Expected value of perfect information-based importance sampling solution procedures for multistage stochastic linear programs on parallel MIMD architectures

Multistage stochastic linear programming has many practical applications for problems whose current decisions have to be made under future uncertainty. There are a variety of methods for solving the deterministic equivalent forms of these dynamic problems, including the simplex and interior-point methods and nested Benders decomposition, which decomposes the original problem into a set of smaller linear programming problems and has recently been shown to be superior to the alternatives for large problems. The Benders subproblems can be visualised as being attached to the nodes of a tree which is formed from the realisations of the random data process determining the uncertainty in the problem. This paper describes a parallel implementation of the nested Benders algorithm which employs a farming technique to parallelize nodal subproblem solutions. Differing structures of the test problems cause differing levels of speed-up on a variety of multicomputing platforms: problems with few variables and constraints per node do not gain from this parallelisation. We therefore employ stage aggregation to such problems to improve their parallel solution efficiency by increasing the size of the nodes and therefore the time spent calculating relative to the time spent communicating between processors. A parallel version of a sequential importance sampling solution algorithm based on local expected value of perfect information (EVPI) is developed which is applicable to extremely large multistage stochastic linear programs which either have too many data paths to solve directly or a continuous distribution of possible realisations. It utilises the parallel nested Benders algorithm and a parallel version of an algorithm designed to calculate the local EVPI values for the nodes of the tree and achieves near linear speed-up.