  Optimization models with non-convex constraints arise in many tasks in
machine learning, e.g., learning with fairness constraints or Neyman-Pearson
classification with non-convex loss. Although many efficient methods have been
developed with theoretical convergence guarantees for non-convex unconstrained
problems, it remains a challenge to design provably efficient algorithms for
problems with non-convex functional constraints. This paper proposes a class of
subgradient methods for constrained optimization where the objective function
and the constraint functions are weakly convex and nonsmooth. Our methods solve
a sequence of strongly convex subproblems, where a quadratic regularization
term is added to both the objective function and each constraint function. Each
subproblem can be solved by various algorithms for strongly convex
optimization. Under a uniform Slater's condition, we establish the computation
complexities of our methods for finding a nearly stationary point.
