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

3.3 Algorithmic Aspects 31
Input : Mixed knapsack set X MK given in the form (3.7) and
(x ∗ , s ∗ ) ∈ (R n +\Z n +) × R+ fractional vector.
Output: Valid c-MIR inequality for X MK or notification that no such
inequality was found.
/* Use initial partition and initial value of δ. */
1 U ← {j ∈ N : x∗ bj
j ≥ 2 }, T ← N\U and t ← |T |
2 Sort T by nonincreasing value of x∗ bj
j − 2 . (Let {j1, . . . , jt} be the ordered
set.)
3 N ∗ ← {|α ′
j | : j ∈ N, α′ j = 0 and 0 < x∗j < bj}
4 δfound ← F ALSE, vbest ← −∞, and l ← 0
5 repeat
/* Complement an additional variable. */
6 if l > 0 and x∗ jl > 0 then U ← U ∪ {jl}
7
and T
else if l > 0 then l ← l + 1 and continue
← T \{jl}
8 foreach δ ∈ N ∗ do
9 β ← α′
0 −
j∈U α′
j bj
δ
10 if β ∈ Z then continue
11 else
12 δfound ← T RUE
+
j∈U
13 v ←
α′
j
Ffβ j∈T ( δ )x∗ α′
j
Ffβ j (− δ )(bj − x∗ j
14 if v > vbest then δbest ← δ and vbest ← v
15 l ← l + 1
until δfound = T RUE or l = t
17 if δfound = F ALSE then return No inequality found
/* Improve violation by modifying δ. */
18 ¯ δ ← δbest
19 foreach δ ∈ { ¯ δ
2 , ¯ δ
4 , ¯ δ
8
} do
20 β ← α′
0 −
j∈U α′
j bj
δ
+
j∈U
21 v ←
α′
j
Ffβ j∈T ( δ )x∗ α′
j
Ffβ j (− δ )(bj − x∗ j
22 if v > vbest then δbest ← δ and vbest ← v
) − ⌊β⌋ − s∗
δ(1−fβ)
) − ⌊β⌋ − s∗
δ(1−fβ)
23 δ ← δbest
/* Improve violation by complementing additional variables. */
24 Ubest ← U, Tbest ← T and t ← |T |
25 Sort T by nonincreasing value of x∗ bj
j − 2 . (Let {j1, . . . , jt} be the ordered
set.)
26 for l ← 1 to t do
if x∗ 28
> 0 then
jl
U ← U ∪ {jl} and T ← T \{jl}
29 β ← α′
0 −
j∈U α′
j bj
δ
+
j∈U
) − ⌊β⌋ − s∗
δ(1−fβ)
30 v ←
α′
j
Ffβ j∈T ( δ )x∗ α′
j
Ffβ j (− δ )(bj − x∗ j
31 if v > vbest then Ubest ← U, Tbest ← T and vbest ← v
32 U ← Ubest and T ← Tbest
33 return
j∈T
Ffβ ( α′
j
δ )xj +
j∈U
Ffβ (− α′
j
δ )(bj − xj) ≤ ⌊β⌋ +
s
δ(1−fβ)
Algorithm 3.4: Cut generation heuristic. Use Procedure 1 and do not use the extended
candidate set for the value of δ.