# On the solution of nonconvex cardinality Boolean quadratic programming problems: a computational study

## Abstract

This paper addresses the solution of a cardinality Boolean quadratic programming problem using three different approaches. The first transforms the original problem into six mixed-integer linear programming (MILP) formulations. The second approach takes one of the MILP formulations and relies on the specific features of an MILP solver, namely using starting incumbents, polishing, and callbacks. The last involves the direct solution of the original problem by solvers that can accomodate the nonlinear combinatorial problem. Particular emphasis is placed on the definition of the MILP reformulations and their comparison with the other approaches. The results indicate that the data of the problem has a strong influence on the performance of the different approaches, and that there are clear-cut approaches that are better for some instances of the data. A detailed analysis of the results is made to identify the most effective approaches for specific instances of the data.

## Keywords

Integer programming Quadratic programming Computing science## Notes

### Acknowledgments

The first author acknowledges the support of the Center for Uncertainty Quantification in Computational Science & Engineering. This research used resources in the King Abdullah University of Science and Technology Scientific Computing Center, supported by the Information Technology, Research Computing Team. The authors would like to thank the Center for Advanced Process Decision-making at Carnegie Mellon University for their financial support. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under grant agreement n. PCOFUND-GA-2009-246542 and from the Fundação para a Ciência e Tecnologia, Portugal, under Grant agreement n. DFRH/WIIA/67/2011.

## Supplementary material

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