Convergent Bounds for Stochastic Programs with Expected Value Constraints

This article elaborates a bounding approximation scheme for convex multistage stochastic programs (MSP) that constrain the conditional expectation of some decision-dependent random variables. Expected value constraints of this type are useful for modelling a decision maker’s risk preferences, but they may also arise as artefacts of stage-aggregation. It is shown that the gap between certain upper and lower bounds on the optimal objective value can be made smaller than any prescribed tolerance. Moreover, the solutions of some tractable approximate MSP give rise to a policy which is feasible in the (untractable) original MSP, and this policy’s cost differs from the optimal cost at most by the difference between the bounds. The considered problem class comprises models with integrated chance constraints and conditional value-at-risk constraints. No relatively complete recourse is assumed.