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Facial Reduction Algorithms for Conic Optimization Problems

Author

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  • Hayato Waki

    (Kyushu University)

  • Masakazu Muramatsu

    (The University of Electro-Communications)

Abstract

In the conic optimization problems, it is well-known that a positive duality gap may occur, and that solving such a problem is numerically difficult or unstable. For such a case, we propose a facial reduction algorithm to find a primal–dual pair of conic optimization problems having the zero duality gap and the optimal value equal to one of the original primal or dual problems. The conic expansion approach is also known as a method to find such a primal–dual pair, and in this paper we clarify the relationship between our facial reduction algorithm and the conic expansion approach. Our analysis shows that, although they can be regarded as dual to each other, our facial reduction algorithm has ability to produce a finer sequence of faces of the cone including the feasible region. A simple proof of the convergence of our facial reduction algorithm for the conic optimization is presented. We also observe that our facial reduction algorithm has a practical impact by showing numerical experiments for graph partition problems; our facial reduction algorithm in fact enhances the numerical stability in those problems.

Suggested Citation

  • Hayato Waki & Masakazu Muramatsu, 2013. "Facial Reduction Algorithms for Conic Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 188-215, July.
    • Handle: RePEc:spr:joptap:v:158:y:2013:i:1:d:10.1007_s10957-012-0219-y
    DOI: 10.1007/s10957-012-0219-y
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    References listed on IDEAS

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    1. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1997. "Duality Results for Conic Convex Programming," Econometric Institute Research Papers EI 9719/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
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    Cited by:

    1. Yuzhu Wang & Akihiro Tanaka & Akiko Yoshise, 2021. "Polyhedral approximations of the semidefinite cone and their application," Computational Optimization and Applications, Springer, vol. 78(3), pages 893-913, April.
    2. Yoshiyuki Sekiguchi & Hayato Waki, 2021. "Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 52-72, January.
    3. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.

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