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

arXiv: https://arxiv.org/abs/2212.00927 · HTML: https://mlsanity.com/paper/2212.00927/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2022.

## Built on by (most cited first, 17 of 17)

- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Independent Learning in Constrained Markov Potential Games](https://mlsanity.com/paper/2402.17885/) (2024)
- [Stochastic Methods for AUC Optimization subject to AUC-based Fairness Constraints](https://mlsanity.com/paper/2212.12603/) (2022)
- [A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization](https://mlsanity.com/paper/2501.19214/) (2025)
- [Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints](https://mlsanity.com/paper/2504.15243/) (2025)
- [Formation Shape Control using the Gromov-Wasserstein Metric](https://mlsanity.com/paper/2503.21538/) (2025)
- [Enforcing Fair Predicted Scores on Intervals of Percentiles by Difference-of-Convex Constraints](https://mlsanity.com/paper/2505.12530/) (2025)
- [GBM-based Bregman Proximal Algorithms for Constrained Learning](https://mlsanity.com/paper/2308.10767/) (2023)
- [Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions](https://mlsanity.com/paper/2502.19764/) (2025)
- [ASAT: Adaptive Scoring and Thresholding with Human Feedback for Robust Out-of-Distribution Detection](https://mlsanity.com/paper/2505.02299/) (2025)
- [First-Order Softmax Weighted Switching Gradient Method for Distributed Stochastic Minimax Optimization with Stochastic Constraints](https://mlsanity.com/paper/2603.05774/) (2026)
- [Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity](https://mlsanity.com/paper/2607.08954/) (2026)
- [Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization](https://mlsanity.com/paper/2305.17323/) (2023)
- [Stochastic Momentum Methods for Non-smooth Non-Convex Finite-Sum Coupled Compositional Optimization](https://mlsanity.com/paper/2506.02504/) (2025)
- [Global Optimality for Constrained Exploration via Penalty Regularization](https://mlsanity.com/paper/2604.28144/) (2026)
- [Fitting Unknown Number of Hyperplanes with Manifold Optimization](https://mlsanity.com/paper/2605.28501/) (2026)
- [A proximal subgradient method for nonconvex stochastic optimization under the Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2608.05460/) (2026)

## Abstract (arXiv)

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, ε-stationary point at a convergence rate of O(ε-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.