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fonn a nondecreasing senes of lower bowidr on the optimal vahe of P if al1 proposais ore
accumulared and the optimal values of S P fom ~ a ~ nonincreasing series of upper bounds on
the optimal value of P if al1 culs are included.
1 7 < C12.io< - 3.2< 3.3~ - 3.le<
i.e. :1*-~
:12s3~ ... - - ..l - :l -.. - ... < rrklo< :'
< 9' 1.. 5 33~-s
3 3 ' 9 i
5 5 - 3 - S-J < .*- < a2*3< a-*-
Proof : Since is a restriction of SP~~." and the feasible regions of successive
subproblems splk.' include that of previous subproblems at each iteration of k and i. by
17
inclusion of another positive À variables, it gives :lomoS :I'.3S ... S :,
S.. 5
:1 - * .
kt0< ;
2.1. 3.2< 3,3< 3.t0<
I
:I - Zl - ..- Z] -
Similarly. SD:~ is a restriction of the dual of SP:~' and it is loosened at each iteration of k
7 - 3-3, 3.2< J * ~ 7 7
... I 2-j 5 ... 5 3 - - ... pS
and j. by inclusion of mother positive p variables. so the feasible region of S D geü ~ bigger ~
at each iteration. It provides that :'< ab' I
a--m IJ