An Augmented Lagrangian Approach to Conically Constrained Nonmonotone Variational Inequality Problems

Lei Zhao (), Daoli Zhu () and Shuzhong Zhang ()
Additional contact information
Lei Zhao: Institute of Translational Medicine and National Center for Translational Medicine, Shanghai Jiao Tong University, Shanghai 200240, China; and Xiangfu Laboratory, Jiashan 314100, China; and Shanghai Artificial Intelligence Research Institute, Shanghai 201109, China
Daoli Zhu: Antai College of Economics and Management, Shanghai Jiao Tong University, Shanghai 200030, China; and School of Data Science, Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen 518172, China
Shuzhong Zhang: Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, Minnesota 55455

Mathematics of Operations Research, 2025, vol. 50, issue 3, 1868-1900

Abstract: In this paper we consider a nonmonotone (mixed) variational inequality (VI) model with (nonlinear) convex conic constraints. Through developing an equivalent Lagrangian function-like primal-dual saddle point system for the VI model in question, we introduce an augmented Lagrangian primal-dual method, called ALAVI (Augmented Lagrangian Approach to Variational Inequality) in the paper, for solving a general constrained VI model. Under an assumption, called the primal-dual variational coherence condition in the paper, we prove the convergence of ALAVI. Next, we show that many existing generalized monotonicity properties are sufficient—though by no means necessary—to imply the abovementioned coherence condition and thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an o ( 1 / k ) global rate of convergence where k is the iteration count. By introducing a new gap function, this rate further improves to be O ( 1 / k ) if the mapping is monotone. Finally, we show that under a metric subregularity condition, even if the VI model may be nonmonotone, the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and nonmonotone VI problems show the practical efficacy of the newly proposed method.

Keywords: 90C33; 65K15; 90C46; 49K35; 90C26; constrained variational inequality problem; nonmonotonicity; augmented Lagrangian function; metric subregularity; iteration complexity analysis (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://dx.doi.org/10.1287/moor.2023.0167 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:3:p:1868-1900

Access Statistics for this article

More articles in Mathematics of Operations Research from INFORMS Contact information at EDIRC.
Bibliographic data for series maintained by Chris Asher ().