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upper bound subpmblem, refemd to as the lower-lower bound subproblem (Pd and the lower-
upper bound subproblem (Pw), while Pu cm be further decornposed into. the upper-lower bound
subproblem (Pv3 and upper-upper bound subproblem (Puu).
The subproblem Puu gives nonincreasing upper bounds to the original problem and PL
provides nondecreving lower bounds to the original problem as the itentions proceed and the
values of Puu and PL converge to an optimal solution of P. The algorithm proceeds through
itentions of parailei solution of Pu and PLU. by exchanges of primal and dud proposais.
converging towûrds the optimal solution of PL. Simuitmeously. PuL and Pvu are solved
itentively. in pdlel. converpng towxds the optimal solution of Pu The algorithm exchanges
primal and dual propos& rt the first level. i.e. between PL and Pu, either when both level II pain
of subproblems have converged to the optimd values of PL and Pu, or when the optimal value of
PCL, :(PC3. is pater than or equal to the optimal value of PLU. :(PLU). Le.. :(Pu3 2 :(PLU).
We developed a paralle1 decomposition solver for four-part problems, called WrZTPAR
(WATerloo PARallel). It cm extnct multi-pan structure from an original GAIS problem and
divides it into four subproblems on an RSi6000. Then it sends the subproblems to four PCs using
P W and solves hem simuitmeously using CPLEX. The solver is designed to utilize any number
of processon from one to four.
In the preliminq tests. the algorithm converges to an optimal solution in a finite number
of itentions and shows faster convergence than direct methods such as the simplex and interior
point methods for some large test problems.
There are seved benefits of the new decomposiuon method. It cm impmve