First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points

Understand

Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings.

• In this paper, we show for any Lipschitz, weakly convex objectives and constraints, a simple first-order method finds a feasible, $\epsilon$-stationary point at a convergence rate of $O(\epsilon^{-4})$ without relying on compactness or Constraint Qualification (CQ).
• When CQ holds, this convergence is measured by approximately satisfying the Karush-Kuhn-Tucker conditions.
• When CQ fails, we guarantee the attainment of weaker Fritz-John conditions.