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The next result guarantees that in Step 1 of the algorithm. al1 subproblems have
feasible solutions.
Thmrem 1.3 Once step O reports that P is feasible. ail subsequenr subproblem are feaFble.
Roof : In subproblems SPr, SP3 and SP4. the cuts are added to the subproblems of previous
iterations and this addition of cuts can't affect the feasibility of subproblems because the cuts
cm always be satisfied by adjusting the value of the free variable 0.
In the subproblems SPI, SPt and SP3, the primal proposais are added to the
subproblems of previous iteration and this addition of primal proposais does not change the
feasibility of subproblerns because the h variables appear with nonzero coefficients only in
the linking constraints and these linking constrainü are always satisfied by artificid variables.
The next theorem shows that the algorithm provides primal and dual feasible solutions
for the original problem P when it proceeds to Step 1 and it justifies the calculabons of pnmal
and dud solutions.
Theorem 4.4 For any k>l. i> l and j> 1. wirh À weights fron SP:', the algorithm gives the
following primal feusible solunon tu the original problem and with the duol p weights of
S P ~ ifprovides ~ , the foilowing dual feasible soiurion for Pr
ntw = p ' k l ' = 4 ' k * 1 ,
pl = ?
nj0= pJIIj-i. mm = p,d*'~{-l, p3' = pt, a'= Q~,
k r k
~2 = lir:-lfIz"', Y' = p$l~:-l, =
k ' k
u'= . p4 = pr -