【作者】 张昕；

【导师】 陶敏；

【作者基本信息】 南京大学 ， 计算数学， 2020， 硕士

【摘要】 在本文中,我们探讨了针对一类非凸非光滑优化问题的逐次凸差分算法,目标函数由一项光滑函数和有限项非负的非凸非光滑闭函数相加组成,特别地,一些非凸非光滑闭函数还复合了线性映射。这类问题在机器学习、图像处理、矩阵填充等领域都有广泛的应用。针对这类问题我们可以用逐次凸差分算法求解:选取一系列特殊的凸差分的光滑函数序列逐次逼近非凸非光滑函数,再利用非单调邻近梯度法对该近似函数序列逐次求解,从而逼近原目标函数的解。在一定的条件下,可以证明逐次凸差分算法产生的序列有界,且该序列的任何聚点都是原目标函数的稳定点。我们把逐次凸差分算法运用到两个具体的应用问题当中,并与现存的一些算法进行比较,进一步数值验证了该算法的收敛性。更多还原

【Abstract】 In this paper,we study a successive difference-of-convex approximation method for a class of nonsmooth nonconvex problems.The objective is the sum of a smooth function and a finite number of nonnegative proper nonsmooth nonconvex closed func-tions,and some of these functions are composed with linear maps.This kind of prob-lems arises from machine learning,etc.We propose a successive difference-of-convex approximation method to solve it.More specifically,we design a sequence of functions,which can approximate the original nonsmooth nonconvex functions asymptotically.Then,we minimize the resulting approximation function by applying nonmonotone proximal gradient method.Under certain assumptions,the sequence generated by this method is bounded,and any accumulation point of the sequence is a stationary point of the original problem.We apply the proposed method to two specific concrete appli-cations,and compare it with some other state-of-the-art algorithms to verify its conver-gence.更多还原