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Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices

Author

Listed:
  • João Gouveia

    (University of Coimbra)

  • Ting Kei Pong

    (The Hong Kong Polytechnic University)

  • Mina Saee

    (University of Coimbra)

Abstract

Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer approximations, with the only inner approximations available being a linear programming based method proposed by Bundfuss and Dür (SIAM J Optim 20(1):30–53, 2009) and also Yıldırım (Optim Methods Softw 27(1):155–173, 2012), and a semidefinite programming based method proposed by Lasserre (Math Program 144(1):265–276, 2014). In this paper, we propose the use of the cone of nonnegative scaled diagonally dominant matrices as a natural inner approximation to the completely positive cone. Using projections of this cone we derive new graph-based second-order cone approximation schemes for completely positive programming, leading to both uniform and problem-dependent hierarchies. This offers a compromise between the expressive power of semidefinite programming and the speed of linear programming based approaches. Numerical results on random problems, standard quadratic programs and the stable set problem are presented to illustrate the effectiveness of our approach.

Suggested Citation

  • João Gouveia & Ting Kei Pong & Mina Saee, 2020. "Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices," Journal of Global Optimization, Springer, vol. 76(2), pages 383-405, February.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:2:d:10.1007_s10898-019-00861-3
    DOI: 10.1007/s10898-019-00861-3
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    References listed on IDEAS

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    1. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
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    Cited by:

    1. E. Alper Yıldırım, 2022. "An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs," Journal of Global Optimization, Springer, vol. 82(1), pages 1-20, January.

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