Analysis of the Primal-Dual Central Path for Nonlinear Semidefinite Optimization Without the Nondegeneracy Condition

• 成果类型: Article; Early Access
• 署名作者: Okuno, Takayuki
• 署名单位: Seikei University
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.0298
• 发表日期: 2025
• 关键词: augmented lagrangian functions; limiting behavior; optimality conditions; CONVERGENCE; point; sdp; algorithm; trajectories; sensitivity

摘要

We study properties of the central path underlying a nonlinear semidefinite optimization problem, called an NSDP for short. The latest radical work on this topic was contributed by Yamashita and Yabe [Yamashita H, Yabe H (2012) Local and superlinear convergence of a primal-dual interior point method for nonlinear semidefinite programming. Mathematical Programming 132(1-2):1-30]: they proved that the Jacobian of a certain equation system derived from the Karush-Kuhn-Tucker (KKT) conditions of the NSDP is nonsingular at a KKT point under the second-order sufficient condition (SOSC), the strict complementarity condition (SC), and the nondegeneracy condition (NC). This yields uniqueness and existence of the central path through the implicit function theorem. In this paper, we consider the following three assumptions on a KKT point: the strong SOSC, the SC, and the Mangasarian-Fromovitz constraint qualification. Under the absence of the NC, the Lagrange multiplier set is not necessarily a singleton, and the nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless, we establish that the central path exists uniquely, and moreover prove that the dual component of the path converges to the so-called analytic center of the Lagrange multiplier set. As another notable result, we clarify a region around the central path where Newton's equations relevant to primal-dual interior-point methods are uniquely solvable.