On the lower semicontinuity and subdifferentiability of the value function for conic linear programming problems

TL;DR

This paper challenges a previous claim by providing a finite-dimensional counterexample where the value function of a linear programming problem is neither lower semicontinuous nor subdifferentiable, highlighting limitations in existing theoretical results.

Contribution

It demonstrates that the lower semicontinuity of the value function does not always hold, even in finite-dimensional spaces, and discusses related assumptions and assertions.

Findings

Counterexample in finite-dimensional space showing non-lower semicontinuity

Value function not subdifferentiable at any point in the domain

Restriction of the value function is unbounded near every domain point

Abstract

Lemma 1 from the paper [N.E. Gretsky, J.M. Ostroy, W.R. Zame, Subdifferentiability and the duality gap, Positivity 6: 261--274, 2002] asserts that the value function $v$ of an infinite dimensional linear programming problem in standard form is lower semicontinuous whenever $v$ is proper and the involved spaces are normed vector spaces. In this note one shows that this statement is false even in finite-dimensional spaces, one provides an example of linear programming problem in Hilbert spaces whose (proper) value function is not lower semicontinuous (hence it is not subdifferentiable) at any point in its domain, one shows that the restriction of the value function to its domain in Kretschmer's gap example is not bounded on any neighborhood of any point of the domain, and discuss other assertions done in the same paper.