On the Optimality of Affine Policies for Budgeted Uncertainty Sets

In this paper, we study the performance of affine policies for two-stage adjustable robust optimization problem with fixed recourse and uncertain right hand side belonging to a budgeted uncertainty set. This is an important class of uncertainty sets widely used in practice where 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 $\Omega \left( \frac{\log n}{\log \log n} \right)$ even for budget of uncertainty sets and fixed non-negative recourse where $n$ is the number of decision variables. Affine policies, where 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 $O\left( \frac{\log n}{\log \log n} \right)$-approximation for the two-stage adjustable robust problem with fixed non-negative 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 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 near-optimal affine solution that is significantly faster than computing the optimal affine policy by solving a large linear program.