A Smoothed Augmented Lagrangian Method for Mathematical Programs with Vertical Complementarity Constraints
Abstract
We propose a smoothed augmented Lagrangian method (ALM) for solving mathematical programs with vertical complementarity constraints (VCCs). The augmented Lagrangian function involving the VCCs is smoothed by the Lasry--Lions double envelope, which is an extension of the Moreau envelope that enjoys enhanced smoothness properties. This construction yields a smooth unconstrained subproblem within the ALM framework that can be efficiently solved by many fast optimization algorithms. In the limiting setting, we establish global convergence to Mordukhovich- and Clarke-stationary points. For the convergence rate, we show that, under appropriate assumptions, the smoothed ALM achieves Q-linear convergence. We further provide a worst-case iteration complexity analysis for computing an approximate stationary point of the original problem or of an associated feasibility problem. Preliminary numerical experiments demonstrate the effectiveness of the proposed method.
Related
- Augmented Lagrangian Method for Mathematical Programs with Second-Order Cone Complementarity Constraints
- Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity
- ALDER: A Single-Loop Augmented-Lagrangian Damped Extragradient Method for Nonconvex--Concave Minimax Problems with Nonlinear Coupled Constraints
- Augmented Lagrangian methods with telescopic subproblems for MPCCs and cardinality constraints with general constraints: constraint-preserving swaps and B-stationarity
- McADMM: A Multi-Clique Augmented Lagrangian-Based Algorithm for Large-Scale Sparse SDPs with Bound Constraints
BibTeX
@misc{wang2026smoothed,
title = {{A Smoothed Augmented Lagrangian Method for Mathematical Programs with Vertical Complementarity Constraints}},
author = {Jia Wang and Andreas Themelis and Panagiotis Patrinos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18048},
eprint = {2610.18048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18048},
}
RIS
TY - UNPB
TI - A Smoothed Augmented Lagrangian Method for Mathematical Programs with Vertical Complementarity Constraints
AU - Jia Wang
AU - Andreas Themelis
AU - Panagiotis Patrinos
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18048
UR - https://arxiv.org/abs/2610.18048
AB - We propose a smoothed augmented Lagrangian method (ALM) for solving mathematical programs with vertical complementarity constraints (VCCs). The augmented Lagrangian function involving the VCCs is smoothed by the Lasry--Lions double envelope, which is an extension of the Moreau envelope that enjoys enhanced smoothness properties. This construction yields a smooth unconstrained subproblem within the ALM framework that can be efficiently solved by many fast optimization algorithms. In the limiting setting, we establish global convergence to Mordukhovich- and Clarke-stationary points. For the convergence rate, we show that, under appropriate assumptions, the smoothed ALM achieves Q-linear convergence. We further provide a worst-case iteration complexity analysis for computing an approximate stationary point of the original problem or of an associated feasibility problem. Preliminary numerical experiments demonstrate the effectiveness of the proposed method.
ER -