Conditional value-at-risk in stochastic programs with mixed-integer recourse

Abstract

In classical two-stage stochastic programming the expected value of the total costs is minimized. Recently, mean-risk models -- studied in mathematical finance for several decades -- have attracted attention in stochastic programming. We consider Conditional Value-at-Risk as risk measures in the framework of two-stage stochastic integer programming. The paper addresses structure, stability, and algorithms for this class of models. In particular, we study continuity properties of the objective function, both with respect to the first-stage decisions and the integrating probability measure. Further, we present an explicit mixed-integer linear programming formulation of the problem when the probability distribution is discrete and finite. Finally, a solution algorithm based on Lagrangean relaxation of nonanticipativity is proposed.