On the Optimality of Affine Policies for Budgeted Uncertainty Sets

Omar El Housni () and Vineet Goyal ()
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Omar El Housni: Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027
Vineet Goyal: Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027

Mathematics of Operations Research, 2021, vol. 46, issue 2, 674-711

Abstract: In this paper, we study the performance of affine policies for a two-stage, adjustable, robust optimization problem with a fixed recourse and an uncertain right-hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets, widely used in practice, in which we can specify a budget on the adversarial deviations of the uncertain parameters from the nominal values to adjust the level of conservatism. The two-stage adjustable robust optimization problem is hard to approximate within a factor better than Ω ( log n log log n ) even for budget of uncertainty sets in which n is the number of decision variables. Affine policies, in which the second-stage decisions are constrained to be an affine function of the uncertain parameters provide a tractable approximation for the problem and have been observed to exhibit good empirical performance. We show that affine policies give an O ( log n log log n ) -approximation for the two-stage, adjustable, robust problem with fixed nonnegative recourse for budgeted uncertainty sets. This matches the hardness of approximation, and therefore, surprisingly, affine policies provide an optimal approximation for the problem (up to a constant factor). We also show strong theoretical performance bounds for affine policy for a significantly more general class of intersection of budgeted sets, including disjoint constrained budgeted sets, permutation invariant sets, and general intersection of budgeted sets. Our analysis relies on showing the existence of a near-optimal, feasible affine policy that satisfies certain nice structural properties. Based on these structural properties, we also present an alternate algorithm to compute a near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.

Keywords: Primary: 90C15; 90C47; Primary: robust optimization; robust optimization; affine policies; budget of uncertainty (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (7)

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