Analysis On Several Nonsymmetric Matrix Cones

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.