Bounding separable recourse functions with limited distribution information

Abstract

The recourse function in a stochastic program with recourse can be approximated by separable functions of the original random variables or linear transformations of them. The resulting bound then involves summing simple integrals. These integrals may themselves be difficult to compute or may require more information about the random variables than is available. In this paper, we show that a special class of functions has an easily computable bound that achieves the best upper bound when only first and second moment constraints are available.

Second-Order Lower Bounds on the Expectation of a Convex Function

“…Sometimes only two points are used in an optimal solution. When f is continuously differentiable and df /du is strictly convex on a b , an application of Kreȋn and Nudel'man [31, Chapter 4, Theorem 1.1] implies that a unique two-point support solves (1). One of these points is a and the other point and weights are determined by (2), (3), and b a dP = 1.…”

“…One of these points is a and the other point and weights are determined by (2), (3), and b a dP = 1. Birge and Dulá [1] develop a more general condition that is sufficient to ensure the twopoint property when the objective in (1) is to maximize E P f . Under their condition, the two points may be interior to a b and a line search is required to find the points and their weights.…”

“…This bound is illustrated in Figure 4. Table 1 provides the bounds of Jensen [26], Edirisinghe [12], L * , and the optimal value of the GMP defined in (1). As the table indicates, the secondorder bounds, EB and L * , are significantly stronger than the first-order Jensen bound as f becomes "more convex."…”

“…Dupačová [10] incorporates a variance constraint in a GMP for finding an upper bound. Birge and Dulá [1], Dulá [8], Dulá and Murthy [9], and Kall [28] also use second-moment information in computing upper bounds.…”

Barycentric Bounds in Stochastic Programming: Theory and Application

“…Edirisinghe [17] was the first to develop second order lower bounds, which were later extended by Dokov and Morton [12]. Birge and Dulá [4], Dupačová [16], and Kall [42] propose second order upper bounds. Higher order upper bounds are suggested in [11].…”