Linear programming sensitivity measured by the optimal value worst-case analysis

TL;DR

This paper introduces a new sensitivity measure for linear programming by defining a derivative of the optimal value function, which quantifies how the optimal value worsens under data perturbations, with practical computation methods and experiments.

Contribution

It proposes a novel derivative-based sensitivity measure for LP problems, including bounds and characterizations, especially addressing nondegenerate cases and highlighting open issues for degenerate cases.

Findings

Derivatives effectively measure LP sensitivity.

Numerical experiments confirm the usefulness of the sensitivity measure.

Open problems remain for handling degenerate LP problems.

Abstract

This paper introduces a concept of a derivative of the optimal value function in linear programming (LP). Basically, it is the the worst case optimal value of an interval LP problem when the nominal data the data are inflated to intervals according to given perturbation patterns. By definition, the derivative expresses how the optimal value can worsen when the data are subject to variation. In addition, it also gives a certain sensitivity measure or condition number of an LP problem. If the LP problem is nondegenerate, the derivatives are easy to calculate from the computed primal and dual optimal solutions. For degenerate problems, the computation is more difficult. We propose an upper bound and some kind of characterization, but there are many open problems remaining. We carried out numerical experiments with specific LP problems and with real LP data from Netlib repository. They show that the derivatives give a suitable sensitivity measure of LP problems. It remains an open problem how to efficiently and rigorously handle degenerate problems.