On Directional Metric Subregularity and Second-Order Optimality Conditions for a Class of Nonsmooth Mathematical Programs

Abstract

We study infinite dimensional optimization problems where the constraint mapping is given as the sum of a smooth function and a generalized polyhedral multifunction, e.g. the normal cone mapping of a convex polyhedral set. By using advanced techniques of variational analysis we obtain first-order and second-order characterizations, both necessary and sufficient, for directional metric subregularity of the constraint mapping. These results are used to obtain second-order optimality conditions for the optimization problem.

On Some Generalized Polyhedral Convex Constructions

“…The theories of generalized linear programming and quadratic programming in [8,Sections 2.5.7 and 3.4.3] are mainly based on this concept of gpcs. Some applications of gpcs in Banach spaces can be found in the recent papers by Ban, Mordukhovich and Song [4], Ban and Song [5], Gfrerer [14,15]. Proposition 2.197 from [8] tells us that a nonempty gpcs in a locally convex Hausdorff topological vector space has nonempty relative interior.…”

New Sharp Necessary Optimality Conditions for Mathematical Programs with Equilibrium Constraints

“…For verifying the property of metric subregularity there are some sufficient conditions known, see e.g. [8,9,10,11,13]. In this paper polyhedrality will play an important role.…”

Directional Hölder Metric Regularity

“…When γ = 1, one refers to the (Lipschitz) directional metric regularity, as equivalently introduced by Gfrerer [32].…”

“…Later, Ioffe [31] has introduced and investigated an extension called relative metric regularity which covers many notions of metric regularity in the literature. In particular, another version of directional metric regularity/subregularity has been introduced and extensively studied by Gfrerer in [32,33] where some variational characterizations of this concept have been established and successfully applied to study optimality conditions for mathematical programs. In fact, this directional regularity property has been earlier used by Penot [34] to study second order optimality conditions.…”

“…In fact, this directional regularity property has been earlier used by Penot [34] to study second order optimality conditions. In the line of the directional version of metric subregularity considered by Gfrerer [32], Huynh, Nguyen and Tinh [35] have studied directional Hölder metric subregularity in order to investigate tangent cones to zero sets in degenerate cases.…”