Implementation of Cutting Plane Separators for Mixed Integer ... - ZIB

3.3 Algorithmic Aspects 19
where αj =
i∈P ωiai j for all j ∈ N, γj =
i∈P ωici j for all j ∈ M, and α0 =
i∈P ωiai 0 .
Bound substitution: Now, choose for each real variable yj, j ∈ M one of the
following substitutions
yj = uj − ¯yj, yj = ũjxj − ¯yj, yj = lj + ¯yj, or yj = ˜ ljxj + ¯yj
where ¯yj is a nonnegative real variable. Note that the first and the second substitution
are only allowed to be chosen if uj < ∞ and ũj < ∞, respectively. Then, X ′
equivalent to the set
is
X ′′
= {(x, ¯y) ∈ Zn + × Rm + :
α
j∈N
′
jxj +
γ ′
j ¯yj = α ′
0 −
γ ′
j ¯yj,
(3.6)
for appropriate rational numbers α ′
j
relax X ′′
where s = −
j∈M,γ ′
j ≥0
xj ≤ bj for all j ∈ N},
for all j ∈ N, γ′
j
to obtain the mixed knapsack set
XMK = {(x, s) ∈ Zn + × R+ :
j∈M,γ ′ γ′
j