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Integer Programming with Complementarity Constraints

Integer Programming with Complementarity Constraints. Ismael R. de Farias, Jr. 1 Joint work with Ernee Kozyreff 1 and Ming Zhao 2 1 Texas Tech 2 SAS. Outline. Problem definition and f ormulation Valid inequalities Instances tested, P latform and P arameters used

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Integer Programming with Complementarity Constraints

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  1. Integer Programming with Complementarity Constraints Ismael R. de Farias, Jr. 1 Joint work with ErneeKozyreff1and Ming Zhao 2 1Texas Tech 2SAS

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

  3. Problem definition Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias • 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

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

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

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

  7. SOS1 cutting planes Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 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

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

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

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

  11. MIP solver and Parameters tested Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias • 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

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

  13. Results 82% 12% Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias Continuous instances: solution time

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

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

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

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

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

  19. Summary of results Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 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.

  20. Continued Research Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias 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

  21. Acknowledgement Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias We are grateful to the Office of Naval Research for partial support to this work through grant N000141310041

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