UV-decomposition Theory Of A Class Of The Nonsmooth Joint Functions And Its Applications

This paper is concerned with the joint function which is the sum of the composite function of the maximum eigenvalue function and affine mapping R" to Sn and a finite-valued convex function. Specifically, F(x)=λ1(A(x))+h(x),x∈R" where λ1(·) is the maximum eigenvalue function, A:x→A0+A(x) is an affine mapping. Let A0∈Sn is given and A:Rn→Sn is a linear operator. Sn is the space of n×n real symmetric matrics and h(x) is a nonsmooth finite-valued convex function. The basic idea of the UV-decomposition theory is to decompose R" into two orthogonal subspaces 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. Considering the nondifferentiability of maximum eigenvalue function and h(x), which inevitably gives rise to extreme difficulties in studying the UV-decomposition theory of the function F(x). Firstly, we consider the smooth convex approximations θε(x) to the function λ1{A(x)) to get the approximate function Fε(x) of F(x). Moreover, we gives the UV-space decomposition, the U-Lagrangian and the UV-algorithm which is based on the UV-decomposition theory of Fε(x) to approximately study the UV-decomposition theory of the function F(x). Finally, we apply the theory and algorithm to solve the following problem, where x∈R",λ1(A(x)) is the composite function of λ1(·) and affine mapping A fi:Rn→R are convex and twice continuously differentiable, i.e. fi∈C2.