X - UWSpace - University of Waterloo

with proposals or cuts from other subproblems. even in the first iteration.
When the iteration counter kl, since al1 upper bound subproblems give upper bounds to
the original problem and al1 lower bound subpmblems provide lower bounds for the original
problem. the best upper bound and lower bound can be chosen for the convergence test from the
subproblems at each iteration. Although the subproblems generate the nonincreasing upper
bounds and nondecreasing lower bounds. the algorithm can not be guaranteed to converge within
a presaibed toleruice. The heuristic algorithm cm get snick and repeat the same solution without
improvement after some number of itentions, so it is temiinated with a feasible solution of the
onginai problern when dl the lower bound subprolems and d1 the upper bound subproblems have
the same objective values respective1 y in three consecutive iterations.
45.2 The Heuristic Decomposition Algorithm for the Second Method
In this section, the procedure of the heuristic decomposition algorithm for multi-piut
problems is discussed. Various propenies of this dgorithm will be discussed in the next
subsection.
Step O determines that the whole problem is feasible or not by detecting the infeasibility
of subproblems. as proven in the next subsection. if any subproblem is infeasible. then the
algorithm stops because the original problem is determined to be infeasible, and if each
subproblem h s its own femible solutions. then the algorithm pmceeds to Step 1 because the
original problem is feasible.
In Step 1. the scdu d is defined by the user, and the aigorithm solves each