Branch-and-cut for SOS1 constrained linear problems

Ismael R. de Farias, Jr. 1

Joint work with Ernee Kozyreff 1 and Ming Zhao 2

1Texas Tech2SASInteger Programming with Complementarity ConstraintsOutlineProblem definition and formulationValid inequalitiesInstances tested, Platform and Parameters usedComputational resultsContinued researchAcknowledgement2/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasProblem definitionDefinition 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 SOS1sApplicationsTransportationSchedulingMap display

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

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

5/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasFormulation6/20SOS1 branching Usual MIP formulation (Dantzig, 1960)Log formulation (Vielma and Nemhauser, 2010; also Vielma, Ahmed, and Nemhauser, 2012)

Comparison over 1,260 instancesInteger Programming with Complementarity Constraints MINLP 2014 Ismael de FariasUsual MIPLogInstances solved806503Wins (faster)79981SOS1 cutting planesTwo 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 PlatformTexas Techs High Performance Computer CenterIntel Xeon 2.8 GHz, 24GB RAM, 1024 nodes10/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasMIP solver and Parameters testedGUROBI 5.0.1 inBranch-and-boundBranch-and-bound + SOS1 CutsDefaultDefault + SOS1 Cuts* Branch-and-bound = Default Presolve MIP Cuts HeuristicsMaximum number of cuts derived: 1,000 of each typeMaximum CPU time allowed: 3,600 seconds

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ResultsContinuous instances: number of instances solved12/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasResultsContinuous instances: solution time

Time with Default1800Time with Default + SOS1 Cuts900Time with Default800Time with Default + SOS1 Cuts1000

13/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias82%12%ResultsBinary instances: number of instances solved

14/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasResultsBinary instances: solution time

15/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias13%39%Results10,000-IP instances: number of instances solved

16/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasResults10,000-IP instances: solution time

17/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias96%0.2%

ResultsBetter strategy (with or without SOS1 cuts)

18/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de FariasNumber of instances solved more efficiently with each methodSummary of resultsThe 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 FariasContinued ResearchWhy 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 FariasAcknowledgementWe are grateful to the Office of Naval Research for partial support to this work through grant N00014131004121/20Integer Programming with Complementarity Constraints MINLP 2014 Ismael de Farias