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\title{Test\_suite\_QP\_solver}
\author{Frans Wessendorp}
\begin{document}
\maketitle
\section{Validity check}
All of the validity checks of a solution computed by the solver are done using
the complete set of constraints; since the solver itself works with an active
set method and therefore uses the reduced basis matrix $\check{M}_{B}$
and its inverse we here restate the relationship given in \cite{Frans_Deg}.
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}
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}
and the inverse $M_{B}^{-1}$ in terms of the reduced basis inverse
$\check{M}_{B}^{-1}$
\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}
\subsection{Verifying Feasibility}
\subsection{Verifying Optimality}
\subsection{Verifying Unboundedness}
Since the solver delivers, in case of unboundedness, implicitly a feasible
solution $x^{*}$ and a vector
$w$ such that
\begin{equation}
\label{eq:Unboundedness}
x^{*}-tw \quad \text{for} \quad t>0
\end{equation}
is a feasible solution
we by convention define the single nonzero nonbasic component of the basic
feasible direction $w$ to be negative:
\begin{eqnarray}
\label{def:w_B}
w_{B}&:=&q_{B}=\left(M_{B}^{-1}\right)_{B_{O} \cup B_{S}, \bullet}
\left(
\begin{array}{c}
A_{C,j} \\
\hline
A_{S_{B}, j} \\
\hline
2D_{B_{O}, j} \\
\hline
2D_{B_{S}, j}
\end{array}
\right) \\
\label{def:w_N}
w_{N}&:=&-e_{\{j\}}
\end{eqnarray}
where $j \in N$ and $e_{\{j\}}$ denotes the unit vector with $\left|N\right|$
entries.
Feasibility of the solution in Equation~(\ref{eq:Unboundedness}) requires
$w <0$ and $Aw=0$. That the latter is true for $w$ defined by
Equations~(\ref{def:w_B}) and~(\ref{def:w_N}) shows the following computation
where we use the
Definitions~(\ref{def:basis_matrix}) and~(\ref{eq:M_B_inv_exp})
\begin{eqnarray}
Aw
&=&
A_{C \cup S_{B}, B_{O} \cup B_{S}}
\left(M_{B}^{-1}\right)_{B_{O} \cup B_{S}, \bullet}
\left(
\begin{array}{c}
A_{C,j} \\
\hline
A_{S_{B}, j} \\
\hline
2D_{B_{O}, j} \\
\hline
0
\end{array}
\right)
-A_{C \cup S_{B}, j}
\nonumber \\
&=&
\left(M_{B}\right)_{C \cup S_{B}, B_{O} \cup B_{S}}
\left(M_{B}^{-1}\right)_{B_{O} \cup B_{S}, \bullet}
\left(
\begin{array}{c}
A_{C,j} \\
\hline
A_{S_{B}, j} \\
\hline
2D_{B_{O}, j} \\
\hline
0
\end{array}
\right)
-A_{C \cup S_{B}, j}
\nonumber \\
&=&
\left(M_{B}\right)_{C \cup S_{B}, \bullet}
M_{B}^{-1}
\left(
\begin{array}{c}
A_{C,j} \\
\hline
A_{S_{B}, j} \\
\hline
2D_{B_{O}, j} \\
\hline
0
\end{array}
\right)
-A_{C \cup S_{B}, j}
\nonumber \\
&=&
\left[
I_{C\cup S_{B},C\cup S_{B}} \left|\right.
\mathbf{0}_{C\cup S_{B},B_{O}\cup B_{S}}
\right]
\left(
\begin{array}{c}
A_{C,j} \\
\hline
A_{S_{B}, j} \\
\hline
2D_{B_{O}, j} \\
\hline
0
\end{array}
\right)
-A_{C \cup S_{B}, j}
\nonumber \\
&=&
0
\end{eqnarray}
\subsubsection{Linear Case}
We are minimizing the objective function
\begin{equation}
f\left(x\right):=c^{T}x,
\end{equation}
so for $t>0$
\begin{eqnarray}
f(x^{*}-tw)
&=&
c^{T}\left(x^{*}-tw\right)
\nonumber \\
&=&
f\left(x^{*}\right) -tc^{T}w
\end{eqnarray}
Since we are minimizing we must require $c^{T}w>0$.
Subsuming we obtain together with the above the necessary conditions for
unboundedness.
\begin{enumerate}
\item $w_{x_{i}} \leq 0$ for $i \in B_{O} \cup B_{S}$
\item $Aw=0$
\item $c^{T}w>0$
\end{enumerate}
\subsubsection{Quadratic Case}
We are minimizing the objective function
\begin{equation}
f\left(x\right):=c^{T}x+x^{T}Dx,
\end{equation}
using the symmetry of $D$ we obtain for $t>0$
\begin{eqnarray}
f(x^{*}-tw)
&=&
c^{T}\left(x^{*}-tw\right)
+\left(x^{*}-tw\right)^{T}D\left(x^{*}-tw\right)
\nonumber \\
&=&
c^{T}x^{*} - tc^{T}w + {x^{*}}^{T}Dx^{*} - tw^{T}Dx^{*}-t{x^{*}}^{T}Dw
+t^{2}w^{T}Dw
\nonumber \\
&=&
c^{T}x^{*} + {x^{*}}^{T}Dx^{*} + t^{2}w^{T}Dw
-t\left[\left(c^{T}+2{x^{*}}^{T}D\right)w\right]
\nonumber \\
&=&
f\left(x^{*}\right) + t^{2}w^{T}Dw
-t\left[\left(c^{T}+2{x^{*}}^{T}D\right)w\right]
\end{eqnarray}
Since $D$ is positive semidefinite $w^{T}Dw \geq 0$ and since we are minimizing
we must require $w^{T}Dw=0$ and $\left(c^{T}+2{x^{*}}^{T}D\right)w>0$.
Subsuming we obtain together with the above the necessary conditions for
unboundedness.
\begin{enumerate}
\item $w_{x_{i}} \leq 0$ for $i \in B_{O} \cup B_{S}$
\item $Aw=0$
\item $w^{T}Dw = 0$
\item $\left(c^{T}+2{x^{*}}^{T}D\right)w>0$
\end{enumerate}
\begin{thebibliography}{99}
\bibitem{Sven} Sven 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
\bibitem{Frans_Deg} Degeneracy
\end{thebibliography}
\end{document}