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

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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.

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Notes

  1. Note that our definition is different from the classical definition of diagonal dominance (see [15, Definition 6.1.9]) in that we require the diagonal entries of A to be nonnegative.

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Authors and Affiliations

  1. CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal

    João Gouveia & Mina Saee

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China

    Ting Kei Pong

Authors
  1. João Gouveia
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  2. Ting Kei Pong
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  3. Mina Saee
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Correspondence to João Gouveia.

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The first author’s research was partially supported by FCT under Grants UID/MAT/00324/2019 through CMUC, and P2020 SAICTPAC/0011/2015. The second author’s research was supported partly by a research Grant (G-UADF) from The Hong Kong Polytechnic University and the third author’s research was supported by a Ph.D. scholarship from FCT, Grant PD/BD/128060/2016.

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Gouveia, J., Pong, T.K. & Saee, M. Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices. J Glob Optim 76, 383–405 (2020). https://doi.org/10.1007/s10898-019-00861-3

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  • DOI: https://doi.org/10.1007/s10898-019-00861-3

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