SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Operations Research
Boosted sampling: approximation algorithms for stochastic optimization
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Adjustable robust solutions of uncertain linear programs
Mathematical Programming: Series A and B
Stochastic Optimization is (Almost) as easy as Deterministic Optimization
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
How to Pay, Come What May: Approximation Algorithms for Demand-Robust Covering Problems
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems
Mathematical Programming: Series A and B
Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems
Mathematical Programming: Series A and B
Stochastic programming approach to optimization under uncertainty
Mathematical Programming: Series A and B
Robust Combinatorial Optimization with Exponential Scenarios
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Optimality of Affine Policies in Multistage Robust Optimization
Mathematics of Operations Research
Mathematics of Operations Research
Theory and Applications of Robust Optimization
SIAM Review
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We consider a two-stage mixed integer stochastic optimization problem and show that a static robust solution is a good approximation to the fully adaptable two-stage solution for the stochastic problem under fairly general assumptions on the uncertainty set and the probability distribution. In particular, we show that if the right-hand side of the constraints is uncertain and belongs to a symmetric uncertainty set (such as hypercube, ellipsoid or norm ball) and the probability measure is also symmetric, then the cost of the optimal fixed solution to the corresponding robust problem is at most twice the optimal expected cost of the two-stage stochastic problem. Furthermore, we show that the bound is tight for symmetric uncertainty sets and can be arbitrarily large if the uncertainty set is not symmetric. We refer to the ratio of the optimal cost of the robust problem and the optimal cost of the two-stage stochastic problem as the stochasticity gap. We also extend the bound on the stochasticity gap for another class of uncertainty sets referred to as positive. If both the objective coefficients and right-hand side are uncertain, we show that the stochasticity gap can be arbitrarily large even if the uncertainty set and the probability measure are both symmetric. However, we prove that the adaptability gap (ratio of optimal cost of the robust problem and the optimal cost of a two-stage fully adaptable problem) is at most four even if both the objective coefficients and the right-hand side of the constraints are uncertain and belong to a symmetric uncertainty set. The bound holds for the class of positive uncertainty sets as well. Moreover, if the uncertainty set is a hypercube (special case of a symmetric set), the adaptability gap is one under an even more general model of uncertainty where the constraint coefficients are also uncertain.