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Chapter 3 Paralle1 Decomposition of the Two-Part Problem
In this chapter. we develop a parallel decomposition aigorithm for NO part linear
programrmng problems, Le. an LP problem which would break into two distinct LPs. except
for a few linking constraints that connect the pans. The new method divides the original
problem into two subproblerns (a lower bound subproblem and an upper bound subproblem).
instead of the traditional master and subproblem. The subproblems are derived in a way that is
similar to the two subproblems in Lm's il9931 scheme. but here we derive them for a two-
pan structure for pyallel computations. whereas Lm's was for the two-stage structure. for a
senal algorithm. Since the subproblems in each part give an upper bound and a lower bound
to the original problem. the algorithm arbiuarily selects an upper bound subproblem from one
part and a lower bound subproblern from the other part and solves them simultaneously in two
diffe~nt processon. By exchanging prima1 and dual information st each itention. these two
bounds are monotonically improved and converge to within the prescribed tolerance of the
optimal value.
3.1 Mode1 Structure and Assumption
The variables and constnints are grouped into two parts. indicated by the subscnpt
r=l. 2. The objective function is the sum of linem functions and each part's objective function
depends only on that pan's variables. Each part contains three sets of variables; nonlinking
variables. .r,. linhng variables. y, and artificial variables. v~. Also. each part consists of three
sets of consuaints: nonlinking constraints. linking consvaints and upper bound constraints on