Differentials Of Matrix-valued Functions And Their Applications

As an important class of mappings, matrix-valued functions play a key role in the analysis for optimization models of matrix variables. The study on semidefinite programming (SDP), semidefinite complementary programming(SDCP) and matrix cone programming(MCP) can not avoid the discussions about the differentiability of matrix-valued functions. Therefore, it is significant to study the differential properties of matrix-valued function. This dissertation is devoted to the study of differential properties of symmetric matrix-valued functions, singular value functions and the symmetric matrix-valued Fischer-Burmeister function.The main results of this dissertation can be summarized as follows:1. Chapter2first presents the perturbation analysis of eigenvector matrices. From the first-order expansion of the eigenvector matrix, the second-order directional differentiability of an eigenvalue is discussed and the explicit formula, for the second-order directional derivative of an eigenvalue function of symmetric matrix, is demonstrated. And two dif-ferent ways for the expression of the second-order directional derivative of a singular value function are provided:●With the help of a linear mapping, the relationship between the eigenvalue functions of a symmetric matrix and singular value functions of an non-symmetric matrix is established, from which the formula for the second-order directional derivative of a singular value function is derived;●Directly from the perturbation properties of eigenvector matrices in the non-symmetric matrix space and the definition of the second-order directional deriva-tive, the formula for of the second-order directional derivative of a singular value functions is derived.Moreover, the relationship between the second-order directional differentiable properties of the real-valued function f:(?)�'(?) and that of the corresponding matrix-valued function F:Sn�'Sn is established, directly from the definition of the second-order directional derivative and the expressions to the second-order directional derivatives of the eigenvalue functions. Importantly, the explicit expressions of the second-order directional derivatives of the matrix-valued function are demonstrated. 2. In Chapter3, we discuss two applications of the theoretical results in Chapter2. We first apply the formulas of the second-order directional derivatives of symmetric matrix-valued function to the projection operator ΠS+n(·), which is the symmetric matrix-valued function with respect to the real valued function f(x):=max{x,0}, and get the first-order and the second-order directional derivatives of ΠS+n(·). The expressions of tangent cone (?)S-n and the second-order tangent set (?)S-n2are given through their closed connections to the first-order and second-order directional derivatives of ΠS+n(·). These expressions are coincide with those in Bonnans and Shapiro. And then the specific expressions of the tangent cone and the second-order tangent set of the epigraph of nuclear norm are established by the first-order and the second-order directional derivatives of singular values, respectively. Furthermore, the "no gap" second-order optimality conditions of the programming induced by the epigraph of nuclear norm are also proved.3. Chapter5focuses on the study of differential properties of the symmetric matrix-valued Fischer-Burmeister (FB) function. As the main results, the formulas for the directional derivative, the B-subdifferential and the generalized Jacobian of the symmetric matrix-valued FB-function are established.