A NEW RELAXATION SCHEME FOR MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS

作     者：Steffensen, Sonja Ulbrich, Michael 

作者机构：Rhein Westfal TH Aachen Fak Math Informat Naturwissensch Fachgrp Math Lehr & Forschungseinheit Kontinuierl D-52056 Aachen Germany Tech Univ Munich Chair Math Optimizat Fak Math D-85747 Munich Germany 

出 版 物：《SIAM JOURNAL ON OPTIMIZATION》 (工业与应用数学会最优化杂志)

年 卷 期：2010年第20卷第5期

页      面：2504-2539页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：DFG SPP [1253 HE5386/8-1] DAAD [D/08/11076] 

主　　题：mathematical program with equilibrium constraints (MPEC) relaxation complementarity constraint Clarke-stationarity Mordukhovich-stationarity strong stationarity constraint qualification nonlinear program 

摘      要：We present a new relaxation scheme for mathematical programs with equilibrium constraints (MPEC), where the complementarity constraints are replaced by a reformulation that is exact for the complementarity conditions corresponding to sufficiently nondegenerate complementarity components and relaxes only the remaining complementarity conditions. A positive parameter determines to what extent the complementarity conditions are relaxed. The relaxation scheme is such that a strongly stationary solution of the MPEC is also a solution of the relaxed problem if the relaxation parameter is chosen sufficiently small. We discuss the properties of the resulting parameterized nonlinear programs and compare stationary points and solutions. We further prove that a limit point of a sequence of stationary points of a sequence of relaxed problems is Clarke-stationary if it satisfies a so-called MPEC-constant rank constraint qualification, and it is Mordukhovich-stationary if it satisfies the MPEC-linear independence constraint qualification and the stationary points satisfy a second order sufficient condition. From this relaxation scheme, a numerical approach is derived that is applied to a comprehensive test set. The numerical results show that the approach combines good efficiency with high robustness.