Aggregation and discretization in multistage stochastic programming

Multistage stochastic programs have applications in many areas and support policy makers in finding rational decisions that hedge against unforeseen negative events. In order to ensure computational tractability, continuous-state stochastic programs are usually discretized; and frequently, the curse of dimensionality dictates that decision stages must be aggregated. In this article we construct two discrete, stage-aggregated stochastic programs which provide upper and lower bounds on the optimal value of the original problem. The approximate problems involve finitely many decisions and constraints, thus principally allowing for numerical solutio

Variants on Dantzig-Wolfe Decomposition with Applications to Multistage Problems

The initial representation of an LP problem to which the Dantzig-Wolfe decomposition procedure is applied, is of the essence. We study this here, and, in particular, we consider two transformations of the problem, by introducing suitable linking rows and variables. We study the application of the Dantzig-Wolfe procedure to these new representations of the original problem and the relationship to previously proposed algorithms. Advantages and disadvantages from a computational viewpoint are discussed. Finally we develop a decomposition algorithm based upon these ideas for solving multistage staircase-structured LP problems

Nonlinear Programming Techniques Applied to Stochastic Programs with Recourse

Stochastic convex programs with recourse can equivalently be formulated as nonlinear convex programming problems. These possess some rather marked characteristics. Firstly, the proportion of linear to nonlinear variables is often large and leads to a natural partition of the constraints and objective. Secondly, the objective function corresponding to the nonlinear variables can vary over a wide range of possibilities; under appropriate assumptions about the underlying stochastic program it could be, for example, a smooth function, a separable polyhedral function or a nonsmooth function whose values and gradients are very expensive to compute. Thirdly, the problems are often large-scale and linearly constrained with special structure in the constraints. This paper is a comprehensive study of solution methods for stochastic programs with recourse viewed from the above standpoint. We describe a number of promising algorithmic approaches that are derived from methods of nonlinear programming. The discussion is a fairly general one, but the solution of two classes of stochastic programs with recourse are of particular interest. The first corresponds to stochastic linear programs with simple recourse and stochastic right-hand-side elements with given discrete probability distribution. The second corresponds to stochastic linear programs with complete recourse and stochastic right-hand-side vectors defined by a limited number of scenarios, each with given probability. A repeated theme is the use of the MINOS code of Murtagh and Saunders as a basis for developing suitable implementations

Sampling Scenario Set Partition Dual Bounds for Multistage Stochastic Programs

We consider multistage stochastic programming problems in which the random parameters have finite support, leading to optimization over a finite scenario set. There has been recent interest in dual bounds for such problems, of two types. One, known as expected group subproblem objective (EGSO) bounds, require solution of a group subproblem, which optimizes over a subset of the scenarios, for all subsets of the scenario set that have a given cardinality. Increasing the subset cardinality in the group subproblem improves bound quality, (EGSO bounds form a hierarchy), but the number of group subproblems required to compute the bound increases very rapidly. Another is based on partitions of the scenario set into subsets. Combining the values of the group subproblems for all subsets in a partition yields a partition bound. In this paper, we consider partitions into subsets of (nearly) equal cardinality. We show that the expected value of the partition bound over all such partitions also forms a hierarchy. To make use of these bounds in practice, we propose random sampling of partitions and suggest two enhancements to the approach: Sampling partitions that align with the multistage scenario tree structure and use of an auxiliary optimization problem to discover new best bounds based on the values of group subproblems already computed. We establish the effectiveness of these ideas with computational experiments on benchmark problems. Finally, we give a heuristic to save computational effort by ceasing computation of a partition partway through if it appears unpromising.