We present NCGD, a method for constrained nonsmooth convex optimization. In each iteration, NCGD finds the best norm-constrained descent direction by considering the worst bound over all local subgradients. We prove a few global convergence rates of NCGD. For well-behaved nonsmooth functions (characterized by the weak smoothness property and being Lipschitz continuous), NCGD converges in $O(\epsilon^{-1})$ iterations, where $\epsilon$ is the desired optimality gap. NCGD converges in $O(\epsilon^{-0.5})$ iterations for strongly convex, weakly smooth functions. Furthermore, if the function is strongly convex and smooth, then NCGD achieves linear convergence (i.e., $O(-\log \epsilon)$). The overall efficiency of NCGD depends on the efficiency of solving a minimax optimization problem involving the subdifferential of the objective function in each iteration.

翻译：我们提出了一种用于约束非光滑凸优化的方法——NCGD。在每次迭代中，NCGD通过考虑所有局部次梯度的最坏上界来寻找最优的范数约束下降方向。我们证明了NCGD的若干全局收敛率。对于具有良好性质的非光滑函数（以弱光滑性和Lipschitz连续性为特征），NCGD在$O(\epsilon^{-1})$次迭代内收敛，其中$\epsilon$为期望的最优性间隙。对于强凸弱光滑函数，NCGD在$O(\epsilon^{-0.5})$次迭代内收敛。此外，若函数为强凸且光滑，则NCGD实现线性收敛（即$O(-\log \epsilon)$）。NCGD的整体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。