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We consider (stochastic) subgradient methods for strongly convex butpotentially nonsmooth non-Lipschitz optimization. We provide new equivalentdual descriptions (in the style of dual averaging) for the classic subgradientmethod, the proximal subgradient method, and the switching subgradient method.These equivalences enable $O(1/T)$ convergence guarantees in terms of boththeir classic primal gap and a not previously analyzed dual gap for stronglyconvex optimization. Consequently, our theory provides these classic methodswith simple, optimal stopping criteria and optimality certificates at no addedcomputational cost. Our results apply under nearly any stepsize selection andfor a range of non-Lipschitz ill-conditioned problems where the earlyiterations of the subgradient method may diverge exponentially quickly (aphenomenon which, to the best of our knowledge, no prior works address). Evenin the presence of such undesirable behaviors, our theory still ensures andbounds eventual convergence.

Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization