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Chapter 4 Paraiiel Decomposition of the Multi-Part Mode1
In this chapter, two parallel decomposition methods for multi-part iinear programrning
problems are presented. The fint is developed by applying a hierarchical decomposition
principle recunively. We first divide the original multi-part problem into two aggregated
subproblems of lower bound type and upper bound type, by applying the basic algorithm of
the two-part method. which was discussed in the previous chapter. Then. the aggregated
subproblems are further divided into two smaller aggregated subproblems of upper bound and
lower bound. This results in some primai subproblems accumulating both proposais and cuts.
This bifurcation process continues until there are no subproblerns left for funher
decomposition. The subproblems are solved in different processon simuItaneously and work
together to reach an optimal point during the iterations by exchanging information in the
hierarchicd way. In Chapter 5, we repon successhl convergence on several test problems.
The second method is less complen than the first. It defines some subproblems to be of
the lower bound type from Chapter 3, Le. utilizing primal proposais from other subproblems,
and the other subproblems are of the upper bound type. Unfortunately, this method fails to
converge in some tests. We include it because it may be useful as a heuristic.
4.1 Definition of a Multi-Part Linear Program
The definition of the two-part problems is now extended to that of multi-part
problems. Consider the following prima1 and dual forms of LP problem of N parts: