Stochastic convex programming: Singular multipliers and extended duality singular multipliers and duality

Abstract

A two-stage stochastic programming problem with recourse is studied here in terms of an extended Lagrangian function which allows certain multipliers to be elements of a dual space (ℒ∞)*, rather than an ℒ1 space. Such multipliers can be decomposed into an ℒ1–component and a “singular” component. The generalization makes it possible to characterize solutions to the problem in terms of a saddle-point, if the problem is strictly feasible. The Kuhn-Tucker conditions for the basic duality framework are modified to admit singular multipliers. It is shown that the optimal multiplier vectors in the extended dual problem are, in at least one broad case, ideal limits of maximizing sequences of multiplier vectors in the basic dual problem. © 1976 Pacific Journal of Mathematics. All rights reserved.