Local convergence of primal-dual interior point methods for nonlinear semi-definite optimization using the family of Monteiro-Tsuchiya directions
Local convergence of primal-dual interior point methods for nonlinear semi-definite optimization using the family of Monteiro-Tsuchiya directions
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发表时间：
2020-09
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通讯作者：
Takayuki Okuno
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作者：
Takayuki Okuno
非线性半定优化问题的算法，称为NSDPs，最近的进展是显着的。Yamashita等人首先提出了一种使用Monteiro-Zhang（MZ）搜索方向族求解NSDP的原始-对偶内点方法（PDIPM）。从那时起，各种各样的PDIPM被提出用于NSDP，但是，据我们所知，它们都是基于MZ家族的。在本文中，我们提出了一个PDIPM配备家庭的蒙特罗-土屋（MT）方向，这最初是为了解决线性半定优化问题的MZ家庭。我们进一步证明了局部超线性收敛到Karush-Kuhn-Tucker点的NSDP在某些一般假设的缩放矩阵，这是用于生产的MT缩放方向的存在。
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.