Approximation Algorithms for 2-Stage Stochastic Optimization Problems

Abstract. Stochastic optimization is a leading approach to model optimization problems in which there is uncertainty in the input data, whether from measurement noise or an inability to know the future. In this survey, we outline some recent progress in the design of polynomialtime algorithms with performance guarantees on the quality of the solutions found for an important class of stochastic programming problems — 2-stage problems with recourse. In particular, we show that for a number of concrete problems, algorithmic approaches that have been applied for their deterministic analogues are also effective in this more challenging domain. More specifically, this work highlights the role of tools from linear programming, rounding techniques, primal-dual algorithms, and the role of randomization more generally.

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 mathematical finance [12] and stochastic programming [14] 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 [10]. 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 we are only allowed to access the underlying probability distribution in a black–box fashion. We also show that when combined with known techniques (e.g. [4, 15]), our result yields new approximation algorithms for many 2–stage stochastic combinatorial optimization problems under the risk–adjusted setting.