Strong duality and sensitivity analysis in semi-infinite linear programming

TL;DR

This paper investigates conditions under which strong duality and dual pricing properties hold in semi-infinite linear programming, extending the feasible constraint space while addressing the limitations of finite-dimensional duality theory.

Contribution

It characterizes the largest constraint space where strong duality and dual pricing hold in semi-infinite linear programs, using advanced functional analysis and an extended Fourier-Motzkin elimination.

Findings

Strong duality and dual pricing hold in a restricted constraint space.

Extension of the constraint space preserves duality properties under certain conditions.

Use of singular and finitely additive linear functionals is key to the analysis.

Abstract

Finite-dimensional linear programs satisfy strong duality (SD) and have the "dual pricing" (DP) property. The (DP) property ensures that, given a sufficiently small perturbation of the right-hand-side vector, there exists a dual solution that correctly "prices" the perturbation by computing the exact change in the optimal objective function value. These properties may fail in semi-infinite linear programming where the constraint vector space is infinite dimensional. Unlike the finite-dimensional case, in semi-infinite linear programs the constraint vector space is a modeling choice. We show that, for a sufficiently restricted vector space, both (SD) and (DP) always hold, at the cost of restricting the perturbations to that space. The main goal of the paper is to extend this restricted space to the largest possible constraint space where (SD) and (DP) hold. Once (SD) or (DP) fail for a given constraint space, then these conditions fail for all larger constraint spaces. We give sufficient conditions for when (SD) and (DP) hold in an extended constraint space. Our results require the use of linear functionals that are singular or purely finitely additive and thus not representable as finite support vectors. The key to understanding these linear functionals is the extension of the Fourier-Motzkin elimination procedure to semi-infinite linear programs.