Optimizing over the Split Closure

1
Optimizing over the Split Closure

• Anureet Saxena
• ACO PhD Student,
• Tepper School of Business,
• Carnegie Mellon University.
• (Joint Work with Egon Balas)

2
MIP Model
Contains xj 0 j2N xj uj j2N1

• min cx
• Ax b
• xj 2 Z 8 j2N1

N1 set of integer variables
Incumbent Fractional Solution
3
Split Disjunctions

• ? 2 ZN, ?0 2 Z
• ?j 0, j2 N2
• ?0 lt ? lt ?0 1

? x ?0
? x ?0 1
Split Disjunction
4
Split Cuts
Ax b ? x ?0
Ax b ? x ?01
u
v
u0
v0
?L x ?L
?R x ?R
? x ?
Split Cut
5
Split Closure

• Elementary Split Closure of P x Ax b
  is the polyhedral set defined by intersecting P
  with the valid rank-1 split cuts.
• How much duality gap can be closed by optimizing
  over the split closure?

Rank-1 Chvatal Closure
Elementary Disjunctive Closure
M. Fischetti A.Lodi
P. Bonami M. Minoux
6
Algorithmic Framework
min cx Ax b ?t x ?t t2?
Solve Master LP
Add Cuts
Integral Sol? Unbounded? Infeasible?
Yes
MIP Solved
No
No Split Cuts Generated
Optimum over Split Closure attained
Split Cuts Generated
Rank-1 Split Cut Separation
7
Algorithmic Framework
min cx Ax b ?t x ?t t2?
Solve Master LP
Add Cuts
Integral Sol? Unbounded? Infeasible?
Yes
MIP Solved
No
No Split Cuts Generated
Optimum over Split Closure attained
Split Cuts Generated
Rank-1 Split Cut Separation
8
SC Separation Theorem

• Theorem lies in the split closure of P if
  and only if the optimal value of the following
  parametric mixed integer linear program is
  non-negative.

Parameter
(u,v,?,?0,?) ? uA - ? ? ? ub - ? ?0 ? x ?
Parametric Mixed Integer Linear Program
Split Cut
9
Deparametrization

• Parameteric Mixed
• Integer Linear
• Program

10
Deparametrization

• Parameteric Mixed
• Integer Linear
• Program

If ? is fixed, then PMILP reduces to a MILP
11
Deparametrization

• MILP( )
• Deparametrized
• Mixed Integer
• Linear Program

Maintain a dynamically updated grid of parameters
12
Separation Algorithm

• Initialize Parameter Grid ( ? )

• For ? 2 ?,
• Solve MILP(?) using CPLEX 9.0
• Enumerate ? branch and bound nodes
• Store all the separating split disjunctions
  which are discovered

Diversification
Strengthening
At least one split disjunction discovered?
Grid Enrichment
no
yes
STOP
Bifurcation
13
Implementation Details

• Processor Details
• Pentium IV
• 2Ghz, 2GB RAM

COIN-OR
CPLEX 9.0
Solving MILP( ? )

• Core Implementation
• Solving Master LP
• Setting up MILP
• Disjunctions/Cuts Management
• LP cut generationstrengthening

14
Computational Results

• MIPLIB 3.0 instances
• OR-Lib (Beasley) Capacitated Warehouse Location
  Problems

15
MIPLIB 3.0 MIP Instances
98-100 Gap Closed
16
MIPLIB 3.0 MIP Instances
98-100 Gap Closed
17
MIPLIB 3.0 MIP Instances
Unsolved MIP Instance In MIPLIB 3.0
75-98 Gap Closed
18
MIPLIB 3.0 MIP Instances
25-75 Gap Closed
19
MIPLIB 3.0 MIP Instances
0-25 Gap Closed
20
MIPLIB 3.0 MIP Instances
Summary of MIP Instances (MIPLIB
3.0) Total Number of Instances 34 Number of
Instances included 33 No duality gap noswot,
dsbmip Instance not included rentacar Results 9
8-100 Gap closed in 14 instances 75-98 Gap
closed in 11 instances 25-75 Gap closed in 3
instances 0-25 Gap closed in 3
instances Average Gap Closed 82.53
21
MIPLIB 3.0 Pure IP Instances
98-100 Gap Closed
22
MIPLIB 3.0 Pure IP Instances
75-98 Gap Closed
23
MIPLIB 3.0 Pure IP Instances
Ceria, Pataki et al closed around 50 of the gap
using 10 rounds of LP cuts
25-75 Gap Closed
24
MIPLIB 3.0 Pure IP Instances
0-25 Gap Closed
25
MIPLIB 3.0 Pure IP Instances
Summary of Pure IP Instances (MIPLIB
3.0) Total Number of Instances 25 Number of
Instances included 24 No duality gap
enigma Instance not included harp2 Results 98-1
00 Gap closed in 9 instances 75-98 Gap closed
in 4 instances 25-75 Gap closed in 6
instances 0-25 Gap closed in 4
instances Average Gap Closed 71.63
26
MIPLIB 3.0 Pure IP Instances
Gap Closed by First Chvatal Closure
(Fischetti-Lodi Bound)
27
MIPLIB 3.0 Pure IP Instances
28
MIPLIB 3.0 Pure IP Instances
29
MIPLIB 3.0 Pure IP Instances
Comparison of Split Closure vs CG
Closure Total Number of Instances 24 CG closure
closes gt98 Gap 9 Results (Remaining 15
Instances) Split Closure closes significantly
more gap in 9 instances Both Closures close
almost same gap in 6 instances
30
OrLib CWLP

• Set 1
• 37 Real-World Instances
• 50 Customers, 16-25-50 Warehouses
• Set 2
• 12 Real-World Instances
• 1000 Customers, 100 Warehouses

31
OrLib CWLP Set 1
Summary of OrLib CWLP Instances (Set
1) Number of Instances 37 Number of Instances
included 37 Results 100 Gap closed in 37
instances
32
OrLib CWLP Set 2
Summary of OrLib CWFL Instances (Set
2) Number of Instances 12 Number of Instances
included 12 Results gt90 Gap closed in 10
instances 85-90 Gap closed in 2
instances Average Gap Closed 92.82
33
Support Size Sparsity

• The support of a split disjunction D(?, ?0) is
  the set of non-zero components of ?

? x ?0
? x ?0 1
34
Support Size Sparsity

• The support of a split disjunction D(?, ?0) is
  the set of non-zero components of ?

• Computationally Faster
• Avoid fill-in

Sparse Split Disjunctions
Basis Factorization Sparse Matrix Op
Disjunctive argument Non-negative row combinations
Sparse Split Cuts
35
Support Size Sparsity
36
Support Size Sparsity
37
Support Size Sparsity
Empirical Observation Substantial Duality gap
can be closed by using split cuts generated
from sparse split disjunctions
38
Support Coefficients

• Practice
• Elementary 0/1 disjunctions
• Mixed Integer Gomory Cuts
• Lift-and-project cuts

• Theory
• Determinants of sub-matrices
• Andersen, Cornuejols Li (05)
• Cook, Kannan Scrhijver (90)

Huge Gap
det (B)
1
39
Support Coefficients
40
Support Coefficients
41
Support Coefficients
Empirical Observation Substantial Duality gap
can be closed by using split cuts generated
from split disjunctions containing small support
coefficients.
42
arki001

• MIPLIB 3.0 2003 instance
• Metallurgical Industry
• Unsolved for the past 10 years 1996-2000-2005

Problem Stats 1048 Rows
1388 Columns 123 Gen Integer Vars 415
Binary Vars 850 Continuous Vars
43
Strengthening CPLEX 9.0
Solved to optimality
Crossover Point (227 rank-1 cuts)
44
CPLEX 9.0
43 million BB nodes 22 million active nodes 12GB
BB Tree
45
Comparison
Crossover Point
46

• Thank You