A Proximal Descent Method for Minimizing Weakly Convex Optimization
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arXiv
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| Format: | Preprint |
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2025
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| author | Liao, Feng-Yi Zheng, Yang |
| author_facet | Liao, Feng-Yi Zheng, Yang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_02804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Proximal Descent Method for Minimizing Weakly Convex Optimization Liao, Feng-Yi Zheng, Yang Optimization and Control Systems and Control 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. |
| title | A Proximal Descent Method for Minimizing Weakly Convex Optimization |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2509.02804 |