Proximally Constrained Methods for Weakly Convex Optimization with Weakly Convex Constraints

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 are weakly convex. Our methods solve a sequence of strongly convex subproblems, where a proximal 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 Slaters condition, we establish the computation complexities of our methods for finding a nearly stationary point.