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.

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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 Slide 2 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 Slide 3 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 Slide 4 Problem definition 4/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 5 Problem definition 5/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 6 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 solved806503 Wins (faster)79981 Slide 7 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 Slide 8 SOS1 Cut 1 8/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 9 SOS1 Cut 2 9/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 10 Instances and Platform Texas Techs High Performance Computer Center Intel Xeon 2.8 GHz, 24GB RAM, 1024 nodes 10/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 11 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 11/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 12 Results Continuous instances: number of instances solved 12/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 13 Results Continuous instances: solution time Time with Default1800 Time with Default + SOS1 Cuts900 Time with Default800 Time with Default + SOS1 Cuts1000 13/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 82% 12% Slide 14 Results Binary instances: number of instances solved 14/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 15 Results Binary instances: solution time 15/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 13% 39% Slide 16 Results 10,000-IP instances: number of instances solved 16/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 17 Results 10,000-IP instances: solution time 17/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 96% 0.2% Slide 18 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 Slide 19 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 Slide 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 20/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Slide 21 Acknowledgement We are grateful to the Office of Naval Research for partial support to this work through grant N000141310041 21/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias