Robust Sensitivity Analysis of the Optimal Value of Linear Programming

TL;DR

This paper introduces a comprehensive framework for sensitivity analysis of linear programs under convex uncertainty, providing new bounds and relaxations that improve understanding of optimal value robustness.

Contribution

It unifies existing sensitivity analysis methods, extends them to convex uncertainty sets, and develops tight relaxations for better bounds on worst- and best-case LP values.

Findings

Bounds on p- and p+ are very strong in practice.

The relaxations are computationally tractable.

The approach generalizes and improves upon previous methods.

Abstract

We propose a framework for sensitivity analysis of linear programs (LPs) in minimization form, allowing for simultaneous perturbations in the objective coefficients and right-hand sides, where the perturbations are modeled in a compact, convex uncertainty set. This framework unifies and extends multiple approaches for LP sensitivity analysis in the literature and has close ties to worst-case linear optimization and two-stage adaptive optimization. We define the minimum (best-case) and maximum (worst-case) LP optimal values, p- and p+, over the uncertainty set, and we discuss issues of finiteness, attainability, and computational complexity. While p- and p+ are difficult to compute in general, we prove that they equal the optimal values of two separate, but related, copositive programs. We then develop tight, tractable conic relaxations to provide lower and upper bounds on p- and p+, respectively. We also develop techniques to assess the quality of the bounds, and we validate our approach computationally on several examples from--and inspired by--the literature. We find that the bounds on p- and p+ are very strong in practice and, in particular, are at least as strong as known results for specific cases from the literature.