mirror of https://github.com/CGAL/cgal
259 lines
6.2 KiB
TeX
259 lines
6.2 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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\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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\newtheorem{lemma}{Lemma}
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\newtheorem{assumption}{Assumption}
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\newtheorem{definition}{Definition}
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\title{Test\_suite\_QP\_solver}
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\author{Frans Wessendorp}
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\begin{document}
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\maketitle
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\section{Validity check}
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All of the validity checks of a solution computed by the solver are done using
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the complete set of constraints; since the solver itself works with an active
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set method and therefore uses the reduced basis matrix $\check{M}_{B}$
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and its inverse we here restate the relationship given in \cite{Frans_Deg}.
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If the basis heading is given as
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$\left[C, S_{B}, B_{O}, B_{S} \right]$ the basis matrix
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$M_{B}$ has the following form
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\begin{equation}
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\label{def:basis_matrix}
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M_{B}:=
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\left(\begin{array}{c|c|c|c}
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0 & 0 & A_{C, B_{O}} & 0 \\
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\hline
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0 & 0 & A_{S_{B}, B_{O}} & A_{S_{B}, B_{S}} \\
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\hline
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A_{C, B_{O}}^{T} & A_{S_{B}, B_{O}}^{T} & D_{B_{O}, B_{O}}
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& 0 \\
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\hline
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0 & A_{S_{B}, B_{S}}^{T} & 0
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& 0 \\
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\end{array}
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\right).
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\end{equation}
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and the inverse $M_{B}^{-1}$ in terms of the reduced basis inverse
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$\check{M}_{B}^{-1}$
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\begin{equation}
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\label{eq:M_B_inv_exp}
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M_{B}^{-1}=
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\left(\begin{array}{c|c|c|c}
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\left(\check{M}_{B}^{-1}\right)_{C,C} &
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0 &
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\left(\check{M}_{B}^{-1}\right)_{C,B_{O}} &
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\left(\check{M}_{B}^{-1}\right)_{C, B_{O}}\alpha^{T} \\
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\hline
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0 &
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0 &
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0 &
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A_{S_{B},B_{S}} \\
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\hline
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\left(\check{M}_{B}^{-1}\right)_{B_{O}, C} &
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0 &
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\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}} &
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\left(\check{M}_{B}^{-1}\right)_{B_{O},B_{O}}\alpha^{T} \\
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\hline
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\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O},
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C} &
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A_{S_{B}, B_{S}}^{T} &
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\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O},
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B_{O}} &
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\alpha\left(\check{M}_{B}^{-1}\right)_{B_{O}, B_{O}}\alpha^{T}
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\end{array}
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\right)
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\end{equation}
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\subsection{Verifying Feasibility}
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\subsection{Verifying Optimality}
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\subsection{Verifying Unboundedness}
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Since the solver delivers, in case of unboundedness, implicitly a feasible
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solution $x^{*}$ and a vector
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$w$ such that
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\begin{equation}
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\label{eq:Unboundedness}
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x^{*}-tw \quad \text{for} \quad t>0
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\end{equation}
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is a feasible solution
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we by convention define the single nonzero nonbasic component of the basic
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feasible direction $w$ to be negative:
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\begin{eqnarray}
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\label{def:w_B}
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w_{B}&:=&q_{B}=\left(M_{B}^{-1}\right)_{B_{O} \cup B_{S}, \bullet}
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\left(
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\begin{array}{c}
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A_{C,j} \\
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\hline
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A_{S_{B}, j} \\
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\hline
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2D_{B_{O}, j} \\
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\hline
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2D_{B_{S}, j}
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\end{array}
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\right) \\
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\label{def:w_N}
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w_{N}&:=&-e_{\{j\}}
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\end{eqnarray}
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where $j \in N$ and $e_{\{j\}}$ denotes the unit vector with $\left|N\right|$
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entries.
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Feasibility of the solution in Equation~(\ref{eq:Unboundedness}) requires
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$w <0$ and $Aw=0$. That the latter is true for $w$ defined by
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Equations~(\ref{def:w_B}) and~(\ref{def:w_N}) shows the following computation
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where we use the
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Definitions~(\ref{def:basis_matrix}) and~(\ref{eq:M_B_inv_exp})
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\begin{eqnarray}
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Aw
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&=&
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A_{C \cup S_{B}, B_{O} \cup B_{S}}
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\left(M_{B}^{-1}\right)_{B_{O} \cup B_{S}, \bullet}
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\left(
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\begin{array}{c}
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A_{C,j} \\
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\hline
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A_{S_{B}, j} \\
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\hline
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2D_{B_{O}, j} \\
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\hline
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0
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\end{array}
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\right)
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-A_{C \cup S_{B}, j}
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\nonumber \\
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&=&
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\left(M_{B}\right)_{C \cup S_{B}, B_{O} \cup B_{S}}
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\left(M_{B}^{-1}\right)_{B_{O} \cup B_{S}, \bullet}
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\left(
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\begin{array}{c}
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A_{C,j} \\
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\hline
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A_{S_{B}, j} \\
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\hline
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2D_{B_{O}, j} \\
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\hline
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0
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\end{array}
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\right)
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-A_{C \cup S_{B}, j}
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\nonumber \\
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&=&
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\left(M_{B}\right)_{C \cup S_{B}, \bullet}
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M_{B}^{-1}
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\left(
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\begin{array}{c}
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A_{C,j} \\
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\hline
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A_{S_{B}, j} \\
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\hline
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2D_{B_{O}, j} \\
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\hline
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0
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\end{array}
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\right)
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-A_{C \cup S_{B}, j}
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\nonumber \\
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&=&
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\left[
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I_{C\cup S_{B},C\cup S_{B}} \left|\right.
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\mathbf{0}_{C\cup S_{B},B_{O}\cup B_{S}}
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\right]
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\left(
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\begin{array}{c}
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A_{C,j} \\
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\hline
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A_{S_{B}, j} \\
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\hline
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2D_{B_{O}, j} \\
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\hline
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0
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\end{array}
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\right)
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-A_{C \cup S_{B}, j}
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\nonumber \\
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&=&
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0
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\end{eqnarray}
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\subsubsection{Linear Case}
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We are minimizing the objective function
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\begin{equation}
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f\left(x\right):=c^{T}x,
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\end{equation}
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so for $t>0$
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\begin{eqnarray}
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f(x^{*}-tw)
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&=&
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c^{T}\left(x^{*}-tw\right)
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\nonumber \\
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&=&
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f\left(x^{*}\right) -tc^{T}w
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\end{eqnarray}
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Since we are minimizing we must require $c^{T}w>0$.
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Subsuming we obtain together with the above the necessary conditions for
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unboundedness.
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\begin{enumerate}
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\item $w_{x_{i}} \leq 0$ for $i \in B_{O} \cup B_{S}$
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\item $Aw=0$
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\item $c^{T}w>0$
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\end{enumerate}
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\subsubsection{Quadratic Case}
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We are minimizing the objective function
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\begin{equation}
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f\left(x\right):=c^{T}x+x^{T}Dx,
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\end{equation}
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using the symmetry of $D$ we obtain for $t>0$
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\begin{eqnarray}
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f(x^{*}-tw)
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&=&
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c^{T}\left(x^{*}-tw\right)
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+\left(x^{*}-tw\right)^{T}D\left(x^{*}-tw\right)
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\nonumber \\
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&=&
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c^{T}x^{*} - tc^{T}w + {x^{*}}^{T}Dx^{*} - tw^{T}Dx^{*}-t{x^{*}}^{T}Dw
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+t^{2}w^{T}Dw
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\nonumber \\
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&=&
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c^{T}x^{*} + {x^{*}}^{T}Dx^{*} + t^{2}w^{T}Dw
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-t\left[\left(c^{T}+2{x^{*}}^{T}D\right)w\right]
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\nonumber \\
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&=&
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f\left(x^{*}\right) + t^{2}w^{T}Dw
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-t\left[\left(c^{T}+2{x^{*}}^{T}D\right)w\right]
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\end{eqnarray}
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Since $D$ is positive semidefinite $w^{T}Dw \geq 0$ and since we are minimizing
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we must require $w^{T}Dw=0$ and $\left(c^{T}+2{x^{*}}^{T}D\right)w>0$.
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Subsuming we obtain together with the above the necessary conditions for
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unboundedness.
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\begin{enumerate}
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\item $w_{x_{i}} \leq 0$ for $i \in B_{O} \cup B_{S}$
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\item $Aw=0$
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\item $w^{T}Dw = 0$
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\item $\left(c^{T}+2{x^{*}}^{T}D\right)w>0$
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\end{enumerate}
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\begin{thebibliography}{99}
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\bibitem{Sven} Sven Sch\"{o}nherr. Quadratic Programming in Geometric Optimization:
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Theory, Implementation, and Applications, Dissertation, Diss. ETH No 14738, ETH
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Z\"{u}rich, Institute of Theoretical Computer Science, 2002.
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\bibitem{Chvatal} Va\v{s}ek Chv\'{a}tal. \textit{Linear Programming}. W. H. Freeman and Company,
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New York, Chapter 8, 1983
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\bibitem{Frans_Deg} Degeneracy
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\end{thebibliography}
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\end{document}
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