New Second-Order Bounds on the Expectation of Saddle Functions with Applications to Stochastic Linear Programming

This paper develops new bounds on the expectation of a convex-concave saddle function of a random vector with compact domains. The bounds are determined by replacing the underlying distribution by unique discrete distributions, constructed using second-order moment information. The results extend directly to new second moment lower bounds in closed-form for the expectation of a convex function. These lower bounds are better than Jensen's bound, the only previously known lower bound for the convex case, under limited moment information. Application of the second moment bounds to two-stage stochastic linear programming is reported. Computational experiments, using randomly generated stochastic programs, indicate that the new bounds may easily outperform the usual first-order bounds.