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 [11] and stochastic programming [13] 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 [9]. In particular, our result generalizes a recent sampling theorem by Charikar et al. [2], 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.


Approximation Algorithm Risk Measure Facility Location Problem Sampling Theorem Stochastic Programming Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Anthony Man–Cho So
    • 1
  • Jiawei Zhang
    • 2
  • Yinyu Ye
    • 3
  1. 1.Department of Computer ScienceStanford UniversityStanfordUSA
  2. 2.Department of Information, Operations, and Management Sciences, Stern School of BusinessNew York UniversityNew YorkUSA
  3. 3.Department of Management Science and Engineering and, by courtesy, Electrical EngineeringStanford UniversityStanfordUSA

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