Integer Programming with
Complementarity Constraints
Ismael R. de Farias, Jr. 1
Joint work with Ernee Kozyreff 1 and Ming Zhao 2
1Texas Tech
2SAS
Outline
 Problem definition and formulation
 Valid inequalities
 Instances tested, Platform and Parameters used
 Computational results
 Continued research
 Acknowledgement
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
2/20
Problem definition
 Definition A set of variables is a special ordered set of type
1, or a SOS1, if, in the problem solution, at most one
variable in the set can be non-zero.
 We will restrict ourselves to nonintersecting SOS1s
 Applications
 Transportation
 Scheduling
 Map display
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
3/20
Problem definition
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
4/20
Problem definition
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
5/20
Formulation
 SOS1 branching
 “Usual” MIP formulation (Dantzig, 1960)
 “Log” formulation (Vielma and Nemhauser, 2010; also
Vielma, Ahmed, and Nemhauser, 2012)
Comparison over 1,260 instances
Usual MIP
Log
Instances solved
806
503
Wins (faster)
799
81
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
6/20
SOS1 cutting planes
 Two families of facet defining Lifted Cover Inequalities
derived in de Farias et al (2002) (not tested
computationally), and improved in de Farias et al
(2014), which are valid for
where
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
7/20
SOS1 Cut 1
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
8/20
SOS1 Cut 2
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
9/20
Instances and Platform
 Texas Tech’s High Performance Computer Center
Intel Xeon 2.8 GHz, 24GB RAM, 1024 nodes
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
10/20
MIP solver and Parameters tested
 GUROBI 5.0.1 in…
 Branch-and-bound
 Branch-and-bound + SOS1 Cuts
 Default
 Default + SOS1 Cuts
* Branch-and-bound = Default – Presolve – MIP Cuts – Heuristics
 Maximum number of cuts derived: 1,000 of each type
 Maximum CPU time allowed: 3,600 seconds
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
11/20
Results
Continuous instances: number of instances solved
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
12/20
Results
Continuous instances: solution time
Time with Default
1800
Time with Default 82%
+ SOS1 Cuts 900
Time with Default
12%
800
Time with Default + SOS1 Cuts 1000
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
13/20
Results
Binary instances: number of instances solved
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
14/20
Results
Binary instances: solution time
13%
39%
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
15/20
Results
10,000-IP instances: number of instances solved
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
16/20
Results
10,000-IP instances: solution time
96%
0.2%
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
17/20
Results
Better strategy (with or without SOS1 cuts)
Number of instances solved more efficiently with each method
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
18/20
Summary of results
 The use of SOS1 cuts was imperative on our continuous
and general integer instances.
 “Usual” MIP formulation for SOS1 performed better
than the “Log” formulation.
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
19/20
Continued Research
 Why were SOS1 cuts so effective for problems with integer
variables with large values of u?
 How can SOS1 cuts be modified to be effective for the case of
binary variables?
 Study branching strategies for SOS1
 Study problems with both positive and negative coefficients in
the constraint matrix
 Study solution approaches to KKT systems, in particular LCP
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
20/20
Acknowledgement
We are grateful to the Office of Naval Research for partial
support to this work through grant N000141310041
Integer Programming with Complementarity Constraints
MINLP 2014
Ismael de Farias
21/20