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Testing copositivity with the help of difference-of-convex optimization

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Abstract

We consider the problem of minimizing an indefinite quadratic form over the nonnegative orthant, or equivalently, the problem of deciding whether a symmetric matrix is copositive. We formulate the problem as a difference of convex functions problem. Using conjugate duality, we show that there is a one-to-one correspondence between their respective critical points and minima. We then apply a subgradient algorithm to approximate those critical points and obtain an efficient heuristic to verify non-copositivity of a matrix.

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Acknowledgments

This research was initiated during a research visit of Mirjam Dür to Université Paul Sabatier in Toulouse. She would like to thank the members of the optimization group for their warm hospitality and inspiring discussions. Her work was partially supported by the Netherlands Organisation for Scientific Research (NWO) through Vici grant no.639.033.907. Both authors wish to thank the referees for highly detailed and valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Trier, 54286 , Trier, Germany

    Mirjam Dür

  2. Institut de Mathématiques, Université Paul Sabatier (Toulouse III), 118, route de Narbonne, 31062 , Toulouse Cedex 9, France

    Jean-Baptiste Hiriart-Urruty

Authors
  1. Mirjam Dür
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  2. Jean-Baptiste Hiriart-Urruty
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Correspondence to Jean-Baptiste Hiriart-Urruty.

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Dedicated to Claude Lemaréchal on the occasion of his 65th birthday and retirement.

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Dür, M., Hiriart-Urruty, JB. Testing copositivity with the help of difference-of-convex optimization. Math. Program. 140, 31–43 (2013). https://doi.org/10.1007/s10107-012-0625-9

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  • DOI: https://doi.org/10.1007/s10107-012-0625-9

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