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solutions.
The next result guanntees that in Step 1 of the algorithm. al1 subproblems have feasible
Theorem 4.9 Once the Prst step reporrs that P is feasible. al1 subsequent subproblems are
feasible.
Proof : In the upper bound subproblems, the cuts are added to the subproblems of previous
itention and this addition of cuts can't affect the feasibiiity of subproblems because the cuts cm
always be satisfied by adjusting the value of €4. which is a free variable.
In the lower bound subproblems. the primal proposais are added to the subproblems of
previous itention and ihis addition of primal proposais does not change the feasibility of
subproblems because the A variables appear with nonzero coefficients only in the Iinking
constraints and these linking constraints are dways satisfied by ani ficial variables.
The next throrem shows that the algorithm provides a primal feasible solution for the
original problem P when it pmceeds to step 1 and it justifies the calculations of primal and dual
solutions in step 3.
Theorem 4.10 For any t and k>l. the huer bound subproblem - S& gives rhr folloing phal
-
feasiblc soiurion ro rlte original problem and the duai form of the upper bound subproblem SP!
provides the folioicing duai feasible solurion for P: