UV-分解在一类具有锥约束的lower-c^2规划中的应用

Application of UV-Decomposition Theory to the Lower-c^2 Function with the Cone Constrained

在非光滑优化中,函数的二阶性质与展开的理论与应用方面的研究是倍受关注的课题.2000年Lemaréchal,Mifflin,Sagastizábal和Oustry等提出的UV-分解理论,给出了非光滑凸函数f在不可微点的二阶性质的新方法.UV-分解理论的基本思想是将Rn分解为2个正交的子空间U和V的直和,使得原函数在U空间上的一阶逼近是线性的,其不光滑特征集中于V空间中,借助于中间函数(U-Lagrange函数),得到函数在切于U空间的某个光滑轨道上的二阶展式.文中主要是将UV-分解理论推广到一类具有锥约束的非凸函数.使用罚函数的方法,讨论了该罚函数的UV-空间分解结构,并得到该罚函数在光滑轨道上的一阶、二阶性质及其展开式.

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 off in U is linear, and f＇s nonsmoothness near the point is coneen-trated 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.