A Simple Adaptive Proximal Gradient Method for Nonconvex Optimization

Abstract. We consider composite nonconvex optimization problems where the objective function consists of a smooth nonconvex term (with Lipschitz-continuous gradient) and a convex (possibly nonsmooth) term. In this paper, we introduce a novel adaptive proximal gradient method (referred to as AdaPGNC) that is both parameter-free and line-search-free for solving this class of problems. It features a simple single-loop structure and achieves the theoretically optimal iteration/gradient evaluation complexity of [Formula: see text] for finding an [Formula: see text]-stationary point. The core innovation lies in designing an adaptive step size strategy that leverages upper and lower curvature estimates. A key technical contribution is the development of a novel Lyapunov function that effectively balances the function value gap and the squared norm of consecutive iterate differences, serving as a central component in our convergence analysis. Preliminary experimental results on several benchmark problems demonstrate that AdaPGNC is competitive against related existing methods.