A tighter variant of Jensen’s lower bound for stochastic programs and separable approximations to recourse functions

In this paper, we propose a new method to compute lower bounds on the optimal objective value of a stochastic program and show how this method can be used to construct separable approximations to the recourse functions. We show that our method yields tighter lower bounds than Jensen’s lower bound and it requires a reasonable amount of computational effort even for large problems. The fundamental idea behind our method is to relax certain constraints by associating dual multipliers with them. This yields a smaller stochastic program that is easier to solve. We particularly focus on the special case where we relax all but one of the constraints. In this case, the recourse functions of the smaller stochastic program are one dimensional functions. We use these one dimensional recourse functions to construct separable approximations to the original recourse functions. Computational experiments indicate that our lower bounds can significantly improve Jensen’s lower bound and our recourse function approximations can provide good solutions.