Parallelization and Aggregation of Nested Benders Decomposition

Dynamic 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 these problems, including nested Benders decomposition. In this method, recently shown to be superior to the alternatives for large problems, the problem is decomposed into a set of smaller linear programming problems. These problems can be visualised as being attached to the nodes of a tree which is formed from the realizations of the random data vectors determining the uncertainty in the problem. The tree is traversed forwards and backwards, with information from the solutions to each nodal linear programming problem being passed to its immediate descendants by the formation of their right hand sides and to its immediate ancestor in the form of cuts. Problems in the same time period can be solved independently and it is this inherent parallelism that is exploited in our parallel nested Benders algorithm. A parallel version of the MSLiP nested Benders code has been developed and tested on various types of MIMD machines. The differing structures of the test problems cause differing levels of speed-up. Results show that problems with few variables and constraints per node do not gain from this parallelization. Stage aggregation has been successfully explored 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.