  We prove the first convergence guarantees for a subgradient method minimizing
a generic Lipschitz function over generic Lipschitz inequality constraints. No
smoothness or convexity (or weak convexity) assumptions are made. Instead, we
utilize a sequence of recent advances in Lipschitz unconstrained minimization,
which showed convergence rates of $O(1/\delta\epsilon^3)$ towards reaching a
"Goldstein" stationary point, that is, a point where an average of gradients
sampled at most distance $\delta$ away has size at most $\epsilon$. We
generalize these prior techniques to handle functional constraints, proposing a
subgradient-type method with similar $O(1/\delta\epsilon^3)$ guarantees on
reaching a Goldstein Fritz-John or Goldstein KKT stationary point, depending on
whether a certain Goldstein-style generalization of constraint qualification
holds.
