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optimal value of P. This is explained further below.
Since it is very diffxcult and long to state the general N part decomposition method in
algebraic terms in detail. we will focus on a Cpart case as shown in Figure 4.2. The extension
to any number of parts Ncan be straightfoward from the demonstration of this Cpart case.
PUU gives an upper bound to the original problem because Pu in the fint level gives an
upper bound to the original problem. and the upper bound subproblem of Pu, which is Puv.
provides an upper bound to Pu. With the same reasoning, Pu. provides a lower bound to the
original problem. The upper bounds provided by Pvv are nonincreasing. as the iterations
proceed. and the lower bounds from PLL are nondecreasing. The algorithm proceeds through
iterations of parailel solution of PLL and PLU, by exchanges of primd and dual proposals.
converging towards the optimal solution of PL. Simultaneously, PLI and Puu are solved
iteratively. in parallel, converging towards the optimal solution of Pu.
Figure 4.2 lpart decomposition principle and information flow.
('Tol" is the predetermined srnall tolemce for judging convergence)