Optimality of Affine Policies in Multistage Robust Optimization

Authors:
Dimitris Bertsimas;Dan A. Iancu;Pablo A. Parrilo
Affiliations:
Sloan School of Management and Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139;Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139;Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Venue:
Mathematics of Operations Research
Year:
2010

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Abstract

In this paper, we prove the optimality of disturbance-affine control policies in the context of one-dimensional, constrained, multistage robust optimization. Our results cover the finite-horizon case, with minimax (worst-case) objective, and convex state costs plus linear control costs. We develop a new proof methodology, which explores the relationship between the geometrical properties of the feasible set of solutions and the structure of the objective function. Apart from providing an elegant and conceptually simple proof technique, the approach also entails very fast algorithms for the case of piecewise-affine state costs, which we explore in connection with a classical inventory management application.