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EVPI-Based Importance Sampling Solution Procedures for Multistage Stochastic Linear Programmes on Parallel MIMD Architectures

By M. A. H. Dempster and R. T. Thompson

Abstract

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 vectors determining the uncertainty in the problem. Parallel versions of the nested Benders algorithm involve two obvious techniques for parallelising the associated tree structure for multiprocessors or multicomputers - subtree parallelisation or a nodal parallelisation - both of which utilise a farming approach. The nodal parallelisation technique is presented in this paper, as it balances load more efficiently than its alternative. 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. Stage aggregation has been successfully employed for 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 an importance sampling solution algorithm based on local EVPI information has been developed for extremely large multistage stochastic linear programmes 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.Linear programming, dynamic stochastic programming, nested Benders decomposition, parallel algorithms, aggregation, MIMD computers

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