Distributionally Robust Stochastic Programming with Binary Random Variables

In this paper, we consider stochastic programming with binary random variables, that is, when the random variable represents random subsets of a set. We consider a distributionally robust model, where the only knowledge about the distribution is the marginal probability of each element to appear in the random set. The objective is to minimize expected cost under the worst case distribution with these marginals. We show the problem can be solved efficiently when the cost function is convex in the decision variable and a) supermodular in the random variable, or b) has (or can be approximated by) a weakly “super-monotonic” cost sharing method. We also provide an approximation algorithm for general convex cost functions that depends on an approximate separating oracle. 1 Distributionally Robust Stochastic Programming (DRSP) Stochastic programming concerns with optimization under uncertain parameters. In general, a stochastic program can be expressed as follows: (SP) minx∈C ED[f(x, ξ)] (1) where x ∈ R is the decision variable within a convex set C, and ξ ∈ R (w.l.o.g., assumed to have the same dimensionality of x) is a random vector whose values Email: [email protected], Computer Science and Engineering, Stanford University, Stanford, CA 94305, USA. Email: [email protected]. Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, USA. Email: [email protected]. Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, USA. Email: [email protected]. Management Science and Engineering and, by courtesy, Electrical Engineering, Stanford, CA 94305, USA.