X - UWSpace - University of Waterloo

END
- othenvise. calculate the optimal (or feasible) primai and dual solutions for part t as
k-1 k 1 k-1 k-1
(X:-'A,:-~,Y~-'A~~-') and II*- 1; Stop-
The heuristic algorithm soives the subproblems simultaneously and searches for the
optimum by broadcasting the primal and dual solutions for k >l. Since the feasibility of the
original problem and subpmblems are ensured by the step 1. each subproblem dways has a
feasible solution for P, (which wiU be discwed in the next subsection). The algorithm terminates
with an optimal solution of the original problem when the difference between the best upper
bound ,' and the best lower bound :: of the whole problern gets less than the predetermined
convergence tolerance ES. However. in the cases of the sarne repeated objective values three
times in a row in dl upper bound and al1 Iower bound subproblems respectively. the heuristic
aigorithm terminates with a feasible but not optimal solution of the original problem.
4S3 Properties of the Algorithm for the Second Method
Several properties of the parailel decomposition algorithm are discussed in this sechon.
The arguments are sirniiar to those in section 4.4.
Theorem 4.7 verifies chat the algorithm des out the possibility of unboundedness of the
problem P and of any of the primal subproblems SP:.
Theorem 4.7 Problem P and al1 subproblem SP: are bounded.