A UNIFIED PROXIMAL GRADIENT METHOD FOR NONCONVEX COMPOSITE OPTIMIZATION WITH EXTRAPOLATION

In this paper, we propose a unified proximal gradient method with extrapolation (UPG-E) to solve a class of nonconvex and nonsmooth composite optimization. UPG-E provides a unified treatment to both convex and nonconvex problems, and adaptively estimates the nonconvexity modulus of the possibly nonconvex component function in the objective function. It is shown that without restarting the extrapolation, UPG-E achieves the optimal convergence rate of the first -order methods for solving convex composite optimization. In the case that the problem is nonconvex, the method performs as a proximal gradient method with extrapolation and guaranteed global convergence. Moreover, a linear convergence rate can be achieved by UPG-E under proper additional regularity assumptions. Our numerical experiments show the performance of UPG-E is very promising compared with other well -established proximal gradient methods in the literature.