A new decomposition method for multistage stochastic linear programming problems is proposed. A multistage stochastic problem is represented in a tree-like form and with each node of the decision tree a certain linear or quadratic subproblem is associated. The subproblems generate proposals for their successors and some backward information for their predecessors. The subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. An analytical illustrative example shows that parallelization can speed up computation over every sequential method. Computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors.

/pdf/parallel-decomposition-of-multistage-stochastic-programming-28qv6alh44.pdf

Parallel decomposition of multistage stochastic programming problems

Multistage stochastic linear programming has many practical applications for problemswhose current decisions have to be made under future uncertainty. There are a variety ofmethods for solving the deterministic equivalent forms of these dynamic problems, includingthe simplex and interior‐point methods and nested Benders decomposition, which decomposesthe original problem into a set of smaller linear programming problems and hasrecently been shown to be superior to the alternatives for large problems. The Benderssubproblems can be visualised as being attached to the nodes of a tree which is formed fromthe realisations of the random data process determining the uncertainty in the problem. Thispaper describes a parallel implementation of the nested Benders algorithm which employsa farming technique to parallelize nodal subproblem solutions. Differing structures of thetest 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 solutionefficiency by increasing the size of the nodes and therefore the time spent calculating relativeto the time spent communicating between processors. A parallel version of a sequentialimportance sampling solution algorithm based on local expected value of perfect information(EVPI) is developed which is applicable to extremely large multistage stochastic linearprogrammes which either have too many data paths to solve directly or a continuous distributionof possible realisations. It utilises the parallel nested Benders algorithm and a parallelversion of an algorithm designed to calculate the local EVPI values for the nodes of the treeand achieves near linear speed‐up.

EVPI‐based importance sampling solution proceduresfor multistage stochastic linear programmeson 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 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 speedup 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 speedup.

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

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.

Parallelization and aggregation ofnested 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.

Parallelization and Aggregation of Nested Benders Decomposition

This thesis introduces a new calculus for manipulating linear-program decomposition schemes. A linear program is represented by a communication network, which is decomposed by splitting nodes in two, and a transformation is defined to recover subproblems from the network. We also define a dual-symmetric oracle that provides solutions to linear programs, and can be performed by the simplex method, nested decomposition, and finally, parallel decomposition.
Two important classes of linear program serve as examples for the above calculus: staircase linear programs and stochastic linear programs. For the former case, a sophisticated yet experimental computer code has been written for an IBM 3090/600E with six processors. The code performs the parallel decomposition algorithm and is tested on twenty-two small to medium sized "real-world" problems. Experiments show that in addition to speedups provided by decomposition alone, performance is improved by using parallel processors.

/pdf/the-parallel-decomposition-of-linear-programs-3zkhfgv6pw.pdf

The parallel decomposition of linear programs

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The parallel decomposition of linear programs