# Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity

arXiv: https://arxiv.org/abs/2607.08954 · HTML: https://mlsanity.com/paper/2607.08954/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 38 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Builds on (most cited first, 20 of 20)

- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems](https://mlsanity.com/paper/2010.15768/) (2020)
- [Stochastic First-order Methods for Convex and Nonconvex Functional Constrained Optimization](https://mlsanity.com/paper/1908.02734/) (2019)
- [Quadratically Regularized Subgradient Methods for Weakly Convex Optimization with Weakly Convex Constraints](https://mlsanity.com/paper/1908.01871/) (2019)
- [Stochastic Inexact Augmented Lagrangian Method for Nonconvex Expectation Constrained Optimization](https://mlsanity.com/paper/2212.09513/) (2022)
- [Complexity of Proximal augmented Lagrangian for nonconvex optimization with nonlinear equality constraints](https://mlsanity.com/paper/1908.00131/) (2019)
- [First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points](https://mlsanity.com/paper/2212.00927/) (2022)
- [Complexity of Single Loop Algorithms for Nonlinear Programming with Stochastic Objective and Constraints](https://mlsanity.com/paper/2311.00678/) (2023)
- [Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis](https://mlsanity.com/paper/2209.10825/) (2022)
- [First-order methods for constrained convex programming based on linearized augmented Lagrangian function](https://mlsanity.com/paper/1711.08020/) (2017)
- [Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence](https://mlsanity.com/paper/1801.03013/) (2018)
- [A Newton-CG based augmented Lagrangian method for finding a second-order stationary point of nonconvex equality constrained optimization with complexity guarantees](https://mlsanity.com/paper/2301.03139/) (2023)
- [Single-loop Algorithms for Stochastic Non-convex Optimization with Weakly-Convex Constraints](https://mlsanity.com/paper/2504.15243/) (2025)
- [A First-Order Primal-Dual Method for Nonconvex Constrained Optimization Based On the Augmented Lagrangian](https://mlsanity.com/paper/2007.12219/) (2020)
- [Smoothed Proximal Lagrangian Method for Nonlinear Constrained Programs](https://mlsanity.com/paper/2408.15047/) (2024)
- [Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints](https://mlsanity.com/paper/2311.09065/) (2023)
- [Smoothing Meets Perturbation: Unified and Tight Analysis for Nonconvex-Concave Minimax Optimization](https://mlsanity.com/paper/2602.14185/) (2026)
- [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)
- [Primal-Dual Methods for Nonsmooth Nonconvex Optimization with Orthogonality Constraints](https://mlsanity.com/paper/2604.04130/) (2026)

## Built on by (most cited first, 1 of 1)

- [Online Optimization of Difference-of-Convex Compositions with Smooth Mappings](https://mlsanity.com/paper/2607.19553/) (2026)

## Abstract (arXiv)

We study nonasymptotic convergence of primal-dual methods for a class of nonconvex constrained optimization problems under local multiplier regularity. Both the objective and functional inequality constraints are compositions of convex Lipschitz outer functions with smooth nonlinear mappings. Unlike existing analyses requiring regularity globally or along the entire iterate sequence, we impose multiplier regularity only near the feasible set. Using an auxiliary minimax problem with a compact dual set, we derive Lyapunov estimates that identify nearly feasible iterates under feasible initialization and a sufficiently large penalty, without assuming multiplier boundedness. Local regularity then bounds the associated multipliers and makes dual truncation inactive, yielding approximate KKT certificates for the original problem. The smoothed prox-linear augmented Lagrangian method with dual regularization yields an O(K-1/3) KKT residual rate. Without regularization, piecewise linear outer functions and additional local structural assumptions yield a local dual error bound and the sharper O(K-1/2) rate.