vpswindows22（VPSWINDOWS）

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\title{\bf On the regularity of the dual solution to a linear program}

\author{Amit Ghosh and Sanjoy Mitter}

\date{}

\begin{document}

\maketitle

\begin{abstract}

We study the regularity properties of the dual solution to a linear program. We present two new results. Our first result is a sufficient condition for the dual solution to be regular in terms of the structure of the corresponding primal Slater point. This result is used to develop a sufficient condition for the dual solution to be regular when the primal solution is regular. The second result is a consequence of the first and it is an upper bound on the number of regions of regularity of the dual solution in terms of the number of regions of regularity of the primal solution.

\end{abstract}

\section{Introduction}

We study the regularity properties of the dual solution to a linear program. We consider the following primal and dual linear programs:

{\it Primal Problem:}

\begin{align}\label{e:primal}

\min \quad & c^T x \\

\mbox{s.t.} \quad & A x \geq b, \nonumber

\end{align}

{\it Dual Problem:}

\begin{align}\label{e:dual}

\max \quad & b^T y\\

\mbox{s.t.} \quad & A^T y \leq c. \nonumber

\end{align}

Let $(x^*,y^*)$ be a pair of primal and dual

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(\bibinfo{year}{2009}).

\bibitem[{\citenamefont{Konig et~al.}(2010)\citenamefont{Konig, Laemmlin,

G{\"o}tze, Schwarz, Thomas, and B{\"o}hme}}]{Konig2010}

\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Konig}},

\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Laemmlin}},

\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{G{\"o}tze}},

\bibinfo{author}{\bibfnamefont{U.~D.} \bibnamefont{Schwarz}},

\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Thomas}}, \bibnamefont{and}

\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{B{\"o}hme}},

\bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{104}},

\bibinfo{pages}{145701} (\bibinfo{year}{2010}).

\bibitem[{\citenamefont{Konig et~al.}(2011)\citenamefont{Konig, G{\"o}tze,

Schwarz, Laemmlin, Thomas, and B{\"o}hme}}]{Konig2011}

\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Konig}},

\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{G{\"o}tze}},

\bibinfo{author}{\bibfnamefont{U.~D.} \bibnamefont{Schwarz}},

\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Laemmlin}},

\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Thomas}}, \bibnamefont{and}

\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{B{\"o}hme}},

\bibinfo{journal}{New J. Phys.} \textbf{\bibinfo{volume}{13}},

\bibinfo{pages}{033021} (\bibinfo{year}{2011}).

\bibitem[{\citenamefont{Boehme}(2011)}]{Boehme2010}

\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Boehme}},

\bibinfo{journal}{Nature Mater.} \textbf{\bibinfo{volume}{10}},

\bibinfo{pages}{459} (\bibinfo{year}{2011}).

\bibitem[{\citenamefont{Konig et~al.}(2012)\citenamefont{