X - UWSpace - University of Waterloo
X - UWSpace - University of Waterloo
X - UWSpace - University of Waterloo
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4.5 A Heuristic Decomposition Algorithm - the Second Method<br />
In this section. a heuristic parallel decomposition algorithm for multi-part linear<br />
propmming problems is presented. The heuristic parallel algorithm divides the original multi-<br />
part problem into several small subproblerns <strong>of</strong> either lower bound type or upper bound type h m<br />
each part. by extending the basic aiprithm <strong>of</strong> the two-pan method without the hierarehical<br />
decomposition principle. The subproblems communicate with each other by sending and<br />
receiving primal and dual solutions. and work together to mach an optimal point during the<br />
itentions. The present approach gives simple subproblem structures and algorithm. however it<br />
does not give any guarantee for convergence; in tests. mentioned briefly in Chapter 5. this<br />
heuristic sometimes fails to converge.<br />
4.5.1 The Structure <strong>of</strong> Subproblems for the Second Method<br />
The heunstic algorithm divides the original multi-part problem into small lower bound<br />
sul'problems and upper bound subprobIems by extending the basic alprithm <strong>of</strong> pdlel<br />
decomposition <strong>of</strong> two-pan models to multi-part models; thus ir has only one iteration counter.<br />
Each pan has a primal form <strong>of</strong> either a lower bound subproblem or an upper bound subproblern.<br />
Each lower bound subproblern consists <strong>of</strong> bat part's variables. plus fractiond weighting variables<br />
for proposals from other parts and artificid variables. and it includes linking consmaints for al1<br />
parts. Each upper bound subproblem has that part's variables. al1 parts's linking variables, and<br />
extra constiiiints (cuts) constructed with dual variable proposals h m al1 other parts. Note that<br />
to proceed with the aigorithm. it should include at least one lower bound subproblem and at least
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