The Price of Robustness

A robust approach to solving linear optimization problems with uncertain data has been 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 tradeoff 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. We thus 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. ∗Boeing Professor of Operations Research, Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, E53-363, Cambridge, MA 02139, [email protected]. The research of the author was partially supported by the Singapore-MIT alliance. †Operations Research Center, Massachusetts Institute of Technology, Cambridge, MA 02139, [email protected]. The research of the author is supported by a graduate scholarship from the National University of Singapore