Constraint Qualifications for Nonsmooth Mathematical Programs with Equilibrium Constraints

Abstract

We study nonsmooth mathematical programs with equilibrium constraints. First we consider a general disjunctive program which embeds a large class of problems with equilibrium constraints. Then, we establish several constraint qualifications for these optimization problems. In particular, we generalize the Abadie and Guignard-type constraint qualifications. Subsequently, we specialize these results to mathematical program with equilibrium constraints. In our investigation, we show that a local minimum results in a so-called M-stationary point under a very weak constraint qualification.

On quasidifferentiable mathematical programs with equilibrium constraints

“…The notion of generalized convexity introduced by Singh and Laha [41] which unifies the functions introduced by Antczak [2] and by Nobakhtian [38] has been used to explore the criteria under which a 𝑊−stationary point becomes a global (or a local) minimizer of the QMPEC. Further, some other constraint qualifications and stationary conditions [16,18,22,36,44] under different generalized convexity assumptions [26,34,35,42] can be investigated for the QMPEC. Moreover, quasidifferentiable analaysis can be used to solve mathematical programs with vanishing constraints [29,30,33,37] involving quasidifferentiable functions.…”

“…A partial precise penalty for MPECs was developed by Liu et al [28]. Movahedian and Nobakhtian [36] studied nonsmooth MPECs. A number of CQs for MPECs were investigated by Guo and Lin [22].…”

Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order σ > 0

“…Flegel et al [12] considered optimization problems with a disjunctive structure of the feasible set and obtained optimality conditions for disjunctive programs with application to MPEC using Guignard-type constraint qualifications. Movahedian and Nobakhtian [33] introduced nonsmooth strong stationarity, M-stationarity and generalized Abadie and Guignard-type constraint qualifications for nonsmooth MPEC. Movahedian and Nobakhtian [34] introduced a nonsmooth type M-stationary condition based on the Michel-Penot subdifferential and established the Kuhn-Tucker, Fritz-John-type, M-stationary necessary conditions for the nonsmooth MPEC.…”

KKT Reformulation and Necessary Conditions for Optimality in Nonsmooth Bilevel Optimization

“…For results on mathematical programs with complementarity constraints (MPCCs) with smooth data, the reader is referred for example to [13,33]. Nonsmooth MPCCs were recently considered in [25,26] while using generalized differentiation tools by Clarke and Michel-Penot. Not only the model in the latter papers does not encompass our problem in (4.1) which contains both set-valued and complementarity constraints, but we rather provide the most natural extensions of the stationarity conditions of (4.1) from a completely different perspective.…”