一类非凸非光滑优化问题的近似uv-分解方法

Approximate uv-decomposition Method for a Class of Nonconvex Nonsmooth Optimization Problems

uv-分解理论是侧重于非光滑函数的光滑信息来研究凸函数的二阶近似,从而得到凸优化问题有效算法的一种新方法.应用uv-分解理论研究一类非光滑优化问题,此问题作为许多随机优化问题的子问题,它的求解方法对处理随机优化问题有重要作用.将所研究的问题适当地转化为一类由两个非光滑函数的和的无约束优化问题,由于无法直接利用uv-分解理论,所以借助其中一个函数的光滑凸近似,得到了目标函数的近似函数.应用uv-分解理论给出该函数的U-lagrangian函数及其基本性质,目标函数的二阶近似,进而给出了求解原问题的近似uv-分解算法以及算法的收敛性证明.

The uv-decomposition theory was a new method for studying the two order approximation of convex functions,and the effective algorithm of convex optimization problems. The application of uv-decomposition theory to a class of the nonsmooth optimization problems was studied,which was the subproblem of many stochastic optimization problems. Its solution method played an important role in dealing with stochastic optimization problems. The problem was transformed into a class of unconstrained optimization problems,which were sums of two nonsmooth functions. With the help of smooth convex approximation of a function that we obtained approximation of the objective function,because of the problem could not directly use the uvdecomposition theory. The u-laragrangian function,its basic properties and two order approximation of the function were given by using the uv-decomposition theory. And then the uv-decomposition algorithm and the convergence of the algorithm were given.