Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity
Abstract
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.
Related
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BibTeX
@misc{zhu2026nonconvex,
title = {{Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity}},
author = {Linglingzhi Zhu and Jiajin Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08954},
eprint = {2607.08954},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08954},
}
RIS
TY - UNPB
TI - Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity
AU - Linglingzhi Zhu
AU - Jiajin Li
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2607.08954
UR - https://arxiv.org/abs/2607.08954
AB - 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.
ER -