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Inscribed ball and enclosing box methods for the convex maximization problem

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Abstract

Many important classes of decision models give rise to the problem of finding a global maximum of a convex function over a convex set. This problem is known also as concave minimization, concave programming or convex maximization. Such problems can have many local maxima, therefore finding the global maximum is a computationally difficult problem, since standard nonlinear programming procedures fail. In this article, we provide a very simple and practical approach to find the global solution of quadratic convex maximization problems over a polytope. A convex function achieves its global maximum at extreme points of the feasible domain. Since an inscribed ball does not contain any extreme points of the domain, we use the largest inscribed ball for an inner approximation while a minimal enclosing box is exploited for an outer approximation of the domain. The approach is based on the use of these approximations along with the standard local search algorithm and cutting plane techniques.

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Acknowledgments

We would like thank to three anonymous referees for their valuable comments and suggestions which greatly improve the readability of the paper. This research was supported by project Energy Positive IT 2.0 led by ALSTOM Energy Management and Bouygues.

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Authors and Affiliations

  1. Laboratoire PRiSM, Université de Versailles, 45, avenue des États-Unis, 78035, Versailles Cedex, France

    Guillaume Guérard & Ider Tseveendorj

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  1. Guillaume Guérard
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  2. Ider Tseveendorj
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Correspondence to Guillaume Guérard.

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Guérard, G., Tseveendorj, I. Inscribed ball and enclosing box methods for the convex maximization problem. Optim Lett 10, 417–432 (2016). https://doi.org/10.1007/s11590-015-0981-5

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  • DOI: https://doi.org/10.1007/s11590-015-0981-5

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