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An algorithm could be defined to exchange prima1 and dual proposals at the first level.
i.e. between PL and Pu. only when both level II pairs of subproblems have converged to the
optimal values of PL and Pu It should be clear. based on the convergence of the two-part case.
that convergence could be proved for such an algorithm. However. we have implemented a
different scheme which requires fewer iterations at the second level before information
exchange at the fint level. A careful examination of the convergence proof for two part
models reveds that convergence is assured if the two parts pass feasible (not necessarily
t
optimal) solutions such that cik 5 r. . Applying this observation to PL and Pu in the
implemented aigonthm. we get a dual feasible solution to PL from PLU. a primai feasible
solution from PuL, and we wait until :(PLU) 5 :(PUù before exchanging proposals between PL
and Pu. Figure 4.7 shows the criteria for the iterations to conrinue with primai and dual
exchmges. at each level: second level exchanges between a pair of subproblems continue if
:(Pc3 c :(PLU) and the pair has not converged to within 3 predetermined tolerance of the
optimal value of its first Ievel problem: fint level exchmges continue as long as the upper
bound. :(Pvc). has not converged to the lower bound. :(PU).
The parailel decomposition rnethod would be bdanced arnong the processon if the
number of pans in the original problem is a power of 2. In other cases. a me like in Figure 4.1
wouId have some end nodes at different levels than other end nodes. This could lead to much
idle time for rhe processon that solve the subproblems at higher level end nodes. However.
one cm consider a balmcing strategy that ~ sips a large or difficult subproblem to a higher
end level node in order to decrease idle time of the processon. Another balancing scheme
could have two or more subproblems of higher level end nodes assigned to one pracessor, to