论文标题
分析非线性半决赛优化的原始双重中心路径，而无需非等级条件
Analysis of the primal-dual central path for nonlinear semidefinite optimization without the nondegeneracy condition
论文作者
论文摘要
我们研究非线性半限定优化问题的基础属性，称为NSDP。关于该主题的最新激进工作是由Yamashita和Yabe（2012）贡献的：他们证明，源自NSDP的Karush-Kuhn-Tucker（KKT）条件的某些方程式系统的Jacobian在第二阶条件（SOSCERENSER）和严格的情况下（Sc）（Sc）（SC），NSDP的NSDP条件是非连接的（kkt）。这产生了通过隐式函数定理的中心路径的独特性和存在。在本文中，我们考虑了KKT点的以下三个假设：强大的SOSC，SC和Mangasarian-Fromovitz的约束资格。在没有NC的情况下，Lagrange乘数集不一定是单身人士，上述Jacobian的非语言性不再有效。尽管如此，我们确定了中心路径的独特存在，此外，路径的双重分量会收敛到Lagrange乘数集的所谓分析中心。作为另一个值得注意的结果，我们阐明了围绕中心路径的区域，牛顿方程与原始双重内点方法相关的方程是唯一可解决的。
We study properties of the central path underlying a nonlinear semidefinite optimization problem, called NSDP for short. The latest radical work on this topic was contributed by Yamashita and Yabe (2012): 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the Lagrange multiplier set is not necessarily a singleton and the nonsingularity of the above-mentioned Jacobian is no longer valid.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.
