2025/09/02 by Liao, Feng-Yi, Zheng, Yang · 2 citations
#FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · doi:10.48550/arxiv.2509.02804
We study the problem of minimizing a m-weakly convex and possibly nonsmooth function. Weak convexity provides a broad framework that subsumes convex, smooth, and many composite nonconvex functions. In this work, we propose a proximal descent method, a simple and efficient first-order algorithm that combines the inexact proximal point method with classical convex bundle techniques. Our analysis establishes explicit non-asymptotic convergence rates in terms of (η,ε)-inexact stationarity. In particular, the method finds an (η,ε)-inexact stationary point using at most O ( (\tfrac1η2 + \tfrac1ε) max \\tfrac1η2, \tfrac1ε\ ) function value and subgradient evaluations. Consequently, the algorithm also achieves the best-known complexity of O(1/δ4) for finding an approximate Moreau stationary point with ‖∇ f2m(x)‖≤ δ. A distinctive feature of our method is its automatic adaptivity: with no parameter tuning or algorithmic modification, it accelerates to O(1/δ2) complexity under smoothness and further achieves linear convergence under quadratic growth. Overall, this work bridges convex bundle methods and weakly convex optimization, while providing accelerated guarantees under structural assumptions.