A UV-decomposition Method For A Class Of The Nonconvex Functions

The currently available algorithms to solve the nonsmooth constrained minimization problems by smoothing the nonsmooth function,without taking the special feature certain structural properties of the problem itself into consideration,which possess a certain smoothness(differentiability).A new method for solving convex optimization problems is the UV-decomposition theory,which gets the smooth approximation of the function by means of the smooth properties of the convex function,while an executable algorithm to solve some nonsmooth unconstrained nonlinear programming problems is the Bundle method.Considering the characteristics of them,we can combine these two methods and study a class of functions,the model problem is as follows:f(x):= h1(x)+ h2(x),x ? Rn,where h1(x)is nonsmooth finite-valued convex function,h2(x)is smooth finite-valued function and twice continuously differentiable.Firstly,considering the nonconvexity of the object function,three kinds of UV-space decomposition of the object function will be introduced by using the proximal subdifferential of lower semi-continous function,the U-Lagrange function and some properties will be shown;Moreover,based on the UV-decomposition theory,the UV-decomposition algorithm combining with the Bundle method will be obtained,meanwhile,the convergence will be solved;At last,we use the theory and algorithm to solve the following problem:where x?Rn,?(x)is smooth nonconvex function,hi are nonsmooth finite-valued convex and twice continuously differentiable.