A Uv-decomposition Method For Minimization Of The Joint Maximum Eigenvalue Functions

With the restraint maximum eigenvalue function optimization question may transform the non-restraint optimization question solution,through certain way as the objective function as the maximum eigenvalue function and a nonsmooth finite-valued convex function.This paper is concerned a kind of maximum eigenvalue function and a nonsmooth finite-valued convex function s UV-decomposition,and obtains solves this kind of optimized question s UV-decomposition algorithm.Considerate the function form is as follows:f(x)=?{A(x))+g(x).where ?(A(x))is the maximum eigenvalue function.A,Rn(?)x?A0 + ?x is an affine:A0 mapping.A0 is a given real n×n symmetric matrix and ? is linear operator from Rn to the space of Rn×n symmetric matrices.g(x)is a nonsmooth finite-valued convex.The basic idea of the UV-decomposition is to decomposition Rn into two orthogonal sub spaces U and V at a nondifferentiable point so that the nonsmoothness of the nonsmooth function is concerned essentially in V and a second-order expansion of the nonsmooth function will be given along some smooth trajectories.Due to in the space decomposition,V is based on the subdifferential generated of(x),to study subdifferential for the function.Considering?(A(x))and g(x)are nonsmooth function.To study f(x)subdifferential dimension will affect the space decomposition.Firstly,we give the smooth convex approximations to the function g(x)to get the approximate function of f(x).Moreover,we give the UV-space decomposition of the approximate function of f(x)and to give to prove,and U-Lagrange,its related to the subdifferential and second-order properties.Finally,our means of UV-decomposition method to get minimization problem solving(P)And the approximateUV-decomposition algorithm.