几类非对称矩阵锥分析
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摘要
以非对称矩阵为决策变量的优化问题在实际生产生活中有着广泛的应用.尤其近些年来,引起了诸多专家学者的浓厚兴趣,已成为当今的研究热点.本文针对几种典型的并且应用广泛的非对称矩阵锥的基本性质展开分析与研究.
本文共分为六章.
第1章,介绍本文的研究背景、意义,所用到的预备知识,以及概括本文的主要研究结果.
第2章,我们分析和刻画了非对称半正定矩阵锥(简称NS-psd)的一些基本性质,包括三个方面的内容.第一,非对称半正定矩阵锥的几何性质,我们揭示了NS-psd不属于齐次锥范畴的事实.第二,我们证明了NS-psd是非凸集合P0-矩阵锥的一个极大凸子锥,同时NS-psd的内部(即非对称正定矩阵锥)却不是P。-矩阵锥内部(即P-矩阵锥)的极大凸子锥,并建立了一些判定非对称矩阵半正定性的充分条件和必要条件.最后,我们给出了NS-psd锥上投影算子的一些性质和结果.第3章,我们针对非对称半定最小二乘(简称NSDLS)问题建立了一个正则化的强对偶模型NSDLS司题是对称半定最小二乘(简称SDLS)问题的拓展,它在机器人与自动控制领域有着重要的应用.借助线性锥约束区域的“极小”表示,我们得到NSDLS问题的一个正则化强对偶模型,该模型涉及一个更低维空间上的投影.在此基础上,我们进一步对这个强对偶问题的最优性条件的广义微分性质以及广义Jacobian的非奇异性进行了分析.所有这些理论结果都阐释了通过Lagrangian对偶方法求解NSDLS问题和SDLS问题一样有效.
第4章,二阶锥是一个典型的非多面对称锥,它可以视为半定锥的一个特殊截面,并且在著名的“二阶锥规划”中扮演着最基本的角色.本章,我们通过建立一个边界充分光滑的闭凸集上的投影函数的新性质,以及对称轴加权二阶锥投影函数的表达式及其微分性质,证明了任意二阶锥截面投影函数的强半光滑性,并刻画该投影函数的Clarke-广义Jacobian和方向导数的表达式.
第5章,l1和l∞范数上图锥分别是非对称矩阵核范数与算子范数上图锥的两个特殊截面.在本章中,我们主要给出了l1和l∞范数上图锥投影函数的明晰表达式,并指明该投影函数的强半光滑性.
第6章,我们总结了本文的主要贡献,同时对进一步可能的研究方向进行了展望.
Optimization problems with nonsymmetric matrix variable have wide applications in real world. Especially during these years, it attracts the interest of many famous researchers and is becoming a focus in the field of mathematical programming. In this thesis, we consider several typical nonsymmetric matrix cones which are frequently used in practice, and study their variational properties.
There are six chapters in this thesis.
In Chapter 1, we give a brief introduction to the background, motivation and sig-nificance of studying nonsymmetric matrix cone, summarize the main results of this thesis, and review some preliminaries.
In Chapter 2, we analyze and characterize the cone of nonsymmetric positive semidefinite matrices (NS-psd). Firstly, we study basic properties of the geometry of the NS-psd cone and show that it is a hyperbolic but not homogeneous cone. Secondly, we prove that the NS-psd cone is a maximal convex subcone of Po-matrix cone which is not convex. But the interior of the NS-psd cone is not a maximal convex subcone of P-matrix cone. As byproducts, some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given. Finally, we present some properties of metric projection onto the NS-psd cone.
In Chapter 3, the nonsymmetric semidefinite least squares (NSDLS) problem is to find a nonsymmetric semidefinite matrix which is closest to a given matrix in Frobenius norm. It is an extension of the semidefinite least squares problem (SDLS) and has im-portant application in the area of robotics and automation. By developing the minimal representation of the underlying cone with the linear constraints, we obtain a regular-ized strong duality with low-dimensional projection for NSDLS. Further, we study the generalized differential properties and nonsingularity of the first order optimality sys-tem about the dual problem.
In Chapter 4, second-order cone (SOC) is a typical subclass of non-polyhedral symmetric cones. It can be regarded as a special slice of positive semidefinite matrix cone, and plays a fundamental role in the second-order cone programming. And it's already proven that the metric projection mapping onto SOC is strongly semismooth everywhere. However, whether such property holds for each slice of SOC has not been known yet. In this chapter, by virtue of a new property of projection onto the closed convex set with sufficiently smooth boundary, and the results about projection onto axis-weighted SOC, we give an affirmative answer to this problem. Meanwhile, we also show Clarke's generalized Jacobian and the directional derivative for the projection mapping onto a slice of SOC.
In Chapter 5, the cones of epigraph of weighted l1 and l∞norm are two special slices of the cone of epigraph of weighted nuclear norm and operator norm. In this chapter, we studied the metric projection onto We describe their projection mappings explicitly and show its strong semismoothness.
In Chapter 6, the main contributions of this thesis are concluded and some future research issues are presented.
本文共分为六章.
第1章,介绍本文的研究背景、意义,所用到的预备知识,以及概括本文的主要研究结果.
第2章,我们分析和刻画了非对称半正定矩阵锥(简称NS-psd)的一些基本性质,包括三个方面的内容.第一,非对称半正定矩阵锥的几何性质,我们揭示了NS-psd不属于齐次锥范畴的事实.第二,我们证明了NS-psd是非凸集合P0-矩阵锥的一个极大凸子锥,同时NS-psd的内部(即非对称正定矩阵锥)却不是P。-矩阵锥内部(即P-矩阵锥)的极大凸子锥,并建立了一些判定非对称矩阵半正定性的充分条件和必要条件.最后,我们给出了NS-psd锥上投影算子的一些性质和结果.第3章,我们针对非对称半定最小二乘(简称NSDLS)问题建立了一个正则化的强对偶模型NSDLS司题是对称半定最小二乘(简称SDLS)问题的拓展,它在机器人与自动控制领域有着重要的应用.借助线性锥约束区域的“极小”表示,我们得到NSDLS问题的一个正则化强对偶模型,该模型涉及一个更低维空间上的投影.在此基础上,我们进一步对这个强对偶问题的最优性条件的广义微分性质以及广义Jacobian的非奇异性进行了分析.所有这些理论结果都阐释了通过Lagrangian对偶方法求解NSDLS问题和SDLS问题一样有效.
第4章,二阶锥是一个典型的非多面对称锥,它可以视为半定锥的一个特殊截面,并且在著名的“二阶锥规划”中扮演着最基本的角色.本章,我们通过建立一个边界充分光滑的闭凸集上的投影函数的新性质,以及对称轴加权二阶锥投影函数的表达式及其微分性质,证明了任意二阶锥截面投影函数的强半光滑性,并刻画该投影函数的Clarke-广义Jacobian和方向导数的表达式.
第5章,l1和l∞范数上图锥分别是非对称矩阵核范数与算子范数上图锥的两个特殊截面.在本章中,我们主要给出了l1和l∞范数上图锥投影函数的明晰表达式,并指明该投影函数的强半光滑性.
第6章,我们总结了本文的主要贡献,同时对进一步可能的研究方向进行了展望.
Optimization problems with nonsymmetric matrix variable have wide applications in real world. Especially during these years, it attracts the interest of many famous researchers and is becoming a focus in the field of mathematical programming. In this thesis, we consider several typical nonsymmetric matrix cones which are frequently used in practice, and study their variational properties.
There are six chapters in this thesis.
In Chapter 1, we give a brief introduction to the background, motivation and sig-nificance of studying nonsymmetric matrix cone, summarize the main results of this thesis, and review some preliminaries.
In Chapter 2, we analyze and characterize the cone of nonsymmetric positive semidefinite matrices (NS-psd). Firstly, we study basic properties of the geometry of the NS-psd cone and show that it is a hyperbolic but not homogeneous cone. Secondly, we prove that the NS-psd cone is a maximal convex subcone of Po-matrix cone which is not convex. But the interior of the NS-psd cone is not a maximal convex subcone of P-matrix cone. As byproducts, some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given. Finally, we present some properties of metric projection onto the NS-psd cone.
In Chapter 3, the nonsymmetric semidefinite least squares (NSDLS) problem is to find a nonsymmetric semidefinite matrix which is closest to a given matrix in Frobenius norm. It is an extension of the semidefinite least squares problem (SDLS) and has im-portant application in the area of robotics and automation. By developing the minimal representation of the underlying cone with the linear constraints, we obtain a regular-ized strong duality with low-dimensional projection for NSDLS. Further, we study the generalized differential properties and nonsingularity of the first order optimality sys-tem about the dual problem.
In Chapter 4, second-order cone (SOC) is a typical subclass of non-polyhedral symmetric cones. It can be regarded as a special slice of positive semidefinite matrix cone, and plays a fundamental role in the second-order cone programming. And it's already proven that the metric projection mapping onto SOC is strongly semismooth everywhere. However, whether such property holds for each slice of SOC has not been known yet. In this chapter, by virtue of a new property of projection onto the closed convex set with sufficiently smooth boundary, and the results about projection onto axis-weighted SOC, we give an affirmative answer to this problem. Meanwhile, we also show Clarke's generalized Jacobian and the directional derivative for the projection mapping onto a slice of SOC.
In Chapter 5, the cones of epigraph of weighted l1 and l∞norm are two special slices of the cone of epigraph of weighted nuclear norm and operator norm. In this chapter, we studied the metric projection onto We describe their projection mappings explicitly and show its strong semismoothness.
In Chapter 6, the main contributions of this thesis are concluded and some future research issues are presented.
引文
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