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Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems

Author

Listed:
  • Shun Arahata

    (The University of Tokyo)

  • Takayuki Okuno

    (RIKEN
    Seikei University)

  • Akiko Takeda

    (The University of Tokyo
    RIKEN)

Abstract

We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only ensure convergence to first-order stationary points such as Karush–Kuhn–Tucker points, but without a worst-case iteration complexity. The proposed method generates a sequence approximating SOSPs while minimizing a primal-dual merit function for NSDPs by using scaled gradient directions and directions of negative curvature. Under some assumptions, the generated sequence accumulates at an SOSP with a worst-case iteration complexity. This result is also obtained for a primal IPM with a slight modification. Finally, our numerical experiments show the benefits of using directions of negative curvature in the proposed method.

Suggested Citation

  • Shun Arahata & Takayuki Okuno & Akiko Takeda, 2023. "Complexity analysis of interior-point methods for second-order stationary points of nonlinear semidefinite optimization problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 555-598, November.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:2:d:10.1007_s10589-023-00501-3
    DOI: 10.1007/s10589-023-00501-3
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    References listed on IDEAS

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