VU分解的理论可以精确地提取一个非光滑函数在一个不可微点的光滑性和非光滑性进而提取所需的二阶信息以便设计超线性收敛的算法。目前根据VU分解思想目前设计的算法，假设条件太高，bundle机制导致无效步过多等。本项目针对一般的非光滑凸函数，考虑以一个双变量U-拉格朗日函数作为理论工具，通过一个过渡的概念性的分解算法，分析超线性收敛性所需要的条件，从而有针对性地设计可执行地分解算法。同时，采用双LP bundle机制来近似邻近点和U空间的一组基，从而降低bundle机制的计算消耗。针对亚线性函数和光滑映射的复合函数，本项目分析其VU分解和U-拉格朗日函数的理论链式性质，定制可执行的且具有超线性收敛速率的VU分解算法。通过求解复合函数的模型来近似计算邻近点，并且利用常见的亚线性函数的结构给出其U空间的一组基的计算方法。最后，我们将利用所提出的高效算法求解机器学习中出现的复合结构凸优化问题。

The theory of VU decomposition can distract accurately all the smoothness and nonsmoothness of a function at a point where it is not differentiable, and hence might be used to distract useful second-order information so that a super linearly convergent algorithm can be designed. Currently, all the VU-algorithms suffer from two drawbacks: the assumption of super linear convergence is too high and the employed double QP Bundle routine results too many null steps. This project, first for a general nonsmooth convex function considers using a double variable U-Lagrangian as a theoretical tool to design accordingly a class of implementable VU-algorithms. We analyze the required condition for superlinear convergence via a transit conceptual algorithm. Moreover, we utilize a double LP bundle routine to approximate the proximal point and a basis of the U subspace, thus decreasing the computational cost. Then, for a function composed by a sublinear function and a smooth mapping, we analyze its VU decomposition and U-Lagrangian and their chain rules. We then design implementable VU algorithm for such functions in which we use a composed linear model of the composite function to approximate the approximal point and take advantages of the structures of common sublinear functions to derive the methods for computing the basis of their U subspaces. Finally, we apply our efficient algorithm on some composite convex optimization problems in the area of machine learning.