Abstract: In nonsmooth optimization,the study concerning the theory and application of second-order analysis of nonsmooth function has drawn much attention.Lemaréchal,Mifflin,Sagastizabal and Oustry(2000) introduced the UV-decomposition theory,which opens a way to define a suitable restricted second-order derivative of a convex function f at nondifferentiable point x.The basic idea is to decompose Rn into two orthogonal subspaces U and V depending on x,so that the first approximation of f in U is linear,and f's nonsmoothness near the point is concentrated essentially in V,and obtain second-order expansions.This paper mainly applies the UV-decomposition theory to a series of nonconvex function.Using the method of penalty function,the paper discusses the definition of the UV-decomposition of the penalty function,and the first-order and second-order expansions of the penalty function.

Key words: nonsmooth, cone constrained, uv-decomposition