On Robustness in Nonconvex Optimization with Application to Defense Planning
This work addresses the challenge of cost-effective, parameter-robust decision-making in nonconvex optimization, particularly for defense planning, and is incremental as it builds on existing robust optimization methods.
The paper tackles the problem of estimating the increase in minimum value for robust decisions in nonconvex optimization under parameter perturbations, using subgradients and local Lipschitz moduli derived from the nominal problem, with a median error of 12% across 54 cases in military operations research examples.
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.