Quasiconvex programming with locally starshaped constraint region and applications to quasiconvex MPEC

In this article we prove an existence result, necessary and sufficient conditions for quasiconvex programming problem with a locally starshaped constraint region. Our optimality conditions are different from the usual optimality conditions, in that the subdifferential of the objective function is replaced by a normal cone operator. Such an optimality condition has advantage over the usual one, i.e. it becomes sufficient even when the objective function is only quasiconvex. As a special case we derive the corresponding results for the class of ‘Quasiconvex-quasiaffine’ MPEC which is a class of mathematical programs with complementarity constraints where the objective function is quasiconvex, the inequality constraint is quasiconvex and the rest of constraints are quasiaffine.

[1]  M. Avriel.  Subdifferential Properties of Quasiconvex and Pseudoconvex Functions : Unified Approach , 2003 .

[2]  M. Volle.  Duality for the level sum of quasiconvex functions and applications , 1998 .

[3]  Z. Naniewicz.  Hemivariational inequality approach to constrained problems for star-shaped admissible sets , 1994 .

[4]  Michal Kočvara,et al.  Nonsmooth approach to optimization problems with equilibrium constraints : theory, applications, and numerical results , 1998 .

[5]  Jane J. Ye,et al.  Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints , 2005 .

[6]  Boris S. Mordukhovich,et al.  Nonconvex differential calculus for infinite-dimensional multifunctions , 1996 .

[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]  Jane J. Ye,et al.  Multiplier Rules Under Mixed Assumptions of Differentiability and Lipschitz Continuity , 2000, SIAM J. Control. Optim..

[10]  Jane J. Ye,et al.  Exact Penalization and Necessary Optimality Conditions for Generalized Bilevel Programming Problems , 1997, SIAM J. Optim..

[11]  BORIS S. MORDUKHOVICH,et al.  Necessary Suboptimality and Optimality Conditions via Variational Principles , 2002, SIAM J. Control. Optim..

[12]  Didier Aussel,et al.  Adjusted Sublevel Sets, Normal Operator, and Quasi-convex Programming , 2005, SIAM J. Optim..

[13]  Jonathan M. Borwein,et al.  Necessary conditions for constrained optimization problems with semicontinuous and continuous data , 1998 .

[14]  B. Mordukhovich,et al.  Nonsmooth sequential analysis in Asplund spaces , 1996 .

[15]  Jane J. Ye,et al.  Nondifferentiable Multiplier Rules for Optimization and Bilevel Optimization Problems , 2004, SIAM J. Optim..

[16]  Boris S. Mordukhovich,et al.  Calculus of sequential normal compactness in variational analysis , 2003 .

[17]  Didier Aussel,et al.  Normal Cones to Sublevel Sets: An Axiomatic Approach , 2001 .

[18]  Daniel Goeleven,et al.  Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets , 1996, J. Glob. Optim..

[19]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[20]  Didier Aussel.  Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach , 1998 .

[21]  F. Giannessi.  Variational Analysis and Generalized Differentiation , 2006 .

[22]  J. Pang,et al.  Existence of optimal solutions to mathematical programs with equilibrium constraints , 1988 .

[23]  I. V. Konnov,et al.  On Quasimonotone Variational Inequalities , 1998 .