We present the Trust Region Adversarial Functional Subdifferential (TRAFS) algorithm for constrained optimization of nonsmooth convex Lipschitz functions. Unlike previous methods that assume a subgradient oracle model, we work with the functional subdifferential defined as a set of subgradients that simultaneously captures sufficient local information for effective minimization while being easy to compute for a wide range of functions. In each iteration, TRAFS finds the best step vector in an $\ell_2$-bounded trust region by considering the worst bound given by the functional subdifferential. TRAFS finds an approximate solution with an absolute error up to $\epsilon$ in $\mathcal{O}\left( \epsilon^{-1}\right)$ or $\mathcal{O}\left(\epsilon^{-0.5} \right)$ iterations depending on whether the objective function is strongly convex, compared to the previously best-known bounds of $\mathcal{O}\left(\epsilon^{-2}\right)$ and $\mathcal{O}\left(\epsilon^{-1}\right)$ in these settings. TRAFS makes faster progress if the functional subdifferential satisfies a locally quadratic property; as a corollary, TRAFS achieves linear convergence (i.e., $\mathcal{O}\left(\log \epsilon^{-1}\right)$) for strongly convex smooth functions. In the numerical experiments, TRAFS is on average 39.1x faster and solves twice as many problems compared to the second-best method.

翻译：我们提出了信赖域对抗泛函次微分（TRAFS）算法，用于非光滑凸Lipschitz函数的约束优化。与以往假设次梯度预言机模型的方法不同，我们采用泛函次微分的定义，即一组同时捕获足够局部信息以进行有效最小化、且对广泛函数易于计算的次梯度集合。在每次迭代中，TRAFS通过考虑泛函次微分给出的最坏界，在$\ell_2$有界信赖域内寻找最佳步向量。根据目标函数是否为强凸函数，TRAFS能在$\mathcal{O}\left( \epsilon^{-1}\right)$或$\mathcal{O}\left(\epsilon^{-0.5} \right)$次迭代内找到绝对误差不超过$\epsilon$的近似解，而此前这些设置下的最佳已知界分别为$\mathcal{O}\left(\epsilon^{-2}\right)$和$\mathcal{O}\left(\epsilon^{-1}\right)$。若泛函次微分满足局部二次性质，TRAFS的收敛速度更快；作为推论，对于强凸光滑函数，TRAFS可实现线性收敛（即$\mathcal{O}\left(\log \epsilon^{-1}\right)$）。在数值实验中，TRAFS平均速度比次优方法快39.1倍，且能解决两倍数量的问题。