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no comsponding cuts, nor are there comsponding ji or 8 variables. because the algorithm
begins by solving al1 the subproblems simultaneously, so there are no solutions yet available
from other subproblems. Also. when i and j are reset to i=l and j=l at the start of each
iteration b l. no information is made available from second level iterations because the new
proposals exchanged at the fint level mate different subproblems at the second level (but see
the next subsection about partial use of old second level proposals).
The definitions of various syrnbols are given below.
ek" is o 1 x (k-1) row vector with ail entnes equal to 1.
e'-' and e"' are a 1 x (i- 1) row vector and a 1 x (j- 1) row vector. with dl entries equal to 1.
respective1 y.
O,, and O,, are scalar variables derived from level 1 and level II respectively in the
subproblern of part r, r = 2, 3.4.
tpr,, and %,[, are scalar variables derived from level 1 and level II respectively in the
subproblem of part r. r = 1. 2. 3.
g,;' is a (k - 1) x 1 column vector variable whose componenu weight prima1 proposais h m
the sggregated upper bound (level 0 subproblem in the subproblern of part t. r = 1.2.
i;-;[ md i!-1
1 11 are r (i -1) xl column vector variable and a (j -1) xl column vector variable.
whose components weight primal proposals from the comsponding upper bound
subpmblem at level II in the subpmblem of part 1 and 3, respectively.
p:;' is r 1 x (k -1) row vector variable whose components weight duaI proposais from the
aggegated lower bound subproblem (level 2) in the subproblem of part t, t = 3.4.