一类非光滑凸优化问题的邻近梯度算法

A proximal gradient method for nonsmooth convex optimization problems

考虑求解目标函数为光滑损失函数与非光滑正则函数之和的凸优化问题的一种基于线搜索的邻近梯度算法及其收敛性分析,证明了在梯度局部Lipschitz连续条件下该算法是R-线性收敛的,并在非光滑部分为稀疏块LASSO正则函数情况下给出了误差界条件成立的证明,得到了线性收敛率。最后,数值实验结果验证了方法的有效性。

A Proximal Gradient Method based on linesearch(L-PGM)and its convergence for solving the convex optimization problems which objective function is the sum of smooth loss function and non-smooth regular function are studied in this paper.Considering the loss function’s gradient is locally Lipschitz continuous in the problems,the R-linear convergence rate of the L-PGM method is proved.Then,focusing on the problems regularized by the sparse group Lasso function,we prove that the error bound holds around the optimal solution set,thus,the linear convergence for solving such problems with the L-PGM method is given.Finally,The preliminary experimental results support our theoretical analysis.