The Price of Robustness

TLDR

Robust optimization for linear problems trades nominal optimality for feasibility under data uncertainty, but its conservatism can be excessive. This paper seeks to reduce the price of robustness by making the trade‑off more attractive. We adjust conservatism using probabilistic bounds on constraint violations, yielding a tractable linear formulation that extends to discrete problems. The resulting robust model remains linear and, as shown by numerical tests on portfolio, knapsack, and Net Lib problems, performs well.

Abstract

A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.