Optimization and equilibrium problems with equilibrium constraints

This paper concerns optimization and equilibrium problems with the so-called equilibrium constraints mathematical programs with equilibrium constraint (MPEC) and equilibrium problems with equilibrium constraint (EPEC), which frequently appear in applications to operations research. These classes of problems can be naturally unified in the framework of multiobjective optimization with constraints governed by parametric variational systems (generalized equations, variational inequalities, complementarity problems, etc.). We focus on necessary conditions for optimal solutions to MPECs and EPECs under general assumptions in finite-dimensional spaces. Since such problems are intrinsically nonsmooth, we use advanced tools of generalized differentiation to study optimal solutions by methods of modern variational analysis. The general results obtained are concretized for special classes of MPECs and EPECs important in applications.

[1]  B. Morduhovic.  Calculus of second-order subdifferentials in infinite dimensions , 2002 .

[2]  B. Mordukhovich.  Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings , 1994 .

[3]  Boris S. Mordukhovich.  Coderivative Analysis of Variational Systems , 2004, J. Glob. Optim..

[4]  Boris S. Mordukhovich,et al.  Necessary Conditions in Nonsmooth Minimization via Lower and Upper Subgradients , 2004 .

[5]  Qiji J. Zhu,et al.  Multiobjective optimization problem with variational inequality constraints , 2003, Math. Program..

[6]  Boris S. Mordukhovich,et al.  An Extended Extremal Principle with Applications to Multiobjective Optimization , 2003, SIAM J. Optim..

[7]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[8]  Boris S. Mordukhovich,et al.  The extremal principle and its applications to optimization and economics , 2001 .

[9]  Stefan Scholtes,et al.  Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity , 2000, Math. Oper. Res..

[10]  S Scholtes,et al.  Mathematical programs with equilibrium constraints: stationarity, optimality, and sensitivity , 1997 .

[11]  Barney Glover,et al.  Optimization and related topics , 2001 .

[12]  B. Mordukhovich.  Maximum principle in the problem of time optimal response with nonsmooth constraints PMM vol. 40, n≗ 6, 1976, pp. 1014-1023 , 1976 .

[13]  Michal Kočvara,et al.  Optimization problems with equilibrium constraints and their numerical solution , 2004, Math. Program..

[14]  Boris S. Mordukhovich,et al.  On Second-Order Subdifferentials and Their Applications , 2001, SIAM J. Optim..

[15]  B. Mordukhovich.  Variational analysis and generalized differentiation , 2006 .