On Robustness in Nonconvex Optimization with Application to Defense Planning

TL;DR

This paper develops methods to estimate how much the optimal value in nonconvex optimization problems increases when decisions are made robust to parameter uncertainties, aiding in cost-effective decision-making.

Contribution

It introduces new expressions for subgradients and Lipschitz moduli that estimate value increases under robustness, requiring only the nominal problem solution.

Findings

Median error in estimates is 12% across 54 cases.

Expressions accurately inform about cost-effective robust decisions.

Applicable to mixed-integer nonconvex optimization models.

Abstract

In the context of structured nonconvex optimization, we estimate the increase in minimum value for a decision that is robust to parameter perturbations as compared to the value of a nominal problem. The estimates rely on detailed expressions for subgradients and local Lipschitz moduli of min-value functions in nonconvex robust optimization and require only the solution of the nominal problem. The theoretical results are illustrated by examples from military operations research involving mixed-integer optimization models. Across 54 cases examined, the median error in estimating the increase in minimum value is 12%. Therefore, the derived expressions for subgradients and local Lipschitz moduli may accurately inform analysts about the possibility of obtaining cost-effective, parameter-robust decisions in nonconvex optimization.