X - UWSpace - University of Waterloo

Chapter 2 Literature Review
This chapter provides an overview of topics found in the literature which are related to
this thesis. in the next section, a bief ~ view of decomposition methods for LPs is presentd The
second section discusses the basic parallel computing concepts and methods. In section 3. the
parailel computational tests and algorithm applied to decomposition methods are reviewed aiong
with the description of their characteristics. The summary is given in the final section.
2.1 An Oveniew of the Decomposition Methods
Many decomposition schemes, such as Dantzig-Wolfe decomposition. Benders
decomposition and cross decomposition by Van Roy, have ken developed to solve large-scde
mathematical proCgamming problems. A new decomposi tion algorithm. cdled primal-dual
decomposition. w u suggested by Lan and Fuller [1995aj. Ln this section. these aigonthms are
briefly reviewed with extensions to the nested alprithm for multi-stages.
Dantzig-Wolfe Decomposition Method
h the decomposition method of Dantzig-Wolfe, the master problem determines an
optimal combination of the proposds on hand subrnined by subproblems, by assigning values
to the weights. The optimal duai variables. known as prices, are used to adjust the objective
function in the subproblems which in tum may produce new proposais to improve the global
objective function in the naster problem. This mechanisrn is ofien called the dual decomposition
method.
Dantzig [1963] applied this method in a hiervchical way to a four stage staircase mode1