{"ID":2875626,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.02804","arxiv_id":"2509.02804","title":"A Proximal Descent Method for Minimizing Weakly Convex Optimization","abstract":"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 $\\textit{proximal descent method}$, 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 $\\mathcal{O}\\!\\left( \\Big(\\tfrac{1}{η^2} + \\tfrac{1}ε\\Big) \\max\\!\\left\\{\\tfrac{1}{η^2}, \\tfrac{1}ε\\right\\} \\right)$ function value and subgradient evaluations. Consequently, the algorithm also achieves the best-known complexity of $\\mathcal{O}(1/δ^4)$ for finding an approximate Moreau stationary point with $\\|\\nabla f_{2m}(x)\\|\\leq δ$. A distinctive feature of our method is its \\emph{automatic adaptivity}: with no parameter tuning or algorithmic modification, it accelerates to $\\mathcal{O}(1/δ^2)$ 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.","short_abstract":"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 $\\textit{proximal descent method}$, a simple and efficient first-order algorithm that combine...","url_abs":"https://arxiv.org/abs/2509.02804","url_pdf":"https://arxiv.org/pdf/2509.02804v1","authors":"[\"Feng-Yi Liao\",\"Yang Zheng\"]","published":"2025-09-02T20:15:47Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"eess.SY\"]","methods":"[]","has_code":false}