A Lagrangian Finite Generation Technique for Solving Linear - Quadratic Problems in Stochastic Programming

A new method is proposed for solving two-stage problems in iinear and quadratic stochastic programmjng. Such problems are dualized, and the dual, elthouSht itself of high dimension, is approximated by a sequence of quadratic programming subproblems whose djmensionality can be kept low. These subproblems correspond to rnaximizing the dual objective over the convex hull of finitely many dual feasible solutions. An optimizing sequence is produced for the primal problem that converges at a iinear rate in the strongly quadratic case. An outer algorithm of augmented Lagrangian type can be used to introduce strongly quadratic terms, if des;red. In the recourse model in stochastic programming, a vector n must be chosen optimally with respect to present costs and constraints as well as certain expected costs and induced constraints that are associated with corrective actions available in the future. Such actions may be taken in response to the obseryation of the values of various random variables about which there is only statistical information at the time r. is selected. The actions involve costs and constraints that depend on these observed values and on r. The theory ol this kind of stochastic programming and the numerical methods that have been proposed for it has been surveyed recently by Wets [12]. We aim here at developing a new solution procedure for the case where the first and second stage problems in the recourse model fit the mold of linear or quadratic (convex) programming. We assume for simplicity that the random variables are discretely distributed with only flnitely many values. This restriction is not fully necessary in theory, but it reflects the realities of computation and a natural division among the questions that arise. Every continuous distribution must in practice be replaced by a nnite discrete one, whether empirically, or through sampling, mathematical approximation, or in connection with the numerical calculation of integrals expressing expectations. The efiects of such discretization raise important questions