An Average Curvature Accelerated Composite Gradient Method for Nonconvex Smooth Composite Optimization Problems

Abstract

This paper presents an accelerated composite gradient (ACG) variant, referred to as the AC-ACG method, for solving nonconvex smooth composite minimization problems. As opposed to well-known ACG variants that are based on either a known Lipschitz gradient constant or a sequence of maximum observed curvatures, the current one is based on the average of all past observed curvatures. More specifically, AC-ACG uses a positive multiple of the average of all observed curvatures until the previous iteration as a way to estimate the “function curvature” at the current point and then two resolvent evaluations to compute the next iterate. In contrast to other variable Lipschitz estimation variants, e.g., the ones based on the maximum curvature, AC-ACG always accepts the aforementioned iterate regardless of how poor the Lipschitz estimation turns out to be. Finally, computational results are presented to illustrate the efficiency of AC-ACG on both randomly generated and real-world problem instances.