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New LP-based local and global algorithms for continuous and mixed-integer nonconvex quadratic programming

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Abstract

In this work, we propose a new approach called “Successive Linear Programming Algorithm (SLPA)” for finding an approximate global minimizer of general nonconvex quadratic programs. This algorithm can be initialized by any extreme point of the convex polyhedron of the feasible domain. Furthermore, we generalize the simplex algorithm for finding a local minimizer of concave quadratic programs written in standard form. We prove a new necessary and sufficient condition for local optimality, then we describe the Revised Primal Simplex Algorithm (RPSA). Finally, we propose a hybrid local-global algorithm called “SLPLEX”, which combines RPSA with SLPA for solving general concave quadratic programs. In order to compare the proposed algorithms to the branch-and-bound algorithm of CPLEX12.8 and the branch-and-cut algorithm of Quadproga, we develop an implementation with MATLAB and we present numerical experiments on 139 nonconvex quadratic test problems.

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Data availability statement

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Notes

  1. The quality of a solution can be measured using the gap between the value of the objective function at this solution and the known global optimum.

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Acknowledgements

The authors are indebted to the anonymous referees whose comments and suggestions have considerably improved the quality of this paper.

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Authors and Affiliations

  1. Laboratory of Pure and Applied Mathematics, University of Laghouat, Laghouat, Algeria

    Mohand Bentobache, Mohamed Telli & Abdelkader Mokhtari

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  1. Mohand Bentobache
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  2. Mohamed Telli
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  3. Abdelkader Mokhtari
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Correspondence to Mohand Bentobache.

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Bentobache, M., Telli, M. & Mokhtari, A. New LP-based local and global algorithms for continuous and mixed-integer nonconvex quadratic programming. J Glob Optim 82, 659–689 (2022). https://doi.org/10.1007/s10898-021-01108-w

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