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Optimality Conditions for Piecewise-Convex Maximization Problems

Abstract : A function $F:\Rset^n\rightarrow \Rset$ is called a {\em piecewise convex}-fun- ction if it decomposes as $$ F(x)=\min\{f_j(x)\;\mid\; j \in M=\{1,2...,m\}\}, $$ where $f_j:\Rset^n\rightarrow \Rset$ is convex for all $j\in M$. Let $D$ be a nonempty, compact, and convex subset of $\Rset^n$, $T:\Rset^n\rightarrow \Rset^n$ be a continuous operator and $F:\Rset^n\rightarrow \Rset$ be piecewise convex, the purpose of this article is twofold: to extend necessary and sufficient optimality conditions for convexmaximization problem to piecewise convex maximization problem: \begin{eqnarray} &\mbox{ maximize }& F(x), \mbox{ subject to } x\in D\specialeqnum{PCMP} \end{eqnarray} to apply (PCMP) to a class of nonmonotone variational inequality problem, i.e. find a vector $z\in D$ such that \begin{eqnarray} &&\langle T(z),x-z\rangle\leq 0, \; \mbox{ for all } x\in D. \specialeqnum{VIP- (T,D)} \end{eqnarray} Both problems have many practical and theoretical applications. Solution for the latter has been extensively considered under monotonicity or pseudomonotonicity of the operator $-T(\cdot)$; here we further study the nonmonotone case.
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Submitted on : Wednesday, May 24, 2006 - 10:40:30 AM
Last modification on : Friday, February 4, 2022 - 3:10:14 AM
Long-term archiving on: : Sunday, April 4, 2010 - 11:19:20 PM

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  • HAL Id : inria-00072709, version 1

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Ider Tsevendorj. Optimality Conditions for Piecewise-Convex Maximization Problems. [Research Report] RR-3941, INRIA. 2000. ⟨inria-00072709⟩

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