Two Optimal Value Functions in Parametric Conic Linear Programming

TL;DR

This paper studies the sensitivity and differentiability of optimal value functions in parametric conic linear programming, providing new theoretical insights under strict feasibility conditions.

Contribution

It establishes local Lipschitz continuity and differentiability properties of the optimal value functions with respect to perturbations in the problem data.

Findings

Proves Lipschitz continuity of the optimal value function under right-hand-side changes.

Derives differentiability properties of the optimal value function under linear perturbations.

Provides conditions under which both primal and dual problems are strictly feasible for differentiability.

Abstract

We consider the conic linear program given by a closed convex cone in an Euclidean space and a matrix, where vector on the right-hand-side of the constraint system and the vector defining the objective function are subject to change. Using the strict feasibility condition, we prove the locally Lipschitz continuity and obtain some differentiability properties of the optimal value function of the problem under right-hand-side perturbations. For the optimal value function under linear perturbations of the objective function, similar differentiability properties are obtained under the assumption saying that both primal problem and dual problem are strictly feasible.