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

TL;DR

This paper introduces a simple first-order method for nonsmooth, nonconvex constrained optimization that guarantees convergence to feasible, stationary points without requiring compactness or constraint qualification, applicable even when CQ fails.

Contribution

It provides a novel first-order algorithm with convergence guarantees for weakly convex, nonsmooth constrained problems, handling cases with and without constraint qualification.

Findings

Converges to an $oldsymbol{ ext{ε}}$-stationary point at rate $O( ext{ε}^{-4})$

Works without compactness or constraint qualification assumptions

Demonstrates stable convergence on piecewise quadratic SCAD problems

Abstract

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. As an illustrative example, our method stably converges on piecewise quadratic SCAD regularized problems despite frequent violations of constraint qualification. The considered algorithm is similar to those of "Quadratically regularized subgradient methods for weakly convex optimization with weakly convex constraints" by Ma et al. and "Stochastic first-order methods for convex and nonconvex functional constrained optimization" by Boob et al. (whose guarantees further assume compactness and CQ), iteratively taking inexact proximal steps, computed via an inner loop applying a switching subgradient method to a strongly convex constrained subproblem. Our non-Lipschitz analysis of the switching subgradient method appears to be new and may be of independent interest.