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驻点。目标函数形式为$f+h$，其中$h$是正常闭凸函数，$f$是定义在$h$定义域上的可微函数，且$\nabla f$在$h$定义域上满足Lipschitz连续。该方法的主要优势在于其“无参数”特性——无需预先知晓$\nabla f$的Lipschitz常数或$f$的全局拓扑性质。理论证明表明，所提方法能够在凸与非凸场景下，以最优的迭代复杂度（关于$\varepsilon$的对数项内最优）获得$\varepsilon$-近似驻点。本文还探讨了如何将该方法应用于现有优化框架（如极小极大平滑框架及约束规划的惩罚框架）以构建更专用的无参数方法。最后通过数值实验验证了该方法的实际可行性。