X - UWSpace - University of Waterloo
X - UWSpace - University of Waterloo
X - UWSpace - University of Waterloo
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type subproblem. The lower bound subproblem with al1 new linking prima1 variables, called<br />
the lowest bound subproblem (PLL..J. consists <strong>of</strong> that pan's variables. plus fractional<br />
weighting variables for proposals from other pans, and anificial variables. and it includes<br />
linking constnints for al1 parts. The upper bound subproblem with al1 new linking dual<br />
variables. called the uppermost subproblem (Pcu...v). has that part's variables, ail parts'<br />
linking variables. and extra constraints (cuts) constructed with dual variable proposals from<br />
al1 other parts. Other subproblems except the rwo subproblerns mentioned above include that<br />
part's variables. some fractionai weighting variables for pnmd proposals from some other<br />
pans and artificial variables, and they also include some linking variables in some linking<br />
constraints for some other parts. and extra constnints (cuts) consuucted with dual variable<br />
proposals from some other pans.<br />
Each subproblern is solved simultaneously in each processor and exchanges pnrnal<br />
and dual information with the imrnediate neighbor subproblem. which is bifurcated from the<br />
same aggegated subproblem. until each pair <strong>of</strong> subproblems reach an optimal solution <strong>of</strong><br />
each aggregated subproblem. Then. the algorithm checks the pain <strong>of</strong> aggregated subproblems<br />
at the previous level in the bifurcation tree. If they converge to the sme value. the algorithm<br />
checks again the optimdity <strong>of</strong> the pain <strong>of</strong> the next level and if not, the algorithm perfoms<br />
information exchange. updates and simuItaneously solves al1 subproblems descended fmm<br />
chat bifurcation. This hiemhical process itentes until the mot level subproblems. PL and PU,<br />
converge to an optimal solution <strong>of</strong> P. Le. the optimal value <strong>of</strong> the lowest bound subpmblem<br />
(PL L) sets close enough to the optimal value <strong>of</strong> the uppermost subproblem (Puv...u) to<br />
satisfy r prescribed tolennce. It tums out that it is possible to stop short <strong>of</strong> optimdity for al1<br />
pain beyond the mot bifurcation. yet still have convergence <strong>of</strong> the whole process to the<br />
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