This paper presents a subgradient-based algorithm for constrained nonsmooth
convex optimization that does not require projections onto the feasible set.
While the well-established Frank-Wolfe algorithm and its variants already avoid
projections, they are primarily designed for smooth objective functions. In
contrast, our proposed algorithm can handle nonsmooth problems with general
convex functional inequality constraints. It achieves an $\epsilon$-suboptimal
solution in $\mathcal{O}(\epsilon^{-2})$ iterations, with each iteration
requiring only a single (potentially inexact) Linear Minimization Oracle (LMO)
call and a (possibly inexact) subgradient computation. This performance is
consistent with existing lower bounds. Similar performance is observed when
deterministic subgradients are replaced with stochastic subgradients. In the
special case where there are no functional inequality constraints, our
algorithm competes favorably with a recent nonsmooth projection-free method
designed for constraint-free problems. Our approach utilizes a simple
separation scheme in conjunction with a new Lagrange multiplier update rule.

提出了一种基于次梯度的算法，用于处理具有通用凸不等式约束的非光滑问题，通过使用线性最小化预言机和拉格朗日乘子更新规则，在不需要对可行域进行投影的情况下实现了 ε- 次优解。