Proximal variable smoothing method for three-composite nonconvex nonsmooth minimization with a linear operator - Liu, Yuncheng; Xia, Fuquan (Numerical algorithms, 2024)
In this paper, we consider a class of three-composite nonconvex nonsmooth optimization problems, where one of nonsmooth functions is further composed with linear operator. Based on the variable smoothing method, as well as first-order methods with suitable majorization techniques, we propose a proximal variable smoothing gradient (ProxVSG) method for solving this kind of problem. The ProxVSG can be implemented efficiently, thanks to the fact that at each iteration, one just separately computes the proximal mapping of each nonsmooth function, rather than that of the sum of these nonsmooth functions. Furthermore, within our broad and flexible analysis framework, we propose a new proximal variable smoothing incremental aggregated gradient (ProxVSIAG) generalizations of the ProxVSG. In ProxVSIAG, an incremental aggregated estimate of the gradient is used, instead of the full gradient. Under suitable assumptions, we prove a complexity of O ( ϵ - 3 ) to achieve an ϵ -approximate solution. Preliminary numerical experiments show the efficiency of the proposed method.