A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

TL;DR

This paper introduces a new constraint qualification and optimality conditions for nonsmooth mathematical programming problems using quasidifferentials, which are more precise and less restrictive than previous methods, improving the detection of nonoptimal points.

Contribution

It develops a novel description of convex subcones of the contingent cone and derives the strongest optimality conditions based on quasidifferentials under minimal assumptions.

Findings

New constraint qualification depends on individual quasidifferential elements.

Optimality conditions are more sensitive and can detect nonoptimal points where subdifferential conditions fail.

Examples demonstrate the effectiveness of the new conditions in identifying nonoptimal solutions.

Abstract

The paper is devoted to an analysis of a new constraint qualification and a derivation of the strongest existing optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials under the minimal possible assumptions. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints with the use of a new constraint qualification. We utilize these description and constraint qualification to derive the strongest existing optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials under less restrictive assumptions than in previous studies. The main feature of the new constraint qualification and related optimality conditions is the fact that they depend on individual elements of quasidifferentials of the objective function and constraints and are not invariant with respect to the choise of quasidifferentials. To illustrate the theoretical results, we present two simple examples in which optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) are satisfied at a nonoptimal point, while the optimality conditions obtained in this paper do not hold true at this point, that is, optimality conditions in terms of quasidifferentials, unlike the ones in terms of subdifferentials, detect the nonoptimality of this point.