X - UWSpace - University of Waterloo

Roof : For t= 2. ... .4, each x,' and y; is a convex combination of known solutions ofx, and
y, in the previous iterations. which saiisfies the nonlinking constraints A, x, + 4 yi I b, and
upper bound constraints y, 5 ut of part t. Since xlS and together with x; and y,' solves SPI.
al1 linking constraints in P are also satisfied. So. SPI gives a feasible solution to the original
problem P. The proof of the duai part is sirnilar. 1
The following theorem States that. at each iteration b l. the optimal values of SP?"
and SP~"' give nonincreasing upper bounds and nondecreasing lower bounds to the original
problem P.
Theorem 1.5 In the processor I and 4 with k>l. the optimal values of SP,"' fonn a
nondecreasing series of iower boundr on the optimal value of P and the oprimal values of
SPJ~ fonn a nonincreasing series of upper boundr on the optimal value of P. i.e. zI"'*< cl'%
* * '< < 3 ~ ~ ~ 2 ~ '
... Cl - .. 4 -*. - 3
Proof : Since spiL' is a restriction of the whoie problem P and the feasible regions of
successive subpro blems SP include that of previous subproblems at each iteration by
inclusion of another positive A variable. it gives ci'% zi3.'k ... ~~~''5 cm.
Similarly. SD:J' is a restriction of the dud of the whole problem P and it is loosened
at each itention by inclusion of another positive p variable, so the feasible region of SD~J'
gels bigger a[ each iteration. It provides that ,-*c-."'~.. S S3~'S& so proves the theorem.
CoroUq 11 In the processor I and 1 with k>l and i>l, j>I. the optimal values of SP~"