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

Shun Arahata (), Takayuki Okuno () and Akiko Takeda ()
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Shun Arahata: The University of Tokyo
Takayuki Okuno: RIKEN
Akiko Takeda: The University of Tokyo

Computational Optimization and Applications, 2023, vol. 86, issue 2, No 5, 555-598

Abstract: 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.

Keywords: Nonlinear semidefinite programming; Primal-dual interior-point method; Negative curvature direction; Second-order stationary points; 90C22; 90C26; 90C51 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10589-023-00501-3

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