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 $f+h$ where $h$ is a proper closed convex function, $f$ is a differentiable function on the domain of $h$, and $\nabla f$ is Lipschitz continuous on the domain of $h$. 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 $\nabla f$ or of any global topological properties of $f$. It is shown that the proposed method can obtain an $\varepsilon$-approximate stationary point with iteration complexity bounds that are optimal, up to logarithmic terms over $\varepsilon$, 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.

翻译：本文开发并分析了一种加速近端下降方法，用于寻找非凸复合优化问题的稳定点。目标函数形式为$f+h$，其中$h$为真闭凸函数，$f$为在$h$定义域上可微的函数，且$\nabla f$在$h$定义域上满足Lipschitz连续性。该方法的主要优势在于其"无参数"特性，即无需知晓$\nabla f$的Lipschitz常数或$f$的任何全局拓扑性质。研究表明，所提方法在凸与非凸设定下均可获得$\varepsilon$-近似稳定点，其迭代复杂度界（除关于$\varepsilon$的对数项外）均达到最优。本文还讨论了如何将该方法应用于其他现有优化框架（如极小极大平滑与约束规划的惩罚框架）以构建更专门化的无参数方法。最后，通过数值实验验证了该方法的实际可行性。