The Abridged Nested Decomposition Method for Multistage Stochastic Linear Programs with Relatively Complete Recourse

This paper considers large-scale multistage stochastic linear programs. Sampling is incorporated into the nested decomposition algorithm in a manner which proves to be significantly more efficient than a previous approach. The main advantage of the method arises from maintaining a restricted set of solutions that substantially reduces computation time in each stage of the procedure. Dedicated to the memory of George Dantzig who inspired us to pursue the challenge of finding optimal decisions under uncertainty. 1. Multistage Stochastic Linear Programs C onsider a multistage dynamic decision process under uncertainty that can be modeled as a linear program. Assume that the stochastic elements of the problem can be found in the objective coefficients, the right-hand side values, the technology matrices, or any combination of these. Further, assume that the stochastic elements are defined over a discrete probability space (Ξ, σ(Ξ),P), where Ξ = Ξ ⊗ · · · ⊗ Ξ is the support of the random data in stages two through N , with Ξ = {ξ i = (h (ξ i ), c (ξ i ), T t−1 ·,1 (ξ t i ), . . . , T t−1 ·,n (ξ i ))}. The stage t nodes of the scenario tree are defined by a realization ξ i of the stage t random parameters and a history of realizations (ξ i , . . . , ξ t−1 i ) of the random parameters through stage t− 1. A multistage stochastic linear programming problem can then be formulated (Table 1).