Inequalities for Stochastic Nonlinear Programming Problems

Many actual situations can be represented in a realistic manner by the two-stage stochastic nonlinear programming problem Min x Emin y [φ(x) + ψ(y)] subject to g(x) + h(y) ≧ b, where b is a random vector with a known distribution, and E denotes expectation taken with respect to the distribution of b. Madansky has obtained upper and lower bounds on the optimum solution to this two-stage problem for the completely linear case. In the present paper these results are extended, under appropriate convexity, concavity, and continuity conditions, to the two-stage nonlinear problem. In many cases of practical interest the calculation of these bounds will require only slightly more effort than two solutions of a deterministic problem of the same size, that is, a problem with a known constant value for the vector b. A small nonlinear numerical example illustrates the calculation of these bounds. For this example the bounds closely bracket the optimum solution to the two-stage problem.