Comparing different subgradient methods for solving convex optimization problems with functional constraints

We consider the problem of minimizing a convex, nonsmooth function subject to a closed convex constraint domain. The methods that we propose are reforms of subgradient methods based on Metel–Takeda’s paper [Optimization Letters 15.4 (2021): 1491-1504] and Boyd’s works [Lecture notes of EE364b, Stanford University, Spring 2013-14, pp. 1-39]. While the former has complexity $\mathcal{O}({\epsilon }^{-2r})$ for all $r>1$, the complexity of the latter is $\mathcal{O}({\epsilon }^{-2})$. We perform some comparisons between these two methods using several test examples.