Ismael R. de Farias, Jr. 1 Joint work with Ernee Kozyreff 1 and Ming Zhao 2 1 Texas Tech 2 SAS Integer Programming with Complementarity Constraints

Outline Problem definition and formulation Valid inequalities Instances tested, Platform and Parameters used Computational results Continued research Acknowledgement 2/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

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 3/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Problem definition 4/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Problem definition 5/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Formulation 6/20 SOS1 branching “Usual” MIP formulation (Dantzig, 1960) “Log” formulation (Vielma and Nemhauser, 2010; also Vielma, Ahmed, and Nemhauser, 2012) Comparison over 1,260 instances Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Usual MIPLog Instances solved Wins (faster)79981

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 7/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

SOS1 Cut 1 8/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

SOS1 Cut 2 9/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Instances and Platform Texas Tech’s High Performance Computer Center Intel Xeon 2.8 GHz, 24GB RAM, 1024 nodes 10/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

MIP solver and Parameters tested GUROBI 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 11/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Results Continuous instances: number of instances solved 12/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Results Continuous instances: solution time Time with Default1800 Time with Default + SOS1 Cuts900 Time with Default800 Time with Default + SOS1 Cuts /20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 82% 12%

Results Binary instances: number of instances solved 14/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Results Binary instances: solution time 15/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 13% 39%

Results 10,000-IP instances: number of instances solved 16/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Results 10,000-IP instances: solution time 17/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 96% 0.2%

Results Better strategy (with or without SOS1 cuts) 18/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Number of instances solved more efficiently with each method

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. 19/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

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 20/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias

Acknowledgement We are grateful to the Office of Naval Research for partial support to this work through grant N /20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias