使用 Monteiro–Tsuchiya 搜索方向族的非线性半定优化原始-对偶内点法的局部收敛性
Local convergence of primal-dual interior point methods for nonlinear semidefinite optimization using the Monteiro-Tsuchiya family of search directions
AI总结：
本文针对非线性半定优化提出采用 Monteiro–Tsuchiya 方向族的原始-对偶内点法，并在缩放矩阵的一般假设下证明其向 KKT 点的局部超线性收敛性。
摘要：
近年来，用于非线性半定优化问题（NSDP）的算法取得了显著进展。Yamashita 等人首先提出了一种原始-对偶内点法（PDIPM），通过使用 Monteiro–Zhang（MZ）搜索方向族来求解 NSDP。此后，人们针对 NSDP 提出了多种 PDIPM，但据我们所知，这些方法全部基于 MZ 方向族。本文提出一种配备 Monteiro–Tsuchiya（MT）方向族的 PDIPM；与 MZ 方向族一样，MT 方向最初也是为求解线性半定优化问题而设计的。我们进一步证明，在对用于生成 MT 缩放方向的缩放矩阵作出某些一般性假设的情况下，该方法向 NSDP 的一个 Karush–Kuhn–Tucker 点具有局部超线性收敛性。
英文摘要：
The recent advance of algorithms for nonlinear semi-definite optimization problems, called NSDPs, is remarkable. Yamashita et al. first proposed a primal-dual interior point method (PDIPM) for solving NSDPs using the family of Monteiro-Zhang (MZ) search directions. Since then, various kinds of PDIPMs have been proposed for NSDPs, but, as far as we know, all of them are based on the MZ family. In this paper, we present a PDIPM equipped with the family of Monteiro-Tsuchiya (MT) directions, which were originally devised for solving linear semi-definite optimization problems as were the MZ family. We further prove local superlinear convergence to a Karush-Kuhn-Tucker point of the NSDP in the presence of certain general assumptions on scaling matrices, which are used in producing the MT scaling directions.
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