An Approximate UV-decomposition Method For Solving A Class Of Nonconvex Nonsmooth Optimization Problems

One of the important branch of nonlinear programming is nonsmooth optimization,and the UV-decomposition theory is a significant way to study the nonsmooth optimization problem,This theory is a new method for studying the two order approximation of convex function,and the effective algorithm of convex optimization problems.In this paper,the application of UV-decomposition theory to a class of the nonsmooth optimization problems,the model problem is as follows: where f（x）,h₁（x,t） and h₂（x） are the convex nonsmooth functions in Rⁿ,X is a compact subset of Rⁿ.This problem is the subproblem of many stochastic optimization problems.Its solving method plays an important role in dealing with stochastic optimization problems.The problem is transformed into a class of unconstrained optimization problems,which are sums of two nonsmooth functions.With the help of smooth convex approximation of a function that we obtain approximation of the objective function,because of the problem can not directly use the UV-decomposition theory.The U-lagrangian,its basic properties and two order approximation of the function are given by using the UV-decomposition theory.And then the UV-decomposition algorithm and the convergence of the algorithm are given.