#### Document Type

Technical Report

#### Abstract

In this paper we study mathematical programs with equilibrium constraints (MPECs) described by generalized equations in the extended form 0 is an element of the set G(x,y) + Q(x,y), where both mappings G and Q are set-valued. Such models arise, in particular, from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish new weak and strong suboptimality conditions for the general MPEC problems under consideration in finite-dimensional and infinite-dimensional spaces that do not assume the existence of optimal solutions. This issue is particularly important for infinite-dimensional optimization problems, where the existence of optimal solutions requires quite restrictive assumptions. Our techriiques are mainly based on modern tools of variational analysis and generalized differentiation revolving around the fundamental extremal principle in variational analysis and its analytic counterpart known as the subdifferential variational principle.

#### Number in Series

2007.07

#### Disciplines

Applied Mathematics | Mathematics

#### Recommended Citation

Bao, Truong Q.; Gupta, Panjak; and Mordukhovich, Boris S., "Suboptimality Conditions for Mathematical Programs with Equilibrium Constraints" (2007). *Mathematics Research Reports*. 51.

http://digitalcommons.wayne.edu/math_reports/51

## Comments

Dedicated to Phan Quoc Khanh. This research was partly supported by the US National Science Foundation under grants DMS-0304989 and DMS-0603846, and partly by the US National Science Foundation under grants DMS-0304989 and DMS-0603846 and the Australian Research Council under grant DP-04511668.