Stochastic Combinatorial Optimization with Controllable Risk Aversion Level
Due to their wide applicability and versatile modeling power, stochastic programming problems have received a lot of attention in many communities. In particular, there has been substantial recent interest in 2–stage stochastic combinatorial optimization problems. Two objectives have been considered in recent work: one sought to minimize the expected cost, and the other sought to minimize the worst–case cost. These two objectives represent two extremes in handling risk — the first trusts the average, and the second is obsessed with the worst case. In this paper, we interpolate between these two extremes by introducing an one–parameter family of functionals. These functionals arise naturally from a change of the underlying probability measure and incorporate an intuitive notion of risk. Although such a family has been used in the mathematical finance  and stochastic programming  literature before, its use in the context of approximation algorithms seems new. We show that under standard assumptions, our risk–adjusted objective can be efficiently treated by the Sample Average Approximation (SAA) method . In particular, our result generalizes a recent sampling theorem by Charikar et al. , and it shows that it is possible to incorporate some degree of robustness even when the underlying probability distribution can only be accessed in a black–box fashion. We also show that when combined with known techniques (e.g. [4,14]), our result yields new approximation algorithms for many 2–stage stochastic combinatorial optimization problems under the risk–adjusted setting.
KeywordsApproximation Algorithm Risk Measure Facility Location Problem Sampling Theorem Stochastic Programming Problem
Unable to display preview. Download preview PDF.
- 4.Dhamdhere, K., Goyal, V., Ravi, R., Singh, M.: How to Pay, Come What May: Approximation Algorithms for Demand–Robust Covering Problems. In: Proc. 46th FOCS, pp. 367–378 (2005)Google Scholar
- 6.Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted Sampling: Approximation Algorithms for Stochastic Optimization. In: Proc. 36th STOC, pp. 417–426 (2004)Google Scholar
- 7.Gupta, A., Ravi, R., Sinha, A.: An Edge in Time Saves Nine: LP Rounding Approximation Algorithms for Stochastic Network Design. In: Proc. 45th FOCS, pp. 218–227 (2004)Google Scholar
- 8.Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.: On the Costs and Benefits of Procrastination: Approximation Algorithms for Stochastic Combinatorial Optimization Problems. In: Proc. 15th SODA, pp. 691–700 (2004)Google Scholar
- 13.Ruszczyński, A., Shapiro, A.: Optimization of Risk Measures. In: Calafiore, G., Dabbene, F. (eds.) Probabilistic and Randomized Methods for Design under Uncertainty. Springer, Heidelberg (2005)Google Scholar
- 14.Shmoys, D.B., Swamy, C.: Stochastic Optimization is (Almost) as Easy as Deterministic Optimization. In: Proc. 45th FOCS, pp. 228–237 (2004)Google Scholar
- 15.Shmoys, D.B., Tardos, É., Aardal, K.I.: Approximation Algorithms for Facility Location Problems. In: Proc. 29th STOC, pp. 265–274 (1997)Google Scholar