Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials
Maksim V. Dolgopolik · arXiv (Cornell University) · 2019
This paper is devoted to an analysis of optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials. 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 this description we derive optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials based on a new constraint qualification. The main feature of these optimality conditions and constraint qualification is the fact that they depend on individual elements of quasidifferentials of constraints. To illustrate the theoretical results, we present two simple examples in which the optimality conditions obtained in this paper are not satisfied at a given point, while optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) fail to disqualify this point as nonoptimal.