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Chapter 1 Introduction
1.1 Brief History of Decomposition
Over the past several decades. linear programming (LP) has been a useful planning and
scheduling tool for economic and management applications. The real world LP problems are very
large and usuaily have some special structures, which could break into several distinct LPs except
for a few linking consuaints. These linking constmints represent relationships among different
periods. regions. stochastic scenarios. etc. The most cornmonly discussed structures are prima1
(or dual) block-anguim. sraircase and block-triangular as shown in Figure 1.1.
l a. bloc k-angular b. staircase c. block-nianmilar
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Figure 1.1 Structures of mulù-stage problem. from Lm and Fuller [1995]
Based on the ideas of utilizing the specid structures, many decomposition algorithms.
which drcompose a very large problem into several subproblems and solve them iteratively
through the exchanges of information, have been proposed since the early 1960's,
There have ken several motivations to study decomposition: the prospect of reducing
computationd time: a rnethod to solve a huge mode1 within a cornputer's memory limits that do
not permit a straightfonvard solution of the whole mode1 (Fragniere et al. [1998]); and to ease