In this paper we propose a proximal subgradient method (Prox-SubGrad) for solving nonconvex and nonsmooth optimization problems without assuming Lipschitz continuity conditions. A number of subgradient upper bounds and their relationships are presented. By means of these upper bounding conditions, we establish some uniform recursive relations for the Moreau envelopes for weakly convex optimization. This uniform scheme simplifies and unifies the proof schemes to establish rate of convergence for Prox-SubGrad without assuming Lipschitz continuity. We present a novel convergence analysis in this context. Furthermore, we propose some new stochastic subgradient upper bounding conditions and establish convergence and iteration complexity rates for the stochastic subgradient method (Sto-SubGrad) to solve non-Lipschitz and nonsmooth stochastic optimization problems. In particular, for both deterministic and stochastic subgradient methods on weakly convex optimization problems without Lipschitz continuity, under any of the subgradient upper bounding conditions to be introduced in the paper, we show that $O(1/\sqrt{T})$ convergence rate holds in terms of the square of gradient of the Moreau envelope function, which further improves to be $O(1/{T})$ if, in addition, the uniform KL condition with exponent $1/2$ holds.

本文提出了一种用于解决非凸、非光滑优化问题的近端次梯度方法（Prox-SubGrad），并通过建立一些子梯度上界及其关系，简化和统一了收敛速度的证明方案，同时还提出了一些新的随机子梯度上界条件，并为随机子梯度方法（Sto-SubGrad）建立了收敛和迭代复杂度。

针对最小化复合非凸、非光滑和非Lipschitz函数的次梯度方法的统一分析