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for the optimum by exchanging the prima1 and dual solutions in the hierarchicai manner.
Since the feasibility of the original problem and subprobiems are ensund by the step O. each
subproblem after step O always has a feasible solution. (this will be discussed in the next
kl*
section). The algorithm terminates when the difference between ab' and ci gets less than
the predetermined convergence tolerance d.
Note that when the algorithm converges within the given tolerance and stops. it may
not give a basic feasible solution to the original problem. but slightly interior, due to the
nature of convex combinations. However. the basic feasible solution could be recovered by
developing a sirnilar scheme used in purification (Konanek and Zhu, 19881 or crossover
facility of CPLEX barrier method, i.e.. the optimai solution of the algorithm is adjusted by
moving some prima1 variables to upper or Iower bounds (to become nonbasic variables) and if
necessuy. the simplex method fin& a basic optimal solution of the original problem in a
small number of itentions since the solution fed into the simplex method is already feasible
and close to an optimal solution of the original problem.
4.4 Properties of the Algorithm for the Fint Method
Sevenl properties of the parailel decomposition algorithm are discussed in this
section. The arguments are sirnilar to those of the serial case of Lan and Fuller [1995b].
The fint theorem verifies that the algorithm mies out the possibility of unboundedness
of the problem P and of any of the prima1 subproblems spit', SP?', sPjb and SPI?
Theorem 4.1 Prublern P and al2 subprublem SP~''. SP?'. SP;' and SP~' are bounded