We present NCCSG, a nonsmooth optimization method. In each iteration, NCCSG finds the best length-constrained descent direction by considering the worst bound over all local subgradients. NCCSG can take advantage of local smoothness or local strong convexity of the objective function. We prove a few global convergence rates of NCCSG. For well-behaved nonsmooth functions (characterized by the weak smooth property), NCCSG converges in $O(\frac{1}{\epsilon} \log \frac{1}{\epsilon})$ iterations, where $\epsilon$ is the desired optimality gap. For smooth functions and strongly-convex smooth functions, NCCSG achieves the lower bound of convergence rates of blackbox first-order methods, i.e., $O(\frac{1}{\epsilon})$ for smooth functions and $O(\log \frac{1}{\epsilon})$ for strongly-convex smooth functions. The efficiency of NCCSG depends on the efficiency of solving a minimax optimization problem involving the subdifferential of the objective function in each iteration.

翻译：我们提出NCCSG，一种非光滑优化方法。在每次迭代中，NCCSG通过考虑所有局部次梯度上的最坏边界，寻找最优的长度约束下降方向。NCCSG能够利用目标函数的局部光滑性或局部强凸性。我们证明了NCCSG的若干全局收敛速率。对于性质良好的非光滑函数（以弱光滑性质为特征），NCCSG在$O(\frac{1}{\epsilon} \log \frac{1}{\epsilon})$次迭代内收敛，其中$\epsilon$为期望的最优性间隙。对于光滑函数和强凸光滑函数，NCCSG达到了黑盒一阶方法的收敛速率下界，即光滑函数为$O(\frac{1}{\epsilon})$，强凸光滑函数为$O(\log \frac{1}{\epsilon})$。NCCSG的效率取决于每次迭代中求解涉及目标函数次微分的极小极大优化问题的效率。