Complexity-Optimal and Parameter-Free First-Order Methods for Finding Stationary Points of Composite Optimization Problems

SIAM Journal on Optimization, Volume 34, Issue 3, Page 3005-3032, September 2024.
Abstract. This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form [math], where [math] is a proper closed convex function, [math] is a differentiable function on the domain of [math], and [math] is Lipschitz continuous on the domain of [math]. The main advantage of this method is that it is “parameter-free” in the sense that it does not require knowledge of the Lipschitz constant of [math] or of any global topological properties of [math]. It is shown that the proposed method can obtain an [math]-approximate stationary point with iteration complexity bounds that are optimal, up to logarithmic terms over [math], in both the convex and nonconvex settings. Some discussion is also given about how the proposed method can be leveraged in other existing optimization frameworks, such as min-max smoothing and penalty frameworks for constrained programming, to create more specialized parameter-free methods. Finally, numerical experiments are presented to support the practical viability of the method.