Parallelization and aggregation ofnested Benders decomposition

Dynamic multistage stochastic linear programming has many practical applications forproblems whose current decisions have to be made under future uncertainty. There are avariety of methods for solving these problems including nested Benders decomposition. Inthis method, recently shown to be superior to the alternatives for large problems, the problemis decomposed into a set of smaller linear programming problems. These problems can bevisualised as being attached to the nodes of a tree which is formed from the realizations ofthe random data vectors determining the uncertainty in the problem. The tree is traversedforwards and backwards, with information from the solutions to each nodal linear programmingproblem being passed to its immediate descendants by the formation of their righthand sides and to its immediate ancestor in the form of cuts. Problems in the same timeperiod can be solved independently and it is this inherent parallelism that is exploited inour parallel nested Benders algorithm. A parallel version of the MSLiP nested Benders codehas been developed and tested on various types of MIMD machines. The differing structuresof the test problems cause differing levels of speed-up. Results show that problems withfew variables and constraints per node do not gain from this parallelization. Stage aggregationhas been successfully exploited for such problems to improve their parallel solutionefficiency by increasing the size of the nodes and therefore the time spent calculating relativeto the time spent communicating between processors.