Relationships between Polyhedral Convex Sets and Generalized Polyhedral Convex Sets

TL;DR

This paper explores the relationship between polyhedral convex sets and their generalized counterparts, clarifies separation conditions, and introduces generalized set-valued mappings with calculus rules.

Contribution

It clarifies the separation conditions between PCS and GPCS and introduces generalized polyhedral set-valued mappings with differentiation rules.

Findings

Counterexample shows separation conditions for PCS do not extend to GPCS.

Introduces generalized polyhedral set-valued mappings.

Provides calculus rules for generalized differentiation.

Abstract

In this paper we study some relationships between polyhedral convex sets (PCS) and generalized polyhedral convex sets (GPCS). In particular, we clarify by a counterexample that the necessary and sufficient conditions for the separation of a convex set and a PCS obtained by Kung Fu Ng and Wen Song in [Fenchel duality in finite-dimensional setting and its applications, Nonlinear Anal. 55(2003), 845--858; Theorem~3.1] are no longer valid when considering GPCS instead of PCS. We also introduce and study the notions of generalized polyhedral set-valued mappings and optimal value functions generated by generalized polyhedral convex set-valued mappings along with their generalized differentiation calculus rules.