Bounding the Expectation of a Saddle Function with Application to Stochastic Programming

Abstract

We previously obtained tight upper and lower bounds to the expectation of a saddle function of multivariate random variables using first and cross moments of the random variables without independence assumptions. These bounds are applicable when domains of the random vectors are compact sets in the euclidean space. In this paper, we extend the results to the case of unbounded domains, similar in spirit to the extensions by Birge and Wets in the pure convex case. The relationship of these bounds to a certain generalized moment problem is also investigated. Finally, for solving stochastic linear programs utilizing the above bounding procedures, a computationally more appealing order-cone decomposition scheme is proposed which behaves quadratically in the number of random variables. Moreover, the resulting upper and lower approximations are amenable to efficient solution techniques.

Robust Optimization with Ambiguous Stochastic Constraints Under Mean and Dispersion Information

“…Dupačová (1966) and Gassmann and Ziemba (1986) give convex upper bounds on the expectation of a convex function under first-moment conditions over a polyhedral support, based on the dual of the related moment problem. Birge and Wets (1987) and Edirisinghe and Ziemba (1994a) extend this approach to distributions with unbounded support. Dulá (1992) provides a bound for the expectation of a simplicial function of a random vector using first moments and the sum of all variances.…”

Multiperiod Portfolio Optimization with Terminal Liability: Bounds for the Convex Case

“…Other methods for solving large stochastic optimization problems are generally addressed within sampling based techniques or bound based techniques. See, for instance, Ermoliev and Wets [15], Ermoliev and Gairvoronsky [14], Higle and Sen [20], Dantzig and Glynn [6] for the former technique; Birge and Wets [3,4], Edirisinghe and Ziemba [11][12][13], Frauendorfer [16,17] for the latter technique. Noting that the problem to be solved in each stage involves a constraint recourse matrix that is random, many of the standard bounding techniques within stochastic programming are inapplicable in this case.…”

23. Refinancing Mortgages in Switzerland

“…To deal with the observed correlations between the risk factors, approximation schemes must take into account cross-moment information in a numerically efficient way. Such techniques have been developed, e.g., by Edirisinghe [6], Edirisinghe and Ziemba [8,9,10], and Frauendorfer [11,13,14]. See also [7] for a general survey on bound-based approximations.…”