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8. tested several models for convergence and speedups of the multi-part decomposition
aigorithm.
The parallel decomposition aigorithm for two-part LPs solves two master-like
subproblems. an upper bound subproblem and a lower bound subproblem, simultaneously. Each
subproblem works as both the master problem and the subproblem in the naditional
decomposition methods: each ûccurnuiates proposais from the other. so is like a master problem
yet each contains full details on only its own part. so is like a subproblem. Each subproblem has
a balanced structure by having the same amount of primai and dual information and has an
equivalent position by conducting the convergence test at the same time. The parailel aiprithm
approximates the optimal value of the original problem by calculating a nonincreving upper
bound and a nondecreasing lower bound 3t each iteration. This pracedure terminates when the
two bounds are considered to be close enough according to the prescribed tolerance.
The two-part decomposition principle cm in eetended to more than two parts by
applying the decomposition principle recursively in a hienrchical way. Fint, the original problem
is divided into an aggregated lower bound subproblem and an aggregated upper bound
subproblem. Then. each aggregated subproblem can be again divided into its aggregated lower
bound subproblem and its aggregated upper bound subproblem respectively. and this hieranihical
decomposing process is repeated until there are no more subproblerns to decompose. in the case
of four p m. the original prublem cm have two aggregated subpmblems in the fint level. an
aggregated lower bound subproblem (PL) and an aggregated upper bound subproblern (Pu), and
in the second level. the PL cm be fiuther decomposed into its lower bound subproblem and its
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