We consider (stochastic) subgradient methods for strongly convex but potentially nonsmooth non-Lipschitz optimization. We provide new equivalent dual descriptions (in the style of dual averaging) for the classic subgradient method, the proximal subgradient method, and the switching subgradient method. These equivalences enable $O(1/T)$ convergence guarantees in terms of both their classic primal gap and a not previously analyzed dual gap for strongly convex optimization. Consequently, our theory provides these classic methods with simple, optimal stopping criteria and optimality certificates at no added computational cost. Our results apply under nearly any stepsize selection and for a range of non-Lipschitz ill-conditioned problems where the early iterations of the subgradient method may diverge exponentially quickly (a phenomenon which, to the best of our knowledge, no prior works address). Even in the presence of such undesirable behaviors, our theory still ensures and bounds eventual convergence.

本文针对强凸但潜在不光滑非Lipschitz的优化问题，提出了新的等价的对偶描述，使得 $O(1/T)$ 收敛保证适用于几乎任何步长选择和一系列非Lipschitz病态问题，并提供了优化证书。

一些针对强凸优化的次梯度方法的原始-对偶理论