X - UWSpace - University of Waterloo
X - UWSpace - University of Waterloo
X - UWSpace - University of Waterloo
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nNk-'.<br />
Pro<strong>of</strong> : For i= 1.2. ... . N and i#t. each .r,' and y' is a convex combination <strong>of</strong> known solutions <strong>of</strong><br />
.Y, and y in the preveious itentions. which satisfies the nonlinking constraints Ai .r, + Di y, S bi and<br />
upper bound constraints y, I ici <strong>of</strong> part i Since .rra and together with xi* and solves - S& , dl<br />
linking constnints in P are dso satisfied. So. - & gives a feasible solution to the original<br />
problem P. The pro<strong>of</strong> <strong>of</strong> dud part is similu.<br />
The following theorem States that, at each iteration bl. the optimal value <strong>of</strong> an upper<br />
bound subproblern and a lower bound subproblem give nonincreasing upper bounds and<br />
nondecreasing lower bounds to the original problem P.<br />
Theorem 4.11 In Step I for any lower boicnd subprobiem (indexed 6y ta). +e oprimai<br />
rdites <strong>of</strong>S& .C . form a nondecrearing series <strong>of</strong> Zower bounds on the optimal value <strong>of</strong> P und for<br />
--<br />
--<br />
an! icpper bound sicbproblem (inde-red by s), the optimal values <strong>of</strong> se . -: . fonn a nonincreacing