X - UWSpace - University of Waterloo

zi - $' 5 z+ - &J-[, and by Theorem 3.8, the right side of the inequality converges to zero.
Therefore, (z/ - :)'')+O as j- through je N.
Cordlary if the parallel primnl-dual decomposition algorithm proceeds to iterution k> 1,
rhen with the given tolerance E > O. it stops in a finite number of iterations.
RooE The corollary follows directly from the stopping criterion uO and Theorem 3.9.
3.5 Summary and Observations on !Le Algorithm
L. The onginai problem is divided into two types of subproblems in each part: a lower
bound type subproblem and an upper bound type subproblem.
2. The lower bound subproblem in primai form is consuucted by resaicting prima1
variables. xz and y, to convex combinations of known values. received from the upper
bound subproblern. thus dropping out redundani prima1 nonlinking conscraints and
primai upper bound constraints. nie upper bound subproblem in dud form is
consuucted by restncting dual variables. iri and al, to convex combinations of known
values. received from the first subproblem. then dropping out redundant dud
nonlinking constraints and dual upper bound constraints.
3. The panIlel aigorithm ha a perfectly bdanced structure by the two master-like
subproblems instead of the master and subproblem structure as in the aaditional
decomposition methods.
4 information on primai and duai solutions is sent and received between the two
subproblems at each iteration. The duai solution of the fint subproblem is passed to