ADMM and Linearized ADMM for Weakly Convex Minimization
Abstract
We study a class of weakly convex optimization problems in which the objective is the sum of a smooth convex term and a weakly convex term that may be nonsmooth. To exploit this structure, we develop a splitting technique based on the alternating direction method of multipliers (ADMM), which decouples the minimization of the two components into tractable subproblems. Because the update associated with the smooth term may require an inner iterative solver, we further linearize this term, yielding a linearized ADMM (LADMM) scheme with an inexpensive one-step update. Under mild conditions, we establish the subsequence convergence of both ADMM and LADMM methods to directional stationary solutions, which are equivalent to critical points and Clarke stationary solutions for our weakly convex problem. Numerical experiments on two low-dimensional test functions and a high-dimensional logarithmic regularized logistic regression model demonstrate that the proposed approaches are computationally efficient and produce solutions of comparable quality to baseline methods.
Related
- On Nonsmooth and Relatively Weakly Convex Minimization
- A Bregman proximal linearized ADMM for fractional programming with nonlinear coupling constraints
- A projected descent method for minimization of weakly semismooth functions over polyhedral sets
- The Method of Ellipcenters for strongly convex minimization
- Exact Theory and Optimal Stepsizes for Stochastic Gradient Descent in Smooth Convex Minimization
BibTeX
@misc{mei2026admm,
title = {{ADMM and Linearized ADMM for Weakly Convex Minimization}},
author = {Shenghan Mei and Chengyu Ke and Yifei Lou and Miju Ahn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12167},
eprint = {2609.12167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12167},
}
RIS
TY - UNPB
TI - ADMM and Linearized ADMM for Weakly Convex Minimization
AU - Shenghan Mei
AU - Chengyu Ke
AU - Yifei Lou
AU - Miju Ahn
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12167
UR - https://arxiv.org/abs/2609.12167
AB - We study a class of weakly convex optimization problems in which the objective is the sum of a smooth convex term and a weakly convex term that may be nonsmooth. To exploit this structure, we develop a splitting technique based on the alternating direction method of multipliers (ADMM), which decouples the minimization of the two components into tractable subproblems. Because the update associated with the smooth term may require an inner iterative solver, we further linearize this term, yielding a linearized ADMM (LADMM) scheme with an inexpensive one-step update. Under mild conditions, we establish the subsequence convergence of both ADMM and LADMM methods to directional stationary solutions, which are equivalent to critical points and Clarke stationary solutions for our weakly convex problem. Numerical experiments on two low-dimensional test functions and a high-dimensional logarithmic regularized logistic regression model demonstrate that the proposed approaches are computationally efficient and produce solutions of comparable quality to baseline methods.
ER -