Tight bounds for stochastic convex programs

Variable and row aggregation as a technique of simplifying a mathematical program is utilized to develop bounds for two-stage stochastic convex programs with random right-hand sides. If one is able to utilize the problem structure along with only first moment information, a tighter bound than the usual mean model bound (based on Jensen’s inequality) may be obtained. Moreover, it is possible to construct examples for which the mean model bound will be arbitrarily poor. Consequently, one can tighten Jensen’s bound for stochastic programs when the distribution has a compact support. This bound may be improved further by partitioning the support using conditional first moments. With regard to first moment upper bounds, the Gassmann-Ziemba inequality is used for the stochastic convex program to seek a model which can be solved using standard convex programming techniques. Moreover, it allows one to easily construct upper bounds using the solution of the lower bounding problem. Finally, the results are extended to multistage stochastic convex programming problems.