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Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization

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  • Hayato Waki
  • Maho Nakata
  • Masakazu Muramatsu

Abstract

We observe that in a simple one-dimensional polynomial optimization problem (POP), the ‘optimal’ values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the strange behaviors of the SDP solvers are given. This result gives a warning to users of the SDP relaxation method for POPs to be careful in believing the results of the SDP solvers. We also demonstrate how SDPA-GMP, a multiple precision SDP solver developed by one of the authors, can deal with this situation correctly. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Hayato Waki & Maho Nakata & Masakazu Muramatsu, 2012. "Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization," Computational Optimization and Applications, Springer, vol. 53(3), pages 823-844, December.
    • Handle: RePEc:spr:coopap:v:53:y:2012:i:3:p:823-844
    DOI: 10.1007/s10589-011-9437-8
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    References listed on IDEAS

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    1. Laurent, M., 2009. "Sums of squares, moment matrices and optimization over polynomials," Other publications TiSEM 9fef820b-69d2-43f2-a501-e, Tilburg University, School of Economics and Management.
    2. de Klerk, E. & Elfadul, G.E.E. & den Hertog, D., 2006. "Optimization of Univariate Functions on Bounded Intervals by Interpolation and Semidefinite Programming," Discussion Paper 2006-26, Tilburg University, Center for Economic Research.
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    Cited by:

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    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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