsilon)D_{B \cup N, B \cup N}
|
|
x_{B \cup N}(\varepsilon) \\
|
|
& s.t. & A_{B}x_{B}(\varepsilon) = b - A_{N}x_{N}(\varepsilon) \\
|
|
&& I_{N}x_{N}(\varepsilon) = \epsilon_{N}^{j}
|
|
\end{eqnarray*}
|
|
with $\epsilon_{N}^{j}:=\epsilon_{N} + te_{j}$, and the regularity of
|
|
$M_{B}$.
|
|
While increasing $t$ starting from zero, either some basic variable $i \in B$
|
|
may become $\xe{i}=-\varepsilon^{i+1}$ or a local minimum of the
|
|
objective function, that is $\mu_{j}(\varepsilon)$ in
|
|
Equation~(\ref{eq:KKT_lagrange_id_j}) becomes zero, is reached.
|
|
We will show later that these two events never happen simultaneously for the
|
|
perturbed problem.
|
|
In order to derive $\mu_{j}\left(\varepsilon, t\right)$, define
|
|
\begin{eqnarray}
|
|
\label{def:sol_eps_zero_I}
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, 0\right)
|
|
\end{array}
|
|
\right)
|
|
&:=&M_{B}^{-1}
|
|
\left[
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B}
|
|
\end{array}
|
|
\right)
|
|
-
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B,N}
|
|
\end{array}
|
|
\right)\epsilon_{N}
|
|
\right]
|
|
\\
|
|
\left(\begin{array}{c}
|
|
q_{\lambda} \\
|
|
\hline
|
|
q_{x}
|
|
\end{array}
|
|
\right)
|
|
&:=&M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{j} \\
|
|
\hline
|
|
2D_{B,j}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
such that Equation~(\ref{eq:UQP_j_t_opt_explicit}) becomes
|
|
\begin{equation}
|
|
\label{eq:UQP_j_t_opt_short}
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, t\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, t\right)
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, 0\right)
|
|
\end{array}
|
|
\right)
|
|
-t
|
|
\left(\begin{array}{c}
|
|
q_{\lambda} \\
|
|
\hline
|
|
q_{x}
|
|
\end{array}
|
|
\right),
|
|
\end{equation}
|
|
$\mu_{j}\left(\varepsilon, t\right)$ can then, by using
|
|
Equations~(\ref{eq:KKT_lagrange_id_j}),(\ref{eq:UQP_j_t_opt_short}) and
|
|
$x_{j}^{*}\left(\varepsilon, t\right)= -\varepsilon^{j+1} + t$ be
|
|
expressed as
|
|
\begin{eqnarray}
|
|
\label{eq:mu_j_eps_t}
|
|
\mu_{j}\left(\varepsilon, t\right) & = & c_{j} +
|
|
A_{j}^{T}\lambda\left(\varepsilon, t\right)
|
|
+ 2D_{B,j}^{T}x_{B}^{*}\left(\varepsilon, t\right) +
|
|
2D_{j,j}x_{j}^{*}\left(\varepsilon, t\right)
|
|
+ 2D_{\hat{N}, j}^{T}\epsilon_{\hat{N}} \nonumber \\
|
|
& = & c_{j} + A_{j}^{T}\lambda\left(\varepsilon, 0\right)
|
|
+ 2D_{B,j}^{T}x_{B}^{*}\left(\varepsilon, 0\right) +
|
|
2D_{j,j}\epsilon_{j}
|
|
+ 2D_{\hat{N}, j}^{T}\epsilon_{\hat{N}} + \nonumber \\
|
|
& & t\left(D_{j,j} - A_{j}^{T}q_{\lambda} - 2D_{B,j}^{T}q_{x}
|
|
\right) \\
|
|
& = & \mu_{j}\left(\varepsilon, 0\right) + t\nu
|
|
\nonumber
|
|
\end{eqnarray}
|
|
where
|
|
\begin{equation}
|
|
\label{def:nu}
|
|
\nu := 2D_{j,j} - A_{j}^{T}q_{\lambda} - 2D_{B,j}^{T}q_{x}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
\mu_{j}\left(\varepsilon, 0\right) :=
|
|
c_{j} + 2D_{N, j}^{T}\epsilon_{N} +
|
|
\left(A_{j}^{T} \left|\right. 2D_{B, j}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, 0\right)
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
Using Definitions~(\ref{def:sol_eps_zero_I}), (\ref{def:nu}) and elementary
|
|
algebraic manipulation
|
|
$\mu_{j}(\varepsilon,0)$ may be written as
|
|
\begin{eqnarray*}
|
|
\mu_{j}\left(\varepsilon, 0\right) &=& c_{j} +
|
|
\left(A_{j}^{T} \left|\right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B}
|
|
\end{array}
|
|
\right) + \\
|
|
&&
|
|
\left[2D_{N, j}^{T} -
|
|
\left(A_{j}^{T} \left|\right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B,N}
|
|
\end{array}
|
|
\right)
|
|
\right]\epsilon_{N}
|
|
\end{eqnarray*}
|
|
Finally, setting $\varepsilon=0$ in the above, we obtain $\mu_{j}(0, 0)$, such
|
|
that $\mu_{j}(\varepsilon, 0)$ can be written in terms of $\mu_{j}(0,0)$ as
|
|
\begin{equation}
|
|
\label{eq:mu_j_eps_zero}
|
|
\mu_{j}\left(\varepsilon, 0\right) =
|
|
\mu_{j}\left(0,0\right) +
|
|
\left[
|
|
2D_{j, N} - \left(A_{j}^{T} \left| \right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B, N}
|
|
\end{array}
|
|
\right)
|
|
\right]\epsilon_{N}
|
|
\end{equation}
|
|
|
|
When $\mu_{j}\left(\varepsilon, t\right)$ becomes zero for some $t > 0$,
|
|
$\lambda\left(\varepsilon, t\right)$ and $\mu_{\hat{B}}(\varepsilon)$ satisfy
|
|
the KKT conditions of Lemma~\ref{lemma:KKT_QP(B)_epsilon}, thus
|
|
$x_{B}^{*}\left(\varepsilon, t\right)$,
|
|
$x_{j}^{*}\left(\varepsilon, t\right)$ is an optimal solution to
|
|
$(QP_{\varepsilon}(\hat{B}))$. Lemma (2.7) in~\cite{Sven} then asserts the
|
|
additional requirements of uniqueness of the solution as well as full row rank
|
|
of $A_{\hat{B}}$ for $\hat{B}$ being the new $QP_{\varepsilon}$-basis.
|
|
|
|
In the first case happening, we implicitly add the constraint
|
|
$\xe{i}=-\varepsilon^{i+1}$
|
|
to $(UQP_{j}^{t}(\hat{B}_{\varepsilon}))$ by removing $i$ from the set $B$.
|
|
If $M_{B \setminus \{i\}}$ is regular,
|
|
we still have a unique optimal solution to
|
|
$(UQP_{j}^{t}(\hat{B}_{\varepsilon} \setminus \{i\}))$ for each value of
|
|
$t$ and Ratio Test Step 1 is
|
|
iterated. Otherwise we proceed with the Ratio Test Step 2.
|
|
|
|
|
|
\subsubsection{Ties in Ratio Test Step 1}
|
|
\label{sec:Ties_ratio_test_step_1}
|
|
Consider two basic variables $i_{1}, i_{2} \in B$ involved in a tie in the
|
|
unperturbed problem, setting $\varepsilon=0$ in
|
|
Equation~(\ref{eq:UQP_j_t_opt_short}), this can be expressed as
|
|
\begin{equation}
|
|
\check{t}\left(0, B\right)=
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(0, 0 \right) \\
|
|
\hline
|
|
x_{B}^{*}\left(0, 0\right)
|
|
\end{array}
|
|
\right)_{x_{i_{1}}}}{q_{x_{i_{1}}}}
|
|
=
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(0, 0 \right) \\
|
|
\hline
|
|
x_{B}^{*}\left(0, 0\right)
|
|
\end{array}
|
|
\right)_{x_{i_{2}}}}{q_{x_{i_{2}}}}
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\check{t}\left(0, B\right):=\min_{i \in B: q_{x_{i}} > 0}
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(0, 0\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(0, 0\right)
|
|
\end{array}
|
|
\right)_{x_{i}}}{q_{x_{i}}}
|
|
\end{equation}
|
|
According to Definition~(\ref{def:QP_eps}) of the perturbed problem,
|
|
Equation~(\ref{eq:UQP_j_t_opt_short}) and
|
|
Definition~(\ref{def:sol_eps_zero_I}),
|
|
$\check{t}(\varepsilon, B)$
|
|
is defined as
|
|
\begin{eqnarray}
|
|
\label{def:t_min_eps}
|
|
\check{t}\left(\varepsilon, B\right) & := &
|
|
\min_{i \in B: q_{x_{i}} > 0}
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0 \right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, 0 \right)
|
|
\end{array}
|
|
\right)_{x_{i}}+ \varepsilon^{i+1}}{q_{x_{i}}} \\
|
|
&=&
|
|
\check{t}\left(0, B \right) +
|
|
\min_{i \in B: q_{x_{i}} > 0} \px{i}{Q_{1}}{B}
|
|
\end{eqnarray}
|
|
where
|
|
\begin{eqnarray}
|
|
\label{def:p_x_i_Q_1}
|
|
\px{i}{Q_{1}}{B} &:=&
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B, N}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{N}}{q_{x_{i}}} \\
|
|
&=&
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{B, N \setminus \{j\}}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{N \setminus \{j\}}}{q_{x_{i}}}
|
|
+ \varepsilon^{j+1}
|
|
\end{eqnarray}
|
|
Therefore, in order to resolve the tie in the perturbed problem, we have
|
|
to compare the polynomials \px{i_{1}}{Q_{1}}{B} and \px{i_{2}}{Q_{1}}{B}.
|
|
Again, as in phaseI we always have either
|
|
$\px{i_{1}}{Q_{1}}{B} < \px{i_{2}}{Q_{1}}{B}$ or
|
|
$\px{i_{1}}{Q_{1}}{B} > \px{i_{2}}{Q_{1}}{B}$, since the
|
|
$\frac{\varepsilon^{i+1}}{q_{x_{i}}}$ terms are unique to each
|
|
\px{i}{Q_{1}}{B}.
|
|
|
|
Ties between a basic variable $x_{i}(0)$ taking its lower bound $0$ value and
|
|
$\mu_{j}\left(0, t\right)$
|
|
becoming zero in the unperturbed problem, that is,
|
|
according to Equation~(\ref{eq:mu_j_eps_t}) with $\varepsilon=0$,
|
|
\begin{equation*}
|
|
\check{t}\left(0, B\right) =
|
|
-\frac{\mu_{j}\left(0, 0\right)}{\nu}
|
|
\end{equation*}
|
|
can be resolved in the perturbed problem,
|
|
given the above equality, by comparing
|
|
\begin{equation*}
|
|
\check{t}(\varepsilon, B)-\check{t}(0, B) = \px{i}{Q_{1}}{B}
|
|
\end{equation*}
|
|
and the expression
|
|
\begin{equation*}
|
|
\frac{-\mu_{j}(\varepsilon, 0)+ \mu_{j}(0,0)}{\nu}.
|
|
\end{equation*}
|
|
Therefore, taking into account Equation~(\ref{eq:mu_j_eps_zero}),
|
|
for the latter expression above, the following two
|
|
polynomials in $\varepsilon$ have to be compared in order to resolve the tie
|
|
\begin{eqnarray}
|
|
\px{i}{Q_{1}}{B} & = &
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{B, N \setminus \{j\}}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{N \setminus \{j\}}}{q_{x_{i}}}
|
|
+ \varepsilon^{j+1} \\
|
|
\label{def:p_mu_j_Q_1}
|
|
\pmu{Q_{1}}{B} & := &
|
|
-\frac{2D_{j, N} -
|
|
\left(A_{j}^{T} \left| \right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B,N}
|
|
\end{array}
|
|
\right)}{\nu}
|
|
\epsilon_{N}
|
|
\nonumber \\
|
|
&=&
|
|
-\frac{2D_{j, N \setminus \{j\}} -
|
|
\left(A_{j}^{T} \left| \right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{B,N \setminus \{j\}}
|
|
\end{array}
|
|
\right)}{\nu}
|
|
\epsilon_{N \setminus \{j\}}
|
|
\nonumber \\
|
|
&&
|
|
+\varepsilon^{j+1}
|
|
\end{eqnarray}
|
|
The term $\frac{\varepsilon^{i+1}}{q_{x_{i}}}$ is again unique to
|
|
\px{i}{Q_{1}}{B}, such that the tie can always
|
|
be resolved.
|
|
|
|
|
|
\subsubsection{Ratio Test Step 2}
|
|
Let $B$ be the set of basic variables after the last iteration of
|
|
Ratio Test Step~1. Since
|
|
$M_{B}$ is no longer regular, Equation~(\ref{eq:UQP_j_t_opt_explicit}) does
|
|
no longer
|
|
determine unique solutions to $(UQP_{j}^{t}(\hat{B_{\epsilon}}))$ for arbitrary
|
|
$t$.
|
|
Reconsidering the KKT conditions for $(QP(\hat{B}_{\epsilon}))$, that is
|
|
Equations~(\ref{eq:KKT_lagrange_id}),(\ref{eq:KKT_compl_slackness}) yields
|
|
\begin{equation}
|
|
\label{eq:QP_j_mu_opt_explicit}
|
|
M_{\hat{B}}
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon\right) \\
|
|
\hline
|
|
x_{j}^{*}\left(\varepsilon\right)
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B} \\
|
|
\hline
|
|
-c_{j}
|
|
\end{array}
|
|
\right)
|
|
-
|
|
\left(\begin{array}{c}
|
|
A_{\hat{N}} \\
|
|
\hline
|
|
2D_{B, \hat{N}} \\
|
|
\hline
|
|
2D_{j, \hat{N}}
|
|
\end{array}
|
|
\right)\epsilon_{\hat{N}}
|
|
+ \mu_{j}\left(\varepsilon\right)
|
|
\left(\begin{array}{c}
|
|
0 \\
|
|
\hline
|
|
0 \\
|
|
\hline
|
|
1
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
In case $M_{\hat{B}}$ is singular, we proceed directly to Step 3. Otherwise,
|
|
the system of linear equations above has a unique solution for each value of
|
|
$\mu_{j}\left(\varepsilon\right)$. The solutions are determined by a linear
|
|
function in $\mu_{j}\left(\varepsilon\right)$, which can be written as
|
|
\begin{equation}
|
|
\label{eq:QP_j_mu_opt_short}
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, \mu_{j}\left(\varepsilon\right)\right) \\
|
|
\hline
|
|
x_{\hat{B}}^{*}\left(\varepsilon,
|
|
\mu_{j}\left(\varepsilon\right)\right)
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{\hat{B}}^{*}\left(\varepsilon, 0\right)
|
|
\end{array}
|
|
\right)
|
|
+ \mu_{j}(\varepsilon)
|
|
\left(\begin{array}{c}
|
|
p_{\lambda} \\
|
|
\hline
|
|
p_{x_{\hat{B}}}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
with
|
|
\begin{eqnarray}
|
|
\label{def:sol_eps_zero_II}
|
|
\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{B}^{*}\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{j}^{*}\left(\varepsilon, 0\right)
|
|
\end{array}
|
|
\right)
|
|
&:=&M_{\hat{B}}^{-1}
|
|
\left[
|
|
\left(\begin{array}{c}
|
|
b \\
|
|
\hline
|
|
-c_{B} \\
|
|
\hline
|
|
-c_{j}
|
|
\end{array}
|
|
\right)
|
|
-
|
|
\left(\begin{array}{c}
|
|
A_{\hat{N}} \\
|
|
\hline
|
|
2D_{B, \hat{N}} \\
|
|
\hline
|
|
2D_{j, \hat{N}}
|
|
\end{array}
|
|
\right)\epsilon_{\hat{N}}
|
|
\right]
|
|
\\
|
|
\left(\begin{array}{c}
|
|
p_{\lambda} \\
|
|
\hline
|
|
p_{x_{B}} \\
|
|
\hline
|
|
p_{x_{j}}
|
|
\end{array}
|
|
\right)
|
|
&:=&M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
0 \\
|
|
\hline
|
|
0 \\
|
|
\hline
|
|
1
|
|
\end{array}
|
|
\right).
|
|
\end{eqnarray}
|
|
Any solution
|
|
$x_{\hat{B}}^{*}\left(\varepsilon,\mu_{j}\left(\varepsilon\right)\right)$ is
|
|
feasible for $(UQP(\hat{B}))$, and it is optimal if
|
|
$\mu_{j}\left(\varepsilon\right)=0$.
|
|
Let $\check{t}_{1}(\varepsilon, \tilde{B})$ be the value of $t$
|
|
for which $M_{B}$ became singular in the last iteration of Ratio Test Step 1
|
|
of the perturbed problem, then
|
|
$x_{\hat{B}}^{*}(\varepsilon,
|
|
\mu_{j}(\varepsilon, \check{t}_{1}(\varepsilon, \tilde{B})))$
|
|
is the current feasible solution at the beginning of Ratio Test Step 2.
|
|
|
|
While growing $\mu_{j}(\varepsilon)$ from
|
|
$\mu_{j}(\varepsilon,\check{t}_{1}(\varepsilon, \tilde{B})
|
|
)$
|
|
towards zero,
|
|
again, either one of the remaining basic variables becomes zero or a local
|
|
minimum of the objective function is reached. In case of the latter happening
|
|
$\mu_{j}(\varepsilon)$ equals zero, we found an optimal solution
|
|
$x_{\hat{B}}^{*}\left(\varepsilon, 0\right)$ to $(UQP(\hat{B_{\varepsilon}}))$,
|
|
which by
|
|
Lemma~\ref{lemma:strict} is also an optimal solution to the constrained
|
|
problem
|
|
$QP(\hat{B_{\varepsilon}})$. Uniqueness of the solution follows from the
|
|
regularity of $M_{\hat{B}}$, which also implies that $\hat{B}$ is the new basis
|
|
in that case.
|
|
|
|
On the other hand, if some basic variable \xe{k} becomes zero, we
|
|
implicitly add the constraint $\xe{k}=-\varepsilon^{k+1}$ to
|
|
$(UQP(\hat{B_{\varepsilon}}))$ by removing $k$ from $\hat{B}$. If
|
|
$M_{\hat{B} \setminus \{k\}}$ stays regular, we still obtain unique solutions
|
|
of Equation~(\ref{eq:QP_j_mu_opt_explicit}) for arbitrary values of
|
|
$\mu_{j}(\varepsilon)$. In this case Ratio Test Step 2 is iterated,
|
|
otherwise we continue with Step 3.
|
|
|
|
\subsubsection{Ties in Ratio Test Step 2}
|
|
\label{sec:Ties_ratio_test_step_2}
|
|
Consider two basic variables $i_{1}, i_{2} \in \hat{B}$ involved in a tie in
|
|
the unperturbed problem, setting $\varepsilon=0$ in
|
|
Equation~(\ref{eq:QP_j_mu_opt_short}), this can be expressed as
|
|
\begin{equation}
|
|
\check{\mu}_{j}(0, \hat{B}) =
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(0,0\right) \\
|
|
\hline
|
|
x_{\hat{B}}^{*}\left(0,0\right)
|
|
\end{array}
|
|
\right)_{x_{i_{1}}}}{p_{x_{i_{1}}}}
|
|
=
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(0,0\right) \\
|
|
\hline
|
|
x_{\hat{B}}^{*}\left(0,0\right)
|
|
\end{array}
|
|
\right)_{x_{i_{2}}}}{p_{x_{i_{2}}}}
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\label{def:hat_mu_j_min_0}
|
|
\check{\mu}_{j}(0, \hat{B}) :=
|
|
\min_{i \in \hat{B}: p_{x_{i}} < 0}
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(0,0\right) \\
|
|
\hline
|
|
x_{\hat{B}}^{*}\left(0,0\right)
|
|
\end{array}
|
|
\right)_{x_{i}}}{p_{x_{i}}}
|
|
\end{equation}
|
|
Again, similar to ties among basic variables in Ratio Test Step 1,
|
|
by Definition~(\ref{def:QP_eps}) of the perturbed problem,
|
|
Equation~(\ref{eq:QP_j_mu_opt_short}) and
|
|
Definition~(\ref{def:sol_eps_zero_II}),
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B})$ is defined as
|
|
\begin{eqnarray}
|
|
\label{def:hat_mu_j_min_eps}
|
|
\check{\mu}_{j}(\varepsilon, \hat{B}) & := &
|
|
\min_{i \in \hat{B}: p_{x_{i}} < 0}
|
|
\frac{\left(\begin{array}{c}
|
|
\lambda\left(\varepsilon, 0\right) \\
|
|
\hline
|
|
x_{\hat{B}}^{*}\left(\varepsilon,0\right)
|
|
\end{array}
|
|
\right)_{x_{i}}+ \varepsilon^{i+1}}{p_{x_{i}}} \\
|
|
&=&
|
|
\check{\mu}_{j}(0, \hat{B}) +
|
|
\min_{i \in \hat{B}: p_{x_{i}} < 0} p_{x_{i}}^{(Q_{2})}
|
|
(\varepsilon, \hat{B})
|
|
\end{eqnarray}
|
|
where
|
|
\begin{eqnarray}
|
|
\label{def:p_x_i_Q_2}
|
|
p_{x_{i}}^{(Q_{2})}(\varepsilon, \hat{B}) &:=&
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{\hat{N}} \\
|
|
\hline
|
|
2D_{\hat{B}, \hat{N}}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{\hat{N}}}{p_{x_{i}}}
|
|
\end{eqnarray}
|
|
Therefore in order to resolve the tie in the perturbed problem, we have to
|
|
compare the polynomials \px{i_{1}}{Q_{2}}{\hat{B}} and
|
|
\px{i_{2}}{Q_{2}}{\hat{B}}. Again, because of the term
|
|
$\frac{\varepsilon^{i+1}}{p_{x_{i}}}$ we always have either
|
|
$\px{i_{1}}{Q_{2}}{\hat{B}} < \px{i_{2}}{Q_{2}}{\hat{B}}$ or
|
|
$\px{i_{1}}{Q_{2}}{\hat{B}} > \px{i_{2}}{Q_{2}}{\hat{B}}$.
|
|
|
|
For ties between a basic variable $x_{i}(0)$ and $\mu_{j}(0)$ becoming zero
|
|
in the unperturbed problem, that is
|
|
\begin{equation}
|
|
\label{eq:tie_unpert_ratio_test_step_2}
|
|
\check{\mu}_{j}(0, \hat{B}) = 0
|
|
\end{equation}
|
|
Therefore, in order to resolve the tie,
|
|
given Equation~(\ref{eq:tie_unpert_ratio_test_step_2}), we have to compare
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B})$ and the constant zero polynomial,
|
|
because of the term $\frac{\varepsilon^{i+1}}{p_{x_{i}}}$ we always have
|
|
either $\px{i}{Q_{2}}{\hat{B}} < 0$ or $\px{i}{Q_{2}}{\hat{B}} > 0$.
|
|
|
|
\subsection{Relaxation of the Definition of a QP-basis}
|
|
The only place where Lemma~(\ref{lemma:strict}) is used in the last subsection
|
|
is Ratio Test Step~2, where it is used to show that the optimal solution
|
|
$x_{\hat{B}}^{*}(\varepsilon, 0)$ to $(UQP(\hat{B}_{\varepsilon}))$
|
|
is also optimal to the constrained problem $QP(\hat{B}_{\varepsilon})$.
|
|
Since only this direction is needed and Lemma~(\ref{lemma:strict}) still holds
|
|
true for this direction if
|
|
$x_{\hat{B}}^{*}(\varepsilon, 0) \geq \epsilon_{\hat{B}}$
|
|
we can relax the definition of $QP_{\varepsilon}$-basis, that is we
|
|
weaken the first requirement on the optimal solution. Since for the
|
|
perturbed problem the first requirement of a $QP_{\varepsilon}$-basis is always
|
|
met with strict inequality we state the relaxed definition of $QP$-basis for
|
|
the unperturbed problem.
|
|
\begin{definition}
|
|
A subset $B$ of the variables of a quadratic program in standard form defines
|
|
a $QP$-basis iff
|
|
\begin{enumerate}
|
|
\item the unconstrained subproblem
|
|
\begin{eqnarray}
|
|
\mbox{(UQP($B$))} & minimize & c_{B}^{T}x_{B}
|
|
+ x_{B}^{T}D_{B, B}
|
|
x_{B} \nonumber\\
|
|
\label{eq:QP_basis_feasibility_B}
|
|
& s.t. &A_{B} x_{B} = b
|
|
\end{eqnarray}
|
|
has an unique optimal solution $x_{B}^{*} \geq 0$ and
|
|
\item $A_{B}$ has full row rank, $rank(A_{B})=m$.
|
|
\end{enumerate}
|
|
\end{definition}
|
|
|
|
\subsection{The lexicographic method in the context of reduced bases}
|
|
The polynomials used in the perturbed problem to decide possible ties use
|
|
the full basis inverses $M_{B}^{-1}$ and $A_{B}^{-1}$ respectively, since these
|
|
are not directly at our disposal, we will express the full basis inverses in
|
|
terms of the reduced ones $\check{M}_{B}^{-1}$ and $\check{A}_{B}^{-1}$.
|
|
\subsubsection{Expanded basis matrix inverse QP-case}
|
|
If the basis heading is given as $\left[C, S_{B}, B_{O}, B_{S} \right]$ the
|
|
basis matrix $M_{B}$ has the following form,
|
|
\begin{equation*}
|
|
\label{def:basis_matrix_form}
|
|
M_{B}:=
|
|
\left(\begin{array}{c|c|c|c}
|
|
0 & 0 & A_{C, B_{O}} & A_{C, B_{S}} \\
|
|
\hline
|
|
0 & 0 & A_{S_{B}, B_{O}} & A_{S_{B}, B_{S}} \\
|
|
\hline
|
|
A_{C, B_{O}}^{T} & A_{S_{B}, B_{O}}^{T} & D_{B_{O}, B_{O}}
|
|
& D_{B_{O}, B_{S}} \\
|
|
\hline
|
|
A_{C, B_{S}}^{T} & A_{S_{B}, B_{S}}^{T} & D_{B_{S}, B_{O}}
|
|
& D_{B_{S}, B_{S}} \\
|
|
\end{array}
|
|
\right),
|
|
\end{equation*}
|
|
since $D_{B_{O}, B_{S}} = D_{B_{S}, B_{O}} = 0$ and
|
|
$D_{B_{S}, B_{S}} = 0$ as well as $A_{C, B_{S}}=0$, this boils down to
|
|
\begin{equation}
|
|
\label{def:basis_matrix}
|
|
M_{B}:=
|
|
\left(\begin{array}{c|c|c|c}
|
|
0 & 0 & A_{C, B_{O}} & 0 \\
|
|
\hline
|
|
0 & 0 & A_{S_{B}, B_{O}} & A_{S_{B}, B_{S}} \\
|
|
\hline
|
|
A_{C, B_{O}}^{T} & A_{S_{B}, B_{O}}^{T} & D_{B_{O}, B_{O}}
|
|
& 0 \\
|
|
\hline
|
|
0 & A_{S_{B}, B_{S}}^{T} & 0
|
|
& 0 \\
|
|
\end{array}
|
|
\right).
|
|
\end{equation}
|
|
Note, that the block $A_{S_{B}, B_{S}}$ is a signed permutation matrix with
|
|
$\pm 1$ nonzero entries, such that
|
|
\begin{equation}
|
|
\label{eq:A_S_BxB_S_inv}
|
|
A_{S_{B}, B_{S}}^{T} = A_{S_{B}, B_{S}}^{-1}
|
|
\end{equation}
|
|
holds, since $A_{S_{B}, B_{S}}$ is orthogonal.
|
|
In order to compute the blocks of $M_{B}^{-1}$ in terms of $A$, $D$ and
|
|
$\check{M}_{B}^{-1}$ we compare the corresponding
|
|
components of
|
|
\begin{equation}
|
|
\check{M}_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b_{C} \\
|
|
\hline
|
|
-c_{B_{O}}
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
\lambda_{C} \\
|
|
\hline
|
|
x_{B_{O}}
|
|
\end{array}
|
|
\right),
|
|
\quad
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b_{C} \\
|
|
\hline
|
|
b_{S_{B}} \\
|
|
\hline
|
|
-c_{B_{O}} \\
|
|
\hline
|
|
-c_{B_{S}}
|
|
\end{array}
|
|
\right)
|
|
=
|
|
\left(\begin{array}{c}
|
|
\lambda_{C} \\
|
|
\hline
|
|
\lambda_{S_{B}} \\
|
|
\hline
|
|
x_{B_{O}} \\
|
|
\hline
|
|
x_{B_{S}}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
for any choice of $b$ and $c$. Note, that $c_{B_{S}}=0$ and by \cite{Sven},
|
|
Section 2.4, $\lambda_{S_{B}}=0$.
|
|
For the first row of blocks of $M_{B}^{-1}$ we obtain, using $c_{B_{S}}=0$,
|
|
\begin{eqnarray*}
|
|
\left(\check{M}_{B}^{-1}\right)_{C, C} b_{C}
|
|
-\left(\check{M}_{B}^{-1}\right)_{C, B_{O}} c_{B_{O}}
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{C, C} b_{C}
|
|
+\left(M_{B}^{-1}\right)_{C, S_{B}} b_{S_{B}} - \\
|
|
&&
|
|
\left(M_{B}^{-1}\right)_{C, B_{O}} c_{B_{O}}
|
|
\end{eqnarray*}
|
|
which yields
|
|
\begin{eqnarray}
|
|
\label{eq:M_B_inv_exp_CxC}
|
|
\left(M_{B}^{-1}\right)_{C,C} &=& \left(\check{M}_{B}^{-1}\right)_{C,C} \\
|
|
\label{eq:M_B_inv_exp_CxS_B}
|
|
\left(M_{B}^{-1}\right)_{C,S_{B}} &=& 0 \\
|
|
\left(M_{B}^{-1}\right)_{C,B_{O}}&=&\left(\check{M}_{B}^{-1}\right)_{C,B_{O}}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
For the third row of blocks of $M_{B}^{-1}$ we obtain likewise
|
|
\begin{eqnarray*}
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O}, C} b_{C}
|
|
-\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}} c_{B_{O}}
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{B_{O}, C} b_{C}
|
|
+\left(M_{B}^{-1}\right)_{B_{O}, S_{B}} b_{S_{B}} - \\
|
|
&&
|
|
\left(M_{B}^{-1}\right)_{B_{O}, B_{O}} c_{B_{O}}
|
|
\end{eqnarray*}
|
|
which yields
|
|
\begin{eqnarray}
|
|
\label{eq:M_B_inv_exp_B_OxC}
|
|
\left(M_{B}^{-1}\right)_{B_{O},C} &=&
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O},C} \\
|
|
\label{eq:M_B_inv_exp_B_OxS_B}
|
|
\left(M_{B}^{-1}\right)_{B_{O},S_{B}} &=& 0 \\
|
|
\label{eq:M_B_inv_exp_B_OxB_O}
|
|
\left(M_{B}^{-1}\right)_{B_{O},B_{O}}&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O},B_{O}}
|
|
\end{eqnarray}
|
|
For the fourth row of blocks of $M_{B}^{-1}$ we use the fact that
|
|
$A_{S_{B},B_{S}}$ is regular, therefore using
|
|
Equation~(\ref{eq:A_S_BxB_S_inv})
|
|
\begin{equation}
|
|
\label{eq:x_B_S}
|
|
x_{B_{S}} = A_{S_{B},B_{S}}^{T}\left(b_{S_{B}} -
|
|
A_{S_{B}, B_{O}}x_{B_{O}}\right)
|
|
\end{equation}
|
|
must hold by definition of the slack variables. Therefore comparing
|
|
corresponding components in
|
|
\begin{eqnarray*}
|
|
\lefteqn{A_{S_{B},B_{S}}^{T}\left(b_{S_{B}} -
|
|
A_{S_{B}, B_{O}}
|
|
\left[\left(\check{M}_{B}^{-1}\right)_{B_{O}, C}b_{C}
|
|
- \left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}} c_{B_{O}}\right]\right)
|
|
= } \\
|
|
& & \quad \quad ,\quad \quad \quad \quad \quad
|
|
\left(M_{B}^{-1}\right)_{B_{S}, C} b_{C}
|
|
+\left(M_{B}^{-1}\right)_{B_{S}, S_{B}} b_{S_{B}}
|
|
-\left(M_{B}^{-1}\right)_{B_{S}, B_{O}} c_{B_{O}}
|
|
\end{eqnarray*}
|
|
yields
|
|
\begin{eqnarray}
|
|
\label{eq:M_B_inv_exp_B_SxC}
|
|
\left(M_{B}^{-1}\right)_{B_{S}, C} &=&
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O},C} \\
|
|
\label{eq:M_B_inv_exp_B_SxS_B}
|
|
\left(M_{B}^{-1}\right)_{B_{S}, S_{B}} &=&
|
|
A_{S_{B}, B_{S}}^{T} \\
|
|
\label{eq:M_B_inv_exp_B_SxB_O}
|
|
\left(M_{B}^{-1}\right)_{B_{S}, B_{O}} &=&
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O},B_{O}}
|
|
\end{eqnarray}
|
|
if we define $\alpha$ as
|
|
\begin{equation}
|
|
\label{def:alpha}
|
|
\alpha := -A_{S_{B}, B_{S}}^{T}A_{S_{B}, B_{O}}
|
|
\end{equation}
|
|
For the second row we take into account that $\lambda_{S_{B}}=0$
|
|
\begin{equation*}
|
|
\left(M_{B}^{-1}\right)_{S_{B}, C} b_{C}
|
|
+\left(M_{B}^{-1}\right)_{S_{B}, S_{B}} b_{S_{B}}
|
|
-\left(M_{B}^{-1}\right)_{S_{B}, B_{O}} c_{B_{O}}
|
|
= 0
|
|
\end{equation*}
|
|
this yields, using Equations~(\ref{eq:M_B_inv_exp_CxS_B})
|
|
and~(\ref{eq:M_B_inv_exp_B_OxS_B}) and the fact that $M_{B}^{-1}$ is symmetric,
|
|
\begin{equation}
|
|
\label{eq:M_B_inv_exp_S_BxS_B}
|
|
\left(M_{B}^{-1}\right)_{S_{B}, S_{B}} = 0
|
|
\end{equation}
|
|
In order to obtain the last yet unknown block
|
|
$\left(M_{B}^{-1}\right)_{B_{S}, B_{S}}$
|
|
we multiply the last row of blocks of $M_{B}^{-1}$ by
|
|
the second column of blocks of $M_{B}$ yielding
|
|
\begin{equation*}
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}} A_{S_{B}, B_{O}}^{T}
|
|
+ \left(M_{B}^{-1}\right)_{B_{S}, B_{S}} A_{S_{B}, B_{S}}^{T}
|
|
= 0
|
|
\end{equation*}
|
|
using Equation~(\ref{eq:A_S_BxB_S_inv}) and
|
|
Definition~(\ref{def:alpha}) we obtain
|
|
\begin{equation}
|
|
\label{eq:M_B_inv_exp_B_SxB_S}
|
|
\left(M_{B}^{-1}\right)_{B_{S}, B_{S}} =
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}}\alpha^{T}
|
|
\end{equation}
|
|
Collecting the different blocks, given by
|
|
Equations~(\ref{eq:M_B_inv_exp_CxC}), (\ref{eq:M_B_inv_exp_CxS_B}),
|
|
(\ref{eq:M_B_inv_exp_S_BxS_B}), (\ref{eq:M_B_inv_exp_B_OxC}),
|
|
(\ref{eq:M_B_inv_exp_B_OxS_B}), (\ref{eq:M_B_inv_exp_B_OxB_O}),
|
|
(\ref{eq:M_B_inv_exp_B_SxC}), (\ref{eq:M_B_inv_exp_B_SxS_B}),
|
|
(\ref{eq:M_B_inv_exp_B_SxB_O}), and~(\ref{eq:M_B_inv_exp_B_SxB_S}),
|
|
and taking into account Equation~(\ref{eq:A_S_BxB_S_inv}),
|
|
we can finally express $M_{B}^{-1}$ in terms
|
|
of $\check{M}_{B}^{-1}$, $A$ and $D$, given the basis heading
|
|
$\left[C, S_{B}, B_{O}, B_{S}\right]$, as
|
|
\begin{equation}
|
|
\label{eq:M_B_inv_exp}
|
|
M_{B}^{-1}=
|
|
\left(\begin{array}{c|c|c|c}
|
|
\left(\check{M}_{B}^{-1}\right)_{C,C} &
|
|
0 &
|
|
\left(\check{M}_{B}^{-1}\right)_{C,B_{O}} &
|
|
\left(\check{M}_{B}^{-1}\right)_{C, B_{O}}\alpha^{T} \\
|
|
\hline
|
|
0 &
|
|
0 &
|
|
0 &
|
|
A_{S_{B},B_{S}} \\
|
|
\hline
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O}, C} &
|
|
0 &
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}} &
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O},B_{O}}\alpha^{T} \\
|
|
\hline
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O},
|
|
C} &
|
|
A_{S_{B}, B_{S}}^{T} &
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O},
|
|
B_{O}} &
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}}\alpha^{T}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
where
|
|
\begin{equation*}
|
|
\alpha=-A_{S_{B}, B_{S}}^{T}A_{S_{B}, B_{O}}
|
|
\end{equation*}
|
|
|
|
\subsubsection{Expanded basis matrix inverse LP-case}
|
|
If the basis heading is given as $\left[B_{O}, B_{S} \right]$ the
|
|
basis matrix $A_{B}$ has the following form,
|
|
\begin{equation}
|
|
A_{B}:=
|
|
\left(\begin{array}{c|c}
|
|
A_{C, B_{O}} & A_{C, B_{S}} \\
|
|
\hline
|
|
A_{S_{B}, B_{O}} & A_{S_{B}, B_{S}}
|
|
\end{array}
|
|
\right).
|
|
\end{equation}
|
|
Again, we compare corresponding components of
|
|
\begin{equation}
|
|
\check{A}_{B}^{-1}b_{C}, \quad \quad
|
|
A_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
b_{C} \\
|
|
\hline
|
|
b_{S_{B}}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
For the first row of blocks of $A_{B}^{-1}$ we obtain,
|
|
\begin{equation*}
|
|
\left(A_{B}^{-1}\right)_{C, B_{O}}b_{C}
|
|
+ \left(A_{B}^{-1}\right)_{C, B_{S}}b_{S_{B}} = \check{A}_{B}^{-1}b_{C}
|
|
\end{equation*}
|
|
which yields,
|
|
\begin{eqnarray}
|
|
\label{eq:A_B_inv_exp_CxB_O}
|
|
\left(A_{B}^{-1}\right)_{C, B_{O}} &=&\check{A}_{B}^{-1} \\
|
|
\label{eq:A_B_inv_exp_CxB_S}
|
|
\left(A_{B}^{-1}\right)_{C, B_{S}} &=& 0
|
|
\end{eqnarray}
|
|
For the second row of blocks of $A_{B}^{-1}$ we obtain,
|
|
using Equation~(\ref{eq:x_B_S}),
|
|
\begin{equation*}
|
|
\left(A_{B}^{-1}\right)_{S_{B}, B_{O}} b_{C}
|
|
+ \left(A_{B}^{-1}\right)_{S_{B}, B_{S}} b_{S_{B}}
|
|
=
|
|
A_{S_{B}, B_{S}}^{T}\left(b_{S_{B}} - A_{S_{B}, B_{O}}
|
|
\check{A}_{B}^{-1} b_{C} \right)
|
|
\end{equation*}
|
|
which yields
|
|
\begin{eqnarray}
|
|
\label{eq:A_B_inv_exp_S_BxB_O}
|
|
\left(A_{B}^{-1}\right)_{S_{B}, B_{O}}
|
|
&=&
|
|
-A_{S_{B}, B_{S}}^{T}A_{S_{B}, B_{O}}\check{A}_{B}^{-1} \\
|
|
\label{eq:A_B_inv_exp_S_BxB_S}
|
|
\left(A_{B}^{-1}\right)_{S_{B}, B_{S}}
|
|
&=&
|
|
A_{S_{B}, B_{S}}^{T}
|
|
\end{eqnarray}
|
|
Collecting the blocks given by Equations~(\ref{eq:A_B_inv_exp_CxB_O}),
|
|
(\ref{eq:A_B_inv_exp_CxB_S}), (\ref{eq:A_B_inv_exp_S_BxB_O})
|
|
and~(\ref{eq:A_B_inv_exp_S_BxB_S}) and using
|
|
Definition~(\ref{def:alpha}) we obtain $A_{B}^{-1}$ in terms of
|
|
$\check{A}_{B}^{-1}$ and $A$ as
|
|
\begin{equation}
|
|
\label{eq:A_B_inv_exp}
|
|
A_{B}^{-1}=
|
|
\left(\begin{array}{c|c}
|
|
\check{A}_{B}^{-1} & 0 \\
|
|
\hline
|
|
\alpha\check{A}_{B}^{-1} & A_{S_{B}, B_{S}}^{T}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
where
|
|
\begin{equation*}
|
|
\alpha = -A_{S_{B}, B_{S}}^{T}A_{S_{B}, B_{O}}
|
|
\end{equation*}
|
|
|
|
\section{Evaluating coefficients of the tie breaking polynomials}
|
|
In order to compute coefficients of the tie breaking polynomials we will need to
|
|
reference entries of the block matrices involved, for technical reasons these
|
|
block matrices may be permuted. We therefore introduce the following
|
|
permutations
|
|
\begin{equation}
|
|
\beta_{O}: B_{O} \rightarrow B_{O}, \quad
|
|
\beta_{S}: B_{S} \rightarrow B_{S}
|
|
\end{equation}
|
|
\begin{equation}
|
|
\gamma_{C}: E \cup S_{N} \rightarrow E \cup S_{N}, \quad
|
|
\gamma_{S_{B}}: S_{B} \rightarrow S_{B}
|
|
\end{equation}
|
|
where the permutations are defined with respect to the headings in ascending
|
|
order.
|
|
|
|
An expression that will be often encountered in the next subsections is the
|
|
following one, given $i \in B_{S}$ we want to compute
|
|
$(\alpha)_{\beta_{S}(i)}$ as a subexpression,
|
|
\begin{eqnarray}
|
|
\left(\alpha\right)_{\beta_{S}(i)}
|
|
&=&
|
|
-\left(A_{S_{B}, B_{S}}^{T}\right)_{\beta_{S}(i)}A_{S_{B}, B_{O}}
|
|
\nonumber \\
|
|
&=&
|
|
-A_{\sigma(i), i}\left(A_{S_{B}, B_{O}}\right)_{\gamma_{S_{B}}(\sigma(i))}
|
|
\nonumber \\
|
|
&=&
|
|
\label{eq:alpha_beta_S}
|
|
-A_{\sigma(i), i}A_{\sigma(i), B_{O}}
|
|
\end{eqnarray}
|
|
|
|
\subsection{PhaseI}
|
|
According to Section~\ref{sec:Res_ties_phaseI} the polynomials
|
|
$\tilde{p}_{x_{i}}^{(L)}(\varepsilon, B)$ and
|
|
$\tilde{p}_{x_{i}}^{(Q)}(\varepsilon, B)$ are equal such that considering
|
|
\begin{equation}
|
|
\label{eq:p_x_i_tilde_ref}
|
|
\tilde{p}_{x_{i}}^{(L)}\left(\varepsilon, B\right) :=
|
|
\frac{\varepsilon^{i+1}
|
|
- \left(A_{B}^{-1}\tilde{A}_{N}\right)_{x_{i}}
|
|
\epsilon_{N}}{q_{x_{i}}}
|
|
=
|
|
\frac{\varepsilon^{i+1}
|
|
- \left(A_{B}^{-1}\tilde{A}_{\hat{N}}\right)_{x_{i}}
|
|
\epsilon_{\hat{N}}}{q_{x_{i}}}
|
|
+ \varepsilon^{j+1}
|
|
\end{equation}
|
|
in light of Equation~(\ref{eq:A_B_inv_exp})
|
|
will be sufficient for the LP-case as well as the QP-case
|
|
in phaseI when evaluating coefficients.
|
|
We will only consider the coefficients of
|
|
$\varepsilon^{k+1}$ for $k \in \hat{N}= N \setminus \{j\}$ in
|
|
$\tilde{p}_{x_{i}}^{(L)}(\varepsilon, B)$,
|
|
since the other cases $k=j$ and $k \in B$
|
|
are trivial. Furthermore we only evaluate entities that are not evaluated in the
|
|
unperturbed problem, such that, taking into account the
|
|
Definition~(\ref{def:epsilon}) of $\epsilon$, we merely consider
|
|
the subexpression
|
|
\begin{equation}
|
|
\label{def:tilde_n_p_x_i_L}
|
|
\tilde{n}_{x_{i}}^{(L)}(B)[i,k]:=
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}}
|
|
\end{equation}
|
|
The correct value
|
|
of the coefficient is obtained by scaling the subexpression with the factor
|
|
$q_{x_{i}}^{-1}$.
|
|
We will distinguish the two
|
|
possibilities $i \in B_{O}$ and $i \in B_{S}$ for $x_{i}$ as well as the
|
|
various possibilities for $k \in \hat{N}$ in phaseI.
|
|
|
|
\subsubsection{LP/QP-case: $\tilde{p}_{x_{i}}^{(L)}(\varepsilon, B)$}
|
|
\paragraph{$\mathbf{i \in B_{O}}$:}
|
|
Assuming $i \in B_{O}$ we distinguish
|
|
according to Equation~(\ref{eq:A_B_inv_exp})
|
|
and the definition of $\tilde{p}_{x_{i}}^{(L)}(\varepsilon, B)$ the
|
|
following cases for $k \in \hat{N}$:
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap S$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{A}_{C \cup S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $\tilde{A}_{S_{B},k}=0$ for $k \in \hat{N} \cap S$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i),C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i), \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\end{eqnarray}
|
|
\item $k \in \hat{N} \cap art \setminus \{\tilde{a}^{s}\}$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{A}_{C \cup S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $\tilde{A}_{S_{B},k}=0$ for
|
|
$k \in \hat{N} \cap art \setminus \{\tilde{a}^{s}\}$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i), C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i), \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\end{eqnarray}
|
|
\item $\tilde{A}_{k}=\tilde{a}^{s}$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}} \tilde{a}_{C \cup S_{B}}^{s}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{a}_{C}^{s} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{a}_{S_{B}}^{s}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:A_B_inv_exp})
|
|
$\left(A_{B}^{-1}\right)_{B_{O}, S_{B}}=0$ holds, so
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{a}_{C}^{s}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i),C}\tilde{a}_{C}^{s}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i), E}\tilde{a}_{E}^{s}
|
|
+\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i), S_{N}}\tilde{a}_{S_{N}}^{s}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $\tilde{a}^{s}_{E}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i), S_{N}}\tilde{a}_{S_N}^{s}
|
|
\end{eqnarray}
|
|
\item $k \in \hat{N} \cap O$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{A}_{C \cup S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:A_B_inv_exp})
|
|
$\left(A_{B}^{-1}\right)_{B_{O}, S_{B}}=0$ holds, so
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{A}_{B}^{-1}\right)_{\beta_{O}(i),C}\tilde{A}_{C,k}
|
|
\end{eqnarray}
|
|
\end{enumerate}
|
|
|
|
\paragraph{$\mathbf{i \in B_{S}}$:}
|
|
Assuming $i \in B_{S}$ we distinguish
|
|
according to Equations~(\ref{eq:A_B_inv_exp})
|
|
and the definition of $\tilde{p}_{x_{i}}^{(L)}(\varepsilon, B)$
|
|
the following cases for $k \in \hat{N}$:
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap S$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{A}_{C \cup S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $\tilde{A}_{S_{B}, k}=0$ for $k \in \hat{N} \cap S$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\check{A}_{B}^{-1}\right)_{\beta_{S}(i), C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\check{A}_{B}^{-1}\right)_{\beta_{S}(i), \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\right)_{\beta_{S}(i)}
|
|
\left(\check{A}_{B}^{-1}\right)_{\bullet, \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
Using Equation~(\ref{eq:alpha_beta_S}) we obtain
|
|
\begin{equation}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} =
|
|
-\tilde{A}_{\sigma(i), i}\tilde{A}_{\sigma(i), B_{O}}
|
|
\left(\check{A}_{B}^{-1}\right)_{\bullet, \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\end{equation}
|
|
|
|
\item $k \in \hat{N} \cap art \setminus \{\tilde{a}^{s}\}$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{A}_{C \cup S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $\tilde{A}_{S_{B}, k}=0$ for
|
|
$k \in \hat{N} \cap art \setminus \{ \tilde{a}^{s} \}$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\check{A}_{B}^{-1}\right)_{\beta_{S}(i),C}\tilde{A}_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\check{A}_{B}^{-1}\right)_{\beta_{S}(i), \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\right)_{\beta_{S}(i)}
|
|
\left(\check{A}_{B}^{-1}\right)_{\bullet, \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
Using Equation~(\ref{eq:alpha_beta_S}) we obtain
|
|
\begin{equation}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} =
|
|
-\tilde{A}_{\sigma(i), i}\tilde{A}_{\sigma(i), B_{O}}
|
|
\left(\check{A}_{B}^{-1}\right)_{\bullet, \gamma_{C}(\sigma(k))}
|
|
\tilde{A}_{\sigma(k), k}
|
|
\end{equation}
|
|
\item $\tilde{A}_{k}=\tilde{a}^{s}$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{a}_{C \cup S_{B}}^{s}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{a}_{C}^{s} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{a}_{S_{B}}^{s}
|
|
\nonumber \\
|
|
&=&
|
|
(\alpha\check{A}_{B}^{-1})_{\beta_{S}(i),C}\tilde{a}_{C}^{s}
|
|
+(A_{S_{B}, B_{S}}^{T})_{\beta_{S}(i), S_{B}}\tilde{a}_{S_{B}}^{s}
|
|
\nonumber \\
|
|
&=&
|
|
(\alpha\check{A}_{B}^{-1})_{\beta_{S}(i), E}\tilde{a}_{E}^{s}
|
|
+(\alpha\check{A}_{B}^{-1})_{\beta_{S}(i), S_{N}}\tilde{a}_{S_{N}}^{s}
|
|
+ \tilde{A}_{\sigma(i), i}\tilde{a}_{\gamma_{S_{B}}(\sigma(i))}^{s}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $\tilde{a}_{E}^{s}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
(\alpha\check{A}_{B}^{-1})_{\beta_{S}(i), S_{N}}\tilde{a}_{S_{N}}^{s}
|
|
+ \tilde{A}_{\sigma(i), i}\tilde{a}_{\gamma_{S_{B}}(\sigma(i))}^{s}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
Using Equation~(\ref{eq:alpha_beta_S}) we obtain
|
|
\begin{equation}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} =
|
|
-\tilde{A}_{\sigma(i), i}\tilde{A}_{\sigma(i), B_{O}}
|
|
\left(\check{A}_{B}^{-1}\right)_{\bullet, S_{N}}\tilde{a}_{S_{N}}^{s}
|
|
+\tilde{A}_{\sigma(i), i}\tilde{a}_{\gamma_{S_{B}}(\sigma(i))}^{s}
|
|
\end{equation}
|
|
\item $k \in \hat{N} \cap O$:
|
|
\begin{eqnarray}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} &=&
|
|
(A_{B}^{-1})_{x_{i}, C \cup S_{B}}\tilde{A}_{C \cup S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(A_{B}^{-1})_{x_{i}, C}\tilde{A}_{C, k} +
|
|
(A_{B}^{-1})_{x_{i}, S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
(\alpha\check{A}_{B}^{-1})_{\beta_{S}(i),C}\tilde{A}_{C, k}
|
|
+ (A_{S_{B}, B_{S}}^{T})_{\beta_{S}(i), S_{B}}\tilde{A}_{S_{B}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\right)_{\beta_{S}(i)}\check{A}_{B}^{-1}\tilde{A}_{C,k}
|
|
+\tilde{A}_{\sigma(i),i}\tilde{A}_{\gamma_{S_{B}}(\sigma(i)),k}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
Using Equation~(\ref{eq:alpha_beta_S}) we obtain
|
|
\begin{equation}
|
|
(A_{B}^{-1}\tilde{A}_{k})_{x_{i}} =
|
|
-\tilde{A}_{\sigma(i), i}\tilde{A}_{\sigma(i), B_{O}}
|
|
\check{A}_{B}^{-1}\tilde{A}_{C, k}
|
|
+\tilde{A}_{\sigma(i), i}\tilde{A}_{\gamma{S_{B}}(\sigma(i)), k}
|
|
\end{equation}
|
|
\end{enumerate}
|
|
|
|
\subsection{PhaseII}
|
|
According to Sections~\ref{sec:Ties_ratio_test_step_1}
|
|
and~\ref{sec:Ties_ratio_test_step_2} we have in the QP-case
|
|
to consider the polynomials $p_{x_{i}}^{(Q_{1})}(\varepsilon, B)$,
|
|
\pmu{Q_{1}}{B} and
|
|
$p_{x_{i}}^{(Q_{2})}(\varepsilon, \hat{B})$ in light of
|
|
Equation~(\ref{eq:M_B_inv_exp}). The LP-case in phaseII can be omitted,
|
|
since only LP-type ties can occur in the LP-case
|
|
and these have already been treated in the last section.
|
|
|
|
\subsubsection{Ratio Test Step 1:
|
|
$\px{i}{Q_{1}}{B}$}
|
|
For ease of reference, we restate the Definition~(\ref{def:p_x_i_Q_1}),
|
|
of $p_{x_{i}}^{(Q_{1})}(\varepsilon, B)$
|
|
\begin{eqnarray*}
|
|
p_{x_{i}}^{(Q_{1})}\left(\varepsilon, B\right) &:=&
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N} \\
|
|
\hline
|
|
2D_{B, N}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{N}}{q_{x_{i}}}
|
|
\\
|
|
&=&
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{B, N \setminus \{j\}}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{N \setminus \{j\}}}{q_{x_{i}}}
|
|
+ \varepsilon^{j+1}
|
|
\end{eqnarray*}
|
|
Again, we will only consider the coefficients of
|
|
$\varepsilon^{k+1}$ for $k \in \hat{N}= N \setminus \{j\}$ in
|
|
$p_{x_{i}}^{(Q_{1})}(\varepsilon, B)$, since the other cases $k=j$ and $k \in B$
|
|
are trivial. Furthermore we only evaluate entities that are not evaluated in the
|
|
unperturbed problem, such that, taking into account the
|
|
Definition~(\ref{def:epsilon}) of $\epsilon$, we merely consider
|
|
the subexpression
|
|
\begin{equation}
|
|
\label{def:n_x_i_Q_1}
|
|
n_{x_{i}}^{(Q_{1})}(B)[i,k]:=
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B,k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
\end{equation}
|
|
The correct value of the coefficient is obtained by scaling the subexpression
|
|
with the factor $q_{x_{i}}^{-1}$.
|
|
According to Equation~(\ref{eq:M_B_inv_exp}) and the definition of
|
|
$p_{x_{i}}^{(Q_{1})}(\varepsilon, B)$ we distinguish $i \in B_{O}$ and
|
|
$i \in B_{S}$.
|
|
|
|
\paragraph{$\mathbf{i \in B_{O}}$:}
|
|
Assuming $i \in B_{O}$ we distinguish
|
|
according to Definition of $p_{x_{i}}^{(Q_{1})}(\varepsilon, B)$
|
|
the following cases for $k \in \hat{N}$:
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap S$:
|
|
\begin{eqnarray}
|
|
\label{eq:r1_i_B_O_k_N_S}
|
|
\lefteqn{\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=} \nonumber \\
|
|
&&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C \cup S_{B}}A_{C \cup S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O} \cup B_{S}}D_{B_{O} \cup B_{S}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
+\left(M_{B}^{-1}\right)_{x_{i}, S_{B}}A_{S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{S}}D_{B_{S}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:M_B_inv_exp}) $\left(M_{B}^{-1}\right)_{B_{O}, S_{B}}=0$,
|
|
and because of $D_{B_{O}, k}=0$ for $k \in N \cap S$
|
|
and $D_{B_{S},k}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{\beta_{O}(i),C}A_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{\beta_{O}(i), \gamma_{C}(\sigma(k))}
|
|
A_{\sigma(k), k}
|
|
\end{eqnarray}
|
|
|
|
\item $k \in \hat{N} \cap O$:
|
|
\begin{eqnarray}
|
|
\label{eq:r1_i_B_O_k_N_O}
|
|
\lefteqn{\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=} \nonumber \\
|
|
&&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C \cup S_{B}}A_{C \cup S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O} \cup B_{S}}D_{B_{O} \cup B_{S}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
+\left(M_{B}^{-1}\right)_{x_{i}, S_{B}}A_{S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{S}}D_{B_{S}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:M_B_inv_exp}) $\left(M_{B}^{-1}\right)_{B_{O}, S_{B}}=0$
|
|
and because of $D_{B_{S},k}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{\beta_{O}(i),C}A_{C,k}
|
|
+2\left(\check{M}_{B}^{-1}\right)_{\beta_{O}(i), B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{\beta_{O}(i)}
|
|
\left(\begin{array}{c}
|
|
A_{C, k} \\
|
|
\hline
|
|
2D_{B_{O}, k}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
\end{enumerate}
|
|
|
|
|
|
\paragraph{$\mathbf{i \in B_{S}}$:}
|
|
Assuming $i \in B_{S}$ we distinguish
|
|
according to Definition of $p_{x_{i}}^{(Q_{1})}(\varepsilon, B)$
|
|
the following two cases for $k \in \hat{N}$:
|
|
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap S$:
|
|
\begin{eqnarray}
|
|
\label{eq:r1_i_B_S_k_N_S}
|
|
\lefteqn{\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=} \nonumber \\
|
|
&&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C \cup S_{B}}A_{C \cup S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O} \cup B_{S}}D_{B_{O} \cup B_{S}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
+\left(M_{B}^{-1}\right)_{x_{i}, S_{B}}A_{S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{S}}D_{B_{S}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $A_{S_{B},k}=0$ and $D_{B_{O}, k}=0$ for
|
|
$k \in N \cap S$ and $D_{B_{S}, k}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{\beta_{S}(i), C}A_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{\beta_{S}(i), \gamma_{C}(\sigma(k))}
|
|
A_{\sigma(k), k}
|
|
\end{eqnarray}
|
|
|
|
\item $k \in \hat{N} \cap O$:
|
|
\begin{eqnarray}
|
|
\lefteqn{\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=} \nonumber \\
|
|
&&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C \cup S_{B}}A_{C \cup S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O} \cup B_{S}}D_{B_{O} \cup B_{S}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
+\left(M_{B}^{-1}\right)_{x_{i}, S_{B}}A_{S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{S}}D_{B_{S}, k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{8cm}
|
|
because of $D_{B_{S}, k}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{x_{i}, C}A_{C, k}
|
|
+\left(M_{B}^{-1}\right)_{x_{i}, S_{B}}A_{S_{B}, k}
|
|
+2\left(M_{B}^{-1}\right)_{x_{i}, B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, C}\right)_{\beta_{S}(i)}
|
|
A_{C, k}
|
|
+\left(A_{S_{B}, B_{S}}^{T}\right)_{\beta_{S}(i)}A_{S_{B},k}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}}\right)_{
|
|
\beta_{S}(i)}D_{B_{O},k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\right)_{\beta_{S}(i)}\left(\check{M}_{B}^{-1}\right)_{B_{O}, C}
|
|
A_{C, k}
|
|
+2\left(\alpha\right)_{\beta_{S}(i)}\left(\check{M}_{B}^{-1}\right)_{B_{O},
|
|
B_{O}}D_{B_{O}, k}
|
|
\nonumber \\
|
|
&&
|
|
+A_{\sigma(i), i}A_{\gamma_{S_{B}}(\sigma(i)), k}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\alpha\right)_{\beta_{S}(i)}\left(\check{M}_{B}^{-1}\right)_{B_{O}}
|
|
\left(\begin{array}{c}
|
|
A_{C,k} \\
|
|
\hline
|
|
2D_{B_{O},k}
|
|
\end{array}
|
|
\right)
|
|
+A_{\sigma(i), i}A_{\gamma_{S_{B}}(\sigma(i)), k}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
Using Equation~(\ref{eq:alpha_beta_S}) we obtain
|
|
\begin{eqnarray}
|
|
\label{eq:r1_i_B_S_k_N_O}
|
|
\left(M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
&=&
|
|
-A_{\sigma(i),i}A_{\sigma(i), B_{O}}\left(\check{M}_{B}^{-1}\right)_{B_{O}}
|
|
\left(\begin{array}{c}
|
|
A_{C,k} \\
|
|
\hline
|
|
2D_{B_{O},k}
|
|
\end{array}
|
|
\right)
|
|
\nonumber \\
|
|
&&
|
|
+A_{\sigma(i), i}A_{\gamma_{S_{B}}(\sigma(i)), k}
|
|
\end{eqnarray}
|
|
\end{enumerate}
|
|
|
|
\subsubsection{Ratio Test Step 1:
|
|
\pmu{Q_{1}}{B}}
|
|
For ease of reference, we restate the Definition~(\ref{def:p_mu_j_Q_1}),
|
|
of \pmu{Q_{1}}{B}
|
|
\begin{eqnarray*}
|
|
\pmu{Q_{1}}{B} &:=&
|
|
-\frac{2D_{j, N \setminus \{j\}} -
|
|
\left(A_{j}^{T} \left| \right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{B,N \setminus \{j\}}
|
|
\end{array}
|
|
\right)}{\nu}
|
|
\epsilon_{N \setminus \{j\}}
|
|
\\
|
|
&&
|
|
+\varepsilon^{j+1}
|
|
\\
|
|
&=&
|
|
-\frac{2D_{j, N \setminus \{j\}} -
|
|
\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{B,N \setminus \{j\}}
|
|
\end{array}
|
|
\right)}{\nu}
|
|
\epsilon_{N \setminus \{j\}}
|
|
+\varepsilon^{j+1}
|
|
\end{eqnarray*}
|
|
Again, we will only consider the coefficients of
|
|
$\varepsilon^{k+1}$ for $k \in \hat{N}= N \setminus \{j\}$ in
|
|
\pmu{Q_{1}}{B}, since the other cases $k=j$ and $k \in B$
|
|
are trivial. Furthermore we only evaluate entities that are not evaluated in the
|
|
unperturbed problem, such that, taking into account the
|
|
Definition~(\ref{def:epsilon}) of $\epsilon$
|
|
and the fact that $j$ and $B$ remain constant during an iteration of Ratio Test
|
|
Step~1 of a given pivot step,
|
|
we merely consider the subexpression
|
|
\begin{equation}
|
|
\label{def:n_mu_j_Q_1}
|
|
n_{\mu_{j}}^{(Q_{1})}(B)[j,k]:=
|
|
2D_{j, k} -
|
|
\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B,k}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
The correct value of the coefficient is obtained by scaling the subexpression
|
|
with the factor $\nu^{-1}$.
|
|
We shall first compute the different components of
|
|
$\left(q_{\lambda}^{T} \left|\right. q_{x}^{T}\right)$ in terms of
|
|
$\check{M}_{B}^{-1}$.
|
|
|
|
\begin{eqnarray}
|
|
\label{eq:q_C}
|
|
q_{\lambda_{C}}
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{C,C}A_{C,j}
|
|
+\left(M_{B}^{-1}\right)_{C, S_{B}}A_{S_{B}, j}
|
|
+2\left(M_{B}^{-1}\right)_{C, B_{O}}D_{B_{O}, j}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{C, B_{S}}D_{B_{S}, j}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:M_B_inv_exp}) $\left(M_{B}^{-1}\right)_{C, S_{B}}=0$
|
|
and because of $D_{B_{S},j}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{C,C}A_{C,j}
|
|
+2\left(\check{M}_{B}^{-1}\right)_{C, B_{O}}D_{B_{O}, j}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{C}
|
|
\left(\begin{array}{c}
|
|
A_{C, j} \\
|
|
\hline
|
|
2D_{B_{O}, j}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
|
|
\begin{eqnarray}
|
|
q_{\lambda_{S_{B}}}
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{S_{B},C}A_{C,j}
|
|
+\left(M_{B}^{-1}\right)_{S_{B}, S_{B}}A_{S_{B}, j}
|
|
+2\left(M_{B}^{-1}\right)_{S_{B}, B_{O}}D_{B_{O}, j}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{S_{B}, B_{S}}D_{B_{S}, j}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:M_B_inv_exp}) $\left(M_{B}^{-1}\right)_{S_{B}, C}=0$,
|
|
$\left(M_{B}^{-1}\right)_{S_{B}, S_{B}}=0$ and
|
|
$\left(M_{B}^{-1}\right)_{S_{B}, B_{O}}=0$
|
|
and because of $D_{B_{S},j}=0$
|
|
this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
0
|
|
\end{eqnarray}
|
|
|
|
\begin{eqnarray}
|
|
\label{eq:q_B_O}
|
|
q_{x_{B_{O}}}
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{B_{O},C}A_{C,j}
|
|
+\left(M_{B}^{-1}\right)_{B_{O}, S_{B}}A_{S_{B}, j}
|
|
+2\left(M_{B}^{-1}\right)_{B_{O}, B_{O}}D_{B_{O}, j}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{B_{O}, B_{S}}D_{B_{S}, j}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
by Equation~(\ref{eq:M_B_inv_exp}) $\left(M_{B}^{-1}\right)_{B_{O}, S_{B}}=0$
|
|
and because of $D_{B_{S},j}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O}, C}A_{C,j}
|
|
+2\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}}D_{B_{O}, j}
|
|
\nonumber \\
|
|
&=&
|
|
\left(\check{M}_{B}^{-1}\right)_{B_{O}}
|
|
\left(\begin{array}{c}
|
|
A_{C, j} \\
|
|
\hline
|
|
2D_{B_{O}, j}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
|
|
\begin{eqnarray}
|
|
q_{x_{B_{S}}}
|
|
&=&
|
|
\left(M_{B}^{-1}\right)_{B_{S},C}A_{C,j}
|
|
+\left(M_{B}^{-1}\right)_{B_{S}, S_{B}}A_{S_{B}, j}
|
|
+2\left(M_{B}^{-1}\right)_{B_{S}, B_{O}}D_{B_{O}, j}
|
|
\nonumber \\
|
|
&&
|
|
+2\left(M_{B}^{-1}\right)_{B_{S}, B_{S}}D_{B_{S}, j}
|
|
\nonumber \\
|
|
&&\begin{minipage}{9cm}
|
|
because of $D_{B_{S},j}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, C}A_{C, j}
|
|
+A_{S_{B}, B_{S}}^{T}A_{S_{B}, j}
|
|
+2\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}}D_{B_{O},j}
|
|
\nonumber \\
|
|
&=&
|
|
\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}}
|
|
\left(\begin{array}{c}
|
|
A_{C,j} \\
|
|
\hline
|
|
2D_{B_{O}, j}
|
|
\end{array}
|
|
\right)
|
|
+A_{S_{B}, B_{S}}^{T}A_{S_{B}, j}
|
|
\nonumber
|
|
\end{eqnarray}
|
|
Using the definition of $\alpha$ and $q_{x_{B_{O}}}$ this can be written as
|
|
\begin{equation}
|
|
q_{x_{B_{S}}}=
|
|
-A_{S_{B}, B_{S}}^{T}A_{S_{B}, B_{O}}q_{x_{B_{O}}}
|
|
+A_{S_{B}, B_{S}}^{T}A_{S_{B}, j}
|
|
\end{equation}
|
|
We are now enabled to compute the coefficients of
|
|
\begin{eqnarray}
|
|
\left(A_{j}^{T} \left| \right. 2D_{B, j}^{T} \right)
|
|
M_{B}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{\hat{N}} \\
|
|
\hline
|
|
2D_{B,\hat{N}}
|
|
\end{array}
|
|
\right)
|
|
&=&
|
|
\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{\hat{N}} \\
|
|
\hline
|
|
2D_{B,\hat{N}}
|
|
\end{array}
|
|
\right)
|
|
\nonumber \\
|
|
&=&
|
|
\left(q_{\lambda_{C}}^{T}\left|\right.
|
|
q_{\lambda_{S_{B}}}^{T}\left|\right.
|
|
q_{x_{B_{O}}}^{T}\left|\right.
|
|
q_{x_{B_{S}}}^{T}
|
|
\right)
|
|
\left(\begin{array}{c}
|
|
A_{C, \hat{N}} \\
|
|
\hline
|
|
A_{S_{B}, \hat{N}} \\
|
|
\hline
|
|
2D_{B_{O}, \hat{N}} \\
|
|
\hline
|
|
2D_{B_{S}, \hat{N}}
|
|
\end{array}
|
|
\right)
|
|
\nonumber \\
|
|
&&\begin{minipage}{6cm}
|
|
because of $D_{B_{S}, \hat{N}}=0$ and $q_{\lambda_{S_{B}}}=0$ this yields
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
q_{\lambda_{C}}^{T}A_{C, \hat{N}} + 2q_{x_{B_{O}}}^{T}D_{B_{O},\hat{N}}
|
|
\nonumber \\
|
|
&=&
|
|
\left(q_{\lambda_{C}}^{T}\left|\right.q_{x_{B_{O}}}^{T}\right)
|
|
\left(\begin{array}{c}
|
|
A_{C, \hat{N}} \\
|
|
\hline
|
|
2D_{B_{O}, \hat{N}}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
According to Equation~(\ref{eq:M_B_inv_exp}) and the definition of
|
|
\pmu{Q_{1}}{B} we distinguish $j \in O \cap N$ and
|
|
$j \in S \cap N$.
|
|
\paragraph{$\mathbf{j \in O \cap N}$:}
|
|
Assuming $j \in O \cap N$ we distinguish according to the definition of
|
|
\pmu{Q_{1}}{B} the following two cases for $k \in
|
|
\hat{N}$:
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap O$:
|
|
\begin{eqnarray}
|
|
\label{eq:r1_j_O_N_k_N_O}
|
|
2D_{j, k}
|
|
-\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B,k}
|
|
\end{array}
|
|
\right)
|
|
&=&
|
|
2D_{j, k} - q_{\lambda_{C}}^{T}A_{C, k} - 2q_{x_{B_{O}}}^{T}D_{B_{O},k}
|
|
\nonumber \\
|
|
&=&
|
|
2D_{j, k}
|
|
-\left(q_{\lambda_{C}}^{T}\left|\right. q_{x_{B_{O}}}^{T}\right)
|
|
\left(\begin{array}{c}
|
|
A_{C, k} \\
|
|
\hline
|
|
2D_{B_{O}, k}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
\item $k \in \hat{N} \cap S$:
|
|
\begin{eqnarray}
|
|
\label{eq:r1_j_O_N_k_N_S}
|
|
2D_{j,k}
|
|
-\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
&=&
|
|
2D_{j,k} - q_{\lambda_{C}}^{T}A_{C, k} - 2q_{x_{B_{O}}}^{T}D_{B_{O},k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{5cm}
|
|
Because of $D_{B_{O},k}=0$, $D_{j,k}=0$ for $k \in \hat{N} \cap S$
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
-q_{\lambda_{C}}^{T}A_{C, k}
|
|
\nonumber \\
|
|
&=&
|
|
-\left(q_{\lambda_{C}}\right)_{\gamma_{C}(\sigma(k))}A_{\sigma(k),k}
|
|
\end{eqnarray}
|
|
\end{enumerate}
|
|
|
|
\paragraph{$\mathbf{j \in S \cap N}$:}
|
|
Assuming $j \in S \cap N$ we distinguish according to the definition of
|
|
\pmu{Q_{1}}{B} the following cases for $k \in \hat{N}$:
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap O:$
|
|
\begin{eqnarray}
|
|
\label{eq:r1_j_S_N_k_N_O}
|
|
2D_{j,k}
|
|
-\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
&=&
|
|
2D_{j,k} - q_{\lambda_{C}}^{T}A_{C, k} - 2q_{x_{B_{O}}}^{T}D_{B_{O},k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{5cm}
|
|
Because of $D_{j,k}=0$ for $j \in S \cap N$
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
- q_{\lambda_{C}}^{T}A_{C, k} - 2q_{x_{B_{O}}}^{T}D_{B_{O},k}
|
|
\nonumber \\
|
|
&=&
|
|
-\left(q_{\lambda_{C}}^{T}\left|\right. q_{x_{B_{O}}}^{T}\right)
|
|
\left(\begin{array}{c}
|
|
A_{C,k} \\
|
|
\hline
|
|
2D_{B_{O},k}
|
|
\end{array}
|
|
\right)
|
|
\end{eqnarray}
|
|
|
|
\item $k \in \hat{N} \cap S:$
|
|
\begin{eqnarray}
|
|
\label{eq:r1_j_S_N_k_N_S}
|
|
2D_{j,k}
|
|
-\left(q_{\lambda}^{T} \left| \right. q_{x}^{T} \right)
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{B, k}
|
|
\end{array}
|
|
\right)
|
|
&=&
|
|
2D_{j,k} - q_{\lambda_{C}}^{T}A_{C, k} - 2q_{x_{B_{O}}}^{T}D_{B_{O},k}
|
|
\nonumber \\
|
|
&&\begin{minipage}{5cm}
|
|
Because of $D_{B_{O},k}=0$, $D_{j,k}=0$ for $j \in S \cap N$
|
|
\end{minipage}
|
|
\nonumber \\
|
|
&=&
|
|
-q_{\lambda_{C}}^{T}A_{C,k}
|
|
\nonumber \\
|
|
&=&
|
|
-\left(q_{\lambda_{C}}\right)_{\gamma_{C}(\sigma(k))}A_{\sigma(k),k}
|
|
\end{eqnarray}.
|
|
\end{enumerate}
|
|
|
|
\subsubsection{Ratio Test Step 2: \px{i}{Q_{2}}{\hat{B}}}
|
|
For ease of reference, we restate the Definition~(\ref{def:p_x_i_Q_2}),
|
|
of $\px{i}{Q_{2}}{\hat{B}}$
|
|
\begin{eqnarray*}
|
|
\px{i}{Q_{2}}{\hat{B}} &:=&
|
|
\frac{\varepsilon^{i+1} -
|
|
\left(M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{N \setminus \{j\}} \\
|
|
\hline
|
|
2D_{\hat{B}, N \setminus \{j\}}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}\epsilon_{N \setminus \{j\}}}{p_{x_{i}}}
|
|
\end{eqnarray*}
|
|
Again, we will only consider the coefficients of
|
|
$\varepsilon^{k+1}$ for $k \in \hat{N}= N \setminus \{j\}$ in
|
|
\px{i}{Q_{2}}{\hat{B}}, since the other case $k \in \hat{B}$
|
|
is trivial. Furthermore we only evaluate entities that are not evaluated in the
|
|
unperturbed problem, such that, taking into account the
|
|
Definition~(\ref{def:epsilon}) of $\epsilon$,
|
|
we merely consider the subexpression
|
|
\begin{equation}
|
|
\label{def:n_x_i_Q_2}
|
|
n_{x_{i}}^{(Q_{2})}(\hat{B})[i,k]:=
|
|
\left(
|
|
M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{\hat{B}, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
\end{equation}
|
|
The correct value of the coefficient is obtained by scaling the subexpression
|
|
with the factor $p_{x_{i}}^{-1}$.
|
|
Since the above subexpression and the corresponding subexpression
|
|
in~(\ref{def:n_x_i_Q_1}) for the polynomial \px{i}{Q_{1}}{B} as
|
|
functions differ only in the sets $B$ and $\hat{B}$ and their respective
|
|
headings, as well as in the scaling factor
|
|
we list the expressions without their derivations.
|
|
|
|
According to Equation~(\ref{eq:M_B_inv_exp}) and the definition of
|
|
\px{i}{Q_{2}}{\hat{B}} we distinguish $i \in \hat{B}_{O}$ and
|
|
$i \in \hat{B}_{S}$.
|
|
|
|
\paragraph{$\mathbf{i \in \hat{B}_{O}}$:}
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap S$:
|
|
According to Equation~(\ref{eq:r1_i_B_O_k_N_S}) and the appropriate changes
|
|
neccessary we obtain
|
|
\begin{equation}
|
|
\label{eq:r2_i_B_O_k_N_S}
|
|
\left(
|
|
M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{\hat{B}, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=
|
|
\left(\check{M}_{\hat{B}}^{-1}\right)_{\hat{\beta}_{O}(i),
|
|
\hat{\gamma}_{\hat{C}}(\sigma(k))}
|
|
A_{\sigma(k), k}
|
|
\end{equation}
|
|
\item $k \in \hat{N} \cap O$:
|
|
According to Equation~(\ref{eq:r1_i_B_O_k_N_O}) and the appropriate changes
|
|
neccessary we obtain
|
|
\begin{equation}
|
|
\label{eq:r2_i_B_O_k_N_O}
|
|
\left(
|
|
M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{\hat{B}, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=
|
|
\left(\check{M}_{\hat{B}}^{-1}\right)_{\hat{\beta}_{O}(i)}
|
|
\left(\begin{array}{c}
|
|
A_{\hat{C}, k} \\
|
|
\hline
|
|
2D_{\hat{B}_{O}, k}
|
|
\end{array}
|
|
\right)
|
|
\end{equation}
|
|
\end{enumerate}
|
|
|
|
\paragraph{$\mathbf{i \in \hat{B}_{S}}$:}
|
|
\begin{enumerate}
|
|
\item $k \in \hat{N} \cap S$:
|
|
According to Equation~(\ref{eq:r1_i_B_S_k_N_S}) and the appropriate changes
|
|
neccessary we obtain
|
|
\begin{equation}
|
|
\label{eq:r2_i_B_S_k_N_S}
|
|
\left(
|
|
M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{\hat{B}, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
=
|
|
\left(\check{M}_{\hat{B}}^{-1}\right)_{\hat{\beta}_{S}(i),
|
|
\hat{\gamma}_{\hat{C}}(\sigma(k))}
|
|
A_{\sigma(k), k}
|
|
\end{equation}
|
|
\item $k \in \hat{N} \cap O$:
|
|
According to Equation~(\ref{eq:r1_i_B_S_k_N_O}) and the appropriate changes
|
|
neccessary we obtain
|
|
\begin{eqnarray}
|
|
\label{eq:r2_i_B_S_k_N_O}
|
|
\left(
|
|
M_{\hat{B}}^{-1}
|
|
\left(\begin{array}{c}
|
|
A_{k} \\
|
|
\hline
|
|
2D_{\hat{B}, k}
|
|
\end{array}
|
|
\right)
|
|
\right)_{x_{i}}
|
|
&=&
|
|
-A_{\sigma(i),i}A_{\sigma(i), \hat{B}_{O}}
|
|
\left(\check{M}_{\hat{B}}^{-1}\right)_{\hat{B}_{O}}
|
|
\left(\begin{array}{c}
|
|
A_{\hat{C},k} \\
|
|
\hline
|
|
2D_{\hat{B}_{O},k}
|
|
\end{array}
|
|
\right)
|
|
\nonumber \\
|
|
&&
|
|
+A_{\sigma(i), i}A_{\hat{\gamma}_{S_{\hat{B}}}(\sigma(i)), k}
|
|
\end{eqnarray}
|
|
\end{enumerate}
|
|
|
|
\section{Ratio Tests for the perturbed problem}
|
|
The setup of the auxiliary problem and the transition from PhaseI to PhaseII
|
|
excepted, Ratio Test Step~1 and Ratio Test
|
|
Step~2 are the only parts of the algorithm that differ for the unperturbed and
|
|
perturbed problem.
|
|
In this section we present a pseudocode description of the ratio tests needed
|
|
for Ratio Test Step~1 and Ratio Test Step~2. We will only consider the most
|
|
general cases for which Ratio Test Step~1 and Ratio Test Step~2 occur,
|
|
that is for both ratio test steps we consider the QP-case with
|
|
inequalities in PhaseII only. Furthermore we will compute the coefficients
|
|
of the involved
|
|
polynomials $\pmu{Q_{1}}{B}$, $\px{i}{Q_{1}}{B}$ and $\px{i}{Q_{2}}{\hat{B}}$
|
|
only when needed. We denote by $\pxz{i}{Q_{1}}{B}[k]$ respectively
|
|
$\pxz{i}{Q_{2}}{\hat{B}}[k]$,
|
|
$0 \leq k \leq \left|O \cup S\right|$,
|
|
the coefficients of $\varepsilon^{k}$ in the polynomials
|
|
defined by Definitions~(\ref{def:t_min_eps}) and~(\ref{def:hat_mu_j_min_eps}),
|
|
such that $\pxz{i}{Q_{1}}{B}[0]=\check{t}(0, B)$ and
|
|
$\pxz{i}{Q_{2}}{\hat{B}}[0]=\check{\mu}_{j}(0, \hat{B})$ respectively.
|
|
Similarly, we denote by $\pmuz{Q_{1}}{B}[k]$,
|
|
$0 \leq k \leq \left|O \cup S\right|$, the coefficient of $\varepsilon^{k}$
|
|
in the polynomial
|
|
defined by Definition~(\ref{def:hat_mu_j_min_eps}), such that
|
|
$\pmuz{Q_{1}}{B}[0]=-\frac{\mu_{j}(0,0)}{\nu}$.
|
|
Note, that subscripted variable names in the pseudocode snippets of the
|
|
following subsections denote a single variable name.
|
|
|
|
\subsection{Ratio Test Step 1}
|
|
Since the Ratio Test Step~1 compares according to
|
|
Equation~(\ref{eq:mu_j_eps_t})
|
|
and Definition~(\ref{def:t_min_eps}) the smallest $t$,
|
|
$\check{t}(\varepsilon, B)$, such that some basic variable is leaving, and
|
|
$t=-\frac{\mu_{j}(\varepsilon,0)}{\nu}$ such that $\mu_{j}(\varepsilon,t)=0$.
|
|
For reasons of efficiency we factored out the most common case, that is, the
|
|
computation of $\check{t}(0, B)$ and $\mu_{j}(0, 0)$.
|
|
We distinguish three cases with respect to $\check{t}(0, B)$ and
|
|
$\mu_{j}(0, t)$, supposing that $T_{k}$ with
|
|
$\emptyset \subset T_{k} \subseteq B$, denotes the set of candidate
|
|
leaving variables after consideration of of
|
|
coefficients $\pxz{i}{Q_{1}}{B}[j]$, $0 \leq j \leq k$:
|
|
|
|
\begin{algorithm}
|
|
\caption{Perturbed Ratio Test 1, $\check{t}(0, B)$}
|
|
\label{alg:ratio_test_step_1_0}
|
|
\begin{algorithmic}
|
|
\Function{ratio\_test\_1\_\_t\_i\_$\varepsilon$}{$B_{O}, B_{S}$}
|
|
\State $T_{0}^{\prime} \gets \emptyset,
|
|
\quad x_{min} \gets 1, \quad q_{min} \gets 0$
|
|
\ForAll{$i \gets 0, \left|B_{O}\right| - 1$}
|
|
\If{$q_{B_{O}}[i] > 0$}
|
|
\If{$x_{min}*q_{B_{O}}[i] < x_{B_{O}}[i]*q_{min}$}
|
|
\State $x_{min} \gets x_{B_{O}}[i],
|
|
\quad q_{min} \gets q_{B_{O}}[i],
|
|
\quad T_{0}^{\prime} \gets \{B_{O}[i]\}$
|
|
\ElsIf{$x_{min}*q_{B_{O}}[i] = x_{B_{O}}[i]*q_{min}$}
|
|
\State $T_{0}^{\prime} \gets T_{0}^{\prime} \cup \{B_{O}[i]\}$
|
|
\EndIf
|
|
\EndIf
|
|
\EndFor
|
|
\ForAll{$i \gets 0, \left|B_{S}\right| - 1$}
|
|
\If{$q_{B_{S}}[i] > 0$}
|
|
\If{$x_{min}*q_{B_{S}}[i] < x_{B_{S}}[i]*q_{min}$}
|
|
\State $x_{min} \gets x_{B_{S}}[i],
|
|
\quad q_{min} \gets q_{B_{S}}[i],
|
|
\quad T_{0}^{\prime} \gets \{B_{S}[i]\}$
|
|
\ElsIf{$x_{min}*q_{B_{S}}[i] = x_{B_{S}}[i]*q_{min}$}
|
|
\State $T_{0}^{\prime} \gets T_{0}^{\prime} \cup \{B_{S}[i]\}$
|
|
\EndIf
|
|
\EndIf
|
|
\EndFor
|
|
\State \textbf{return} $(T_{0}^{\prime}, x_{min}, q_{min})$
|
|
\EndFunction
|
|
%\Function{ratio\_test\_1\_0\_t\_j}{}
|
|
%\EndFunction
|
|
\end{algorithmic}
|
|
\end{algorithm}
|
|
|
|
\begin{itemize}
|
|
\item $-\frac{\mu_{j}(0, 0)}{\nu} < \check{t}(0, B)$:
|
|
According to Equation~(\ref{eq:mu_j_eps_t}) and
|
|
Definition~(\ref{def:t_min_eps}) we then have
|
|
$-\frac{\mu_{j}(\varepsilon, 0)}{\nu} < \check{t}(\varepsilon, B)$ and a local
|
|
optimum is found, according to Lemma~2.7 $B \cup \{j\}$ is the new basis.
|
|
|
|
\item $-\frac{\mu_{j}(0, 0)}{\nu} = \check{t}(0, B)$:
|
|
According to Equation~(\ref{eq:mu_j_eps_t}) and
|
|
Definition~(\ref{def:t_min_eps})
|
|
both $-\frac{\mu_{j}(\varepsilon, 0)}{\nu} < \check{t}(\varepsilon, B)$ and
|
|
$-\frac{\mu_{j}(\varepsilon, 0)}{\nu} > \check{t}(\varepsilon, B)$ are
|
|
possible.
|
|
We continue comparing the coefficients $\pmu{Q_{1}}{B}[k]$ and
|
|
$\px{i}{Q_{1}}{B}[k]$, $1 \leq k \leq \left|O \cup S \right|$,
|
|
until $\pmu{Q_{1}}{B}[k] \neq \px{i}{Q_{1}}{B}[k]$.
|
|
If $-\frac{\mu_{j}(\varepsilon, 0)}{\nu} < \check{t}(\varepsilon, B)$
|
|
a local optimum is found and according to Lemma~2.7 $B \cup \{j\}$ is the new
|
|
basis,
|
|
if $-\frac{\mu_{j}(\varepsilon, 0)}{\nu} > \check{t}(\varepsilon, B)$
|
|
we continue computing coefficients
|
|
$\px{i}{Q_{1}}{B}[k]$, $1 \leq k \leq \left|O \cup S \right|$, until
|
|
$\left|T_{k}\right|=1$. $T_{k}$ then contains the index of the leaving
|
|
variable.
|
|
|
|
\item $-\frac{\mu_{j}(0, 0)}{\nu} > \check{t}(0, B)$:
|
|
According to Equation~(\ref{eq:mu_j_eps_t}) and
|
|
Definition~(\ref{def:t_min_eps}) we then have
|
|
$-\frac{\mu_{j}(\varepsilon, 0)}{\nu} > \check{t}(\varepsilon, B)$. If
|
|
$\left|T_{k}\right| > 1$ we continue computing coefficients
|
|
$\px{i}{Q_{1}}{B}[k]$, $1 \leq k \leq \left|O \cup S \right|$, until
|
|
$\left|T_{k}\right|=1$. $T_{k}$ then contains the index of the leaving
|
|
variable.
|
|
\end{itemize}
|
|
\begin{algorithm}
|
|
\caption{Perturbed Ratio Test 1}
|
|
\label{alg:ratio_test_step_1_pert}
|
|
\begin{algorithmic}
|
|
\Function{ratio\_test\_1\_$\varepsilon$}{$B_{O},B_{S},j$}
|
|
\State $ratio\_test\_1\_\_q(B_{O},B_{S},j)$
|
|
\Comment{Initializes global $q_{B_{O}}$ and $q_{B_{S}}$}
|
|
\State $(T_{k}, c_{min}, q_{min})
|
|
\gets RATIO\_TEST\_1\_\_T\_I\_\varepsilon(B_{O},B_{S})$
|
|
\If{$q_{min}=0$}
|
|
\Comment{$\check{t}(\varepsilon, B)=\infty$}
|
|
\State \textbf{return} \texttt{unbounded}
|
|
\EndIf
|
|
\State $(c_{j,k}, \nu)
|
|
\gets RATIO\_TEST\_1\_\_T\_J(B_{O}, B_{S},j)$
|
|
\If{$c_{j,k}*q_{min} > \nu * c_{min} \wedge \left|T_{k}\right| =1$}
|
|
\Comment{$-\frac{\mu_{j}(0, 0)}{\nu}>\check{t}(0, B)
|
|
\wedge \left|T_{k}\right|=1$}
|
|
\State \textbf{return} $T_{k}$
|
|
\ElsIf{$c_{j,k}*q_{min} < \nu * c_{min}$}
|
|
\Comment{$-\frac{\mu_{j}(0, 0)}{\nu}<\check{t}(0, B)$}
|
|
\State \textbf{return} $\emptyset$
|
|
\Else
|
|
\Comment{$-\frac{\mu_{j}(0,0)}{\nu}>\check{t}(0,B) \wedge
|
|
\left|T_{k}\right|>1 \vee -\frac{\mu_{j}(0,0)}{\nu}=\check{t}(0,B)$}
|
|
\State $k \gets 0, \quad leaving \gets c_{j,k}*q_{min} > \nu * c_{min}$
|
|
\Repeat
|
|
\State $T_{k}^{\prime} \gets \emptyset,
|
|
\quad c_{min} \gets 1,
|
|
\quad q_{min} \gets 0$
|
|
\ForAll{$i \in T_{k}$}
|
|
\State $c_{i,k} \gets n_{x_{i}}^{(Q_{1})}(B)[i,k]$
|
|
\If{$i < n$} \Comment{$x_{i}$ is original variable}
|
|
\If{$c_{min}*q_{B_{O}}[\beta_{O}[i]]<c_{i,k}*q_{min}$}
|
|
\State $c_{min} \gets c_{i,k},
|
|
\quad q_{min} \gets q_{B_{O}}[\beta_{O}[i]],
|
|
\quad T_{k}^{\prime} \gets \{i\}$
|
|
\ElsIf{$c_{min}*q_{B_{O}}[\beta_{O}[i]]=c_{i,k}*q_{min}$}
|
|
\State $T_{k}^{\prime} \gets T_{k}^{\prime} \cup \{i\}$
|
|
\EndIf
|
|
\Else \Comment{$x_{i}$ is slack variable}
|
|
\If{$c_{min}*q_{B_{S}}[\beta_{S}[i]]<c_{i,k}*q_{min}$}
|
|
\State $c_{min} \gets c_{i,k},
|
|
\quad q_{min} \gets q_{B_{S}}[\beta_{S}[i]],
|
|
\quad T_{k}^{\prime} \gets \{i\}$
|
|
\ElsIf{$c_{min}*q_{B_{S}}[\beta_{S}[i]]=c_{i,k}*q_{min}$}
|
|
\State $T_{k}^{\prime} \gets T_{k}^{\prime} \cup \{i\}$
|
|
\EndIf
|
|
\EndIf
|
|
\EndFor
|
|
\If{$ \neg leaving$}
|
|
\State $c_{j,k} \gets n_{\mu_{j}}^{(Q_{1})}(B)[j,k]$
|
|
\If{$c_{j,k} * q_{min} < \nu *c_{min}$}
|
|
\Comment{$-\frac{\mu_{j}(\varepsilon,0)}{\nu}<
|
|
\check{t}(\varepsilon,B)$}
|
|
\State \textbf{return} $\emptyset$
|
|
\EndIf
|
|
\State $leaving \gets c_{j,k} * q_{min} > \nu *c_{min}$
|
|
\EndIf
|
|
\State $k \gets k+1, \quad T_{k} \gets T_{k}^{\prime}$
|
|
\Until{$\left|T_{k}\right|=1 \wedge leaving$}
|
|
\Comment{$-\frac{\mu_{j}(\varepsilon,0)}{\nu}>\check{t}(\varepsilon,B)
|
|
\wedge \left|T_{k}\right|=1$}
|
|
\State \textbf{return} $T_{k}$
|
|
\EndIf
|
|
\EndFunction
|
|
\end{algorithmic}
|
|
\end{algorithm}
|
|
The function that computes $\check{t}(0, B)$ is outlined in
|
|
Algorithm~(\ref{alg:ratio_test_step_1_0}), the keeping of the set of candidate
|
|
leaving variables $T_{0}$ is the only part in which it differs from the
|
|
unperturbed variant.
|
|
The function \texttt{RATIO\_TEST\_1\_\_T\_J}
|
|
which computes $-\frac{\mu_{j}(0,0)}{\nu}$ is omitted here, since it already
|
|
occurs in the unperturbed problem.
|
|
The function that performs the actual Ratio Test Step~1 by comparing
|
|
$\pmuz{Q_{1}}{B}$ and $\pxz{i}{Q_{1}}{B}$
|
|
coefficient by coefficient is outlined in
|
|
Algorithm~(\ref{alg:ratio_test_step_1_pert}). It returns the unique index of
|
|
the leaving variable if
|
|
$-\frac{\mu_{j}(\varepsilon, 0)}{\nu} > \check{t}(\varepsilon,B)$, the empty
|
|
set if $-\frac{\mu_{j}(\varepsilon, 0)}{\nu} < \check{t}(\varepsilon,B)$
|
|
and \texttt{unbounded} if $\check{t}(\varepsilon, B)=\infty$.
|
|
Note, that by the remarks of
|
|
Section~(\ref{sec:Ties_ratio_test_step_1}) \px{i}{Q_{1}}{B} is unique and
|
|
either $\px{i}{Q_{1}}{B} < \pmu{Q_{1}}{B}$ or
|
|
$\px{i}{Q_{1}}{B} > \pmu{Q_{1}}{B}$
|
|
holds, such that Algorithm~(\ref{alg:ratio_test_step_1_pert}) terminates.
|
|
Note, that there is opportunity for improvement; we could distinguish the cases
|
|
$\neg leaving \wedge \left|T_{k}\right|=1$, where the index of the potentially
|
|
leaving variable $x_{i}$ is known and no minimum among basic variables has to
|
|
be determined in order to compute the next coefficient of \px{i}{Q_{1}}{B}, and
|
|
$\neg leaving \wedge \left|T_{k}\right|>1$, where the next coefficient of
|
|
\px{i}{Q_{1}}{B} is to be computed as minimum over the index set $T_{k}$.
|
|
|
|
\subsection{Ratio Test Step 2}
|
|
Since the Ratio Test Step~2 determines according to
|
|
Equation~(\ref{eq:QP_j_mu_opt_short}) the absolute
|
|
$\mu_{j}(\varepsilon)$ for which some
|
|
basic variable is leaving, supposing that $T_{k}$ with
|
|
$\emptyset \subset T_{k} \subseteq \hat{B}$, denotes the set
|
|
of candidate leaving variables after consideration of coefficients
|
|
$\pxz{i}{Q_{2}}{\hat{B}}[j]$,
|
|
$0 \leq j \leq k$,
|
|
we distinguish three cases with respect to
|
|
$\check{\mu}_{j}(0, \hat{B})$:
|
|
\begin{algorithm}
|
|
\caption{Perturbed Ratio Test 2, $\check{\mu}_{j}(0,\hat{B})$}
|
|
\label{alg:ratio_test_step_2_0}
|
|
\begin{algorithmic}
|
|
\Function{ratio\_test\_2\_0\_$\varepsilon$}{$B_{O},B_{S}$}
|
|
\State $T_{0} \gets \emptyset,
|
|
\quad x_{min} \gets 1, \quad p_{min} \gets 0$
|
|
\ForAll{$i \gets 0, \left|B_{O}\right| - 1$}
|
|
\If{$p_{B_{O}}[i] < 0$}
|
|
\If{$x_{min}*p_{B_{O}}[i] < x_{B_{O}}[i]*p_{min}$}
|
|
\State $x_{min} \gets x_{B_{O}}[i],
|
|
\quad p_{min} \gets p_{B_{O}}[i],
|
|
\quad T_{0} \gets \{B_{O}[i]\}$
|
|
\ElsIf{$x_{min}*p_{B_{O}}[i] = x_{B_{O}}[i]*p_{min}$}
|
|
\State $T_{0} \gets T_{0} \cup \{B_{O}[i]\}$
|
|
\EndIf
|
|
\EndIf
|
|
\EndFor
|
|
\ForAll{$i \gets 0, \left|B_{S}\right| - 1$}
|
|
\If{$p_{B_{S}}[i] < 0$}
|
|
\If{$x_{min}*p_{B_{S}}[i] < x_{B_{S}}[i]*p_{min}$}
|
|
\State $x_{min} \gets x_{B_{S}}[i],
|
|
\quad p_{min} \gets p_{B_{S}}[i],
|
|
\quad T_{0} \gets \{B_{S}[i]\}$
|
|
\ElsIf{$x_{min}*p_{B_{S}}[i] = x_{B_{S}}[i]*p_{min}$}
|
|
\State $T_{0} \gets T_{0} \cup \{B_{S}[i]\}$
|
|
\EndIf
|
|
\EndIf
|
|
\EndFor
|
|
\State \textbf{return} $(T_{0}, x_{min}, p_{min})$
|
|
\EndFunction
|
|
\end{algorithmic}
|
|
\end{algorithm}
|
|
\begin{itemize}
|
|
\item $\check{\mu}_{j}(0, \hat{B}) > 0$:
|
|
According to Definition~(\ref{def:hat_mu_j_min_eps})
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B}) > 0$ then holds, so we found an optimal
|
|
solution $x_{\hat{B}}^{*}(\varepsilon, 0)$ to $(UQP(\hat{B}_{\varepsilon}))$
|
|
which by Lemma~(\ref{lemma:strict}) is also an optimal solution to
|
|
$QP(\hat{B}_{\varepsilon})$.
|
|
|
|
\item $\check{\mu}_{j}(0, \hat{B}) = 0$:
|
|
According to Definition~(\ref{def:hat_mu_j_min_eps}) and
|
|
Definition~(\ref{def:p_x_i_Q_2}) both
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B}) > 0$ and
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B}) < 0$ are possible.
|
|
We compute coefficients $\px{i}{Q_{2}}{\hat{B}}[k]$,
|
|
$1 \leq k \leq \left|O \cup S\right|$, until the sign of
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B})$ is known.
|
|
If $\check{\mu}_{j}(\varepsilon, \hat{B}) > 0$, we found an optimal solution
|
|
$x_{\hat{B}}^{*}(\varepsilon, \hat{B})$ to $UQP(\hat{B}_{\varepsilon})$,
|
|
which again by Lemma~(\ref{lemma:strict}) is also an optimal solution to
|
|
$QP(\hat{B}_{\varepsilon})$,
|
|
if $\check{\mu}_{j}(\varepsilon, \hat{B}) < 0$, we continue computing
|
|
coefficients $\px{i}{Q_{2}}{\hat{B}}[k]$ until
|
|
$\left|T_{k}\right|=1$. $T_{k}$ then contains the index of the leaving
|
|
variable.
|
|
|
|
\item $\check{\mu}_{j}(0, \hat{B}) < 0$:
|
|
According to Definition~(\ref{def:hat_mu_j_min_eps})
|
|
$\check{\mu}_{j}(\varepsilon, \hat{B}) < 0$ then holds, so we compute
|
|
coefficients $\px{i}{Q_{2}}{\hat{B}}[k]$,
|
|
$1 \leq k \leq \left|O \cup S\right|$, until $\left|T_{k}\right|=1$.
|
|
$T_{k}$ then contains the index of the leaving
|
|
variable.
|
|
\end{itemize}
|
|
|
|
\begin{algorithm}
|
|
\caption{Perturbed Ratio Test 2}
|
|
\label{alg:ratio_test_step_2_pert}
|
|
\begin{algorithmic}
|
|
\Function{ratio\_test\_2\_$\varepsilon$}{$B_{O},B_{S},j$}
|
|
\State $ratio\_test\_2\_\_p$
|
|
\Comment{Initializes global $p_{B_{O}}$ and $p_{B_{S}}$}
|
|
\State $(T_{k}, c_{min}, p_{min})
|
|
\gets RATIO\_TEST\_2\_0\_\varepsilon(B_{O},B_{S})$
|
|
\If{$p_{min} = 0$}
|
|
\Comment{$\check{\mu}_{j}(\varepsilon, \hat{B}) = \infty$}
|
|
\State \textbf{return} $\emptyset$
|
|
\ElsIf{$c_{min} < 0$}
|
|
\Comment{$\check{\mu}_{j}(0, \hat{B}) > 0$}
|
|
\State \textbf{return} $\emptyset$
|
|
\ElsIf{$c_{min} > 0 \wedge \left|T_{k}\right|=1$}
|
|
\Comment{$\check{\mu}_{j}(0, \hat{B}) < 0 \wedge
|
|
\left|T_{k}\right|=1$}
|
|
\State \textbf{return} $T_{k}$
|
|
\Else
|
|
\Comment{$\check{\mu}_{j}(0, \hat{B}) < 0 \wedge \left|T_{k}\right|>1
|
|
\vee \check{\mu}_{j}(0, \hat{B})=0$}
|
|
\State $k \gets 0$
|
|
\Repeat
|
|
\State $T_{k}^{\prime} \gets \emptyset,
|
|
\quad c_{min} \gets 1,
|
|
\quad p_{min} \gets 0$
|
|
\ForAll{$i \in T_{k}$}
|
|
\State $c_{i,k} \gets n_{x_{i}}^{(Q_{2})}(\hat{B})[i,k]$
|
|
\If{$i < n$} \Comment{$x_{i}$ is original variable}
|
|
\If{$c_{min}*p_{B_{O}}[\beta_{O}[i]]<c_{i,k}*p_{min}$}
|
|
\State $c_{min} \gets c_{i,k},
|
|
\quad p_{min} \gets p_{B_{O}}[\beta_{O}[i]],
|
|
\quad T_{k}^{\prime} \gets \{i\}$
|
|
\ElsIf{$c_{min}*p_{B_{O}}[\beta_{O}[i]]=c_{i,k}*p_{min}$}
|
|
\State $T_{k}^{\prime} \gets T_{k}^{\prime} \cup \{i\}$
|
|
\EndIf
|
|
\Else \Comment{$x_{i}$ is slack variable}
|
|
\If{$c_{min}*p_{B_{S}}[\beta_{S}[i]]<c_{i,k}*p_{min}$}
|
|
\State $c_{min} \gets c_{i,k},
|
|
\quad p_{min} \gets p_{B_{S}}[\beta_{S}[i]],
|
|
\quad T_{k}^{\prime} \gets \{i\}$
|
|
\ElsIf{$c_{min}*p_{B_{S}}[\beta_{S}[i]]=c_{i,k}*p_{min}$}
|
|
\State $T_{k}^{\prime} \gets T_{k}^{\prime} \cup \{i\}$
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|
\EndIf
|
|
\EndIf
|
|
\EndFor
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|
\State $k \gets k+1, \quad T_{k} \gets T_{k}^{\prime}$
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|
% \State $T_{k} \gets T_{k}^{\prime}$
|
|
\Until{$\left|T_{k}\right|=1 \wedge c_{min} > 0 \vee c_{min}<0$}
|
|
\If{$c_{min} < 0$}
|
|
\Comment{$\check{\mu}_{j}(\varepsilon, \hat{B}) > 0$}
|
|
\State \textbf{return} $\emptyset$
|
|
\Else
|
|
\Comment{$\check{\mu}_{j}(\varepsilon, \hat{B}) < 0 \wedge
|
|
\left|T_{k}\right|=1$}
|
|
\State \textbf{return} $T_{k}$
|
|
\EndIf
|
|
\EndIf
|
|
\EndFunction
|
|
\end{algorithmic}
|
|
\end{algorithm}
|
|
The function that computes $\check{\mu}_{j}(0, \hat{B})$ is outlined in
|
|
Algorithm~(\ref{alg:ratio_test_step_2_0}), the keeping of the set of
|
|
candidate leaving variables $T_{0}$ excepted, it does not differ from the
|
|
unperturbed variant.
|
|
The function that computes $\check{\mu}_{j}(\varepsilon, \hat{B})$, based on
|
|
the value of $\check{\mu}_{j}(0, \hat{B})$, is outlined
|
|
in Algorithm~(\ref{alg:ratio_test_step_2_pert}). It returns the unique index of
|
|
the leaving variable if $\check{\mu}_{j}(\varepsilon, \hat{B})<0$ and the empty
|
|
set otherwise.
|
|
Again, by the remarks of
|
|
Section~(\ref{sec:Ties_ratio_test_step_1}) \px{i}{Q_{1}}{\hat{B}} is unique and
|
|
either $\px{i}{Q_{2}}{\hat{B}} < 0$ or $\px{i}{Q_{2}}{\hat{B}} > 0$
|
|
holds, such that Algorithm~(\ref{alg:ratio_test_step_1_pert}) terminates.
|
|
Note, that there is opportunity for improvement here as well;
|
|
we could distinguish the cases $\left|T_{k}\right|=1$, where the index of
|
|
the potentially leaving variable $x_{i}$ is known and no minimum among basic
|
|
variables has to be determined in order to compute the next coefficient of
|
|
\px{i}{Q_{2}}{\hat{B}}, and
|
|
$\left|T_{k}\right|>1$, where the next coefficient of
|
|
\px{i}{Q_{2}}{\hat{B}} is to be computed as minimum over the index set $T_{k}$.
|
|
|
|
\begin{thebibliography}{99}
|
|
\bibitem{Sven} S. Sch\"{o}nherr Quadratic Programming in Geometric Optimization:
|
|
Theory, Implementation, and Applications, Dissertation, Diss. ETH No 14738, ETH
|
|
Z\"{u}rich, Institute of Theoretical Computer Science, 2002.
|
|
\bibitem{Chvatal} Va\v{s}ek Chv\'{a}tal. \textit{Linear Programming}. W. H. Freeman and Company,
|
|
New York, Chapter 8, 1983
|
|
\end{thebibliography}
|
|
\end{document}
|