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nonlinking constraints and upper bound constraints in S# - should be infeasible because the
anificial variables in the linking constraints guarantee that these constraints are always
satisfied.
However. for the aigorithm to proceed to iteration k>l, the sarne set of nonlinking constraints
and upper bound constraints. which was present in SPI - at step O, must be feasible. a
c~n~diction. Thus, SP: - is feasible for bl. Therefore, spf - has a finite optimal solution at
each iteration k>l .
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Theorem 3.26 For k> 1. each subproblem spf has ajinite optimal solution.
Proof: For bl. the feasibility of 3 is discussed fint and then the boundedness will be
shown. There exists a feasible solution ~v{? xi, yd v/) which satisfies dl constraints except
possibly the set of first constraints (or cuts) in 3 because if not, the algorithm would have
stopped ai step O. The first set of consuainu is satisfied if Bz is chosen as
Next. the boundedness cm be shown by the following reasoning: