An Augmented Lagrangian Approach to Conically Constrained Nonmonotone Variational Inequality Problems

• 成果类型: Article
• 署名作者: Zhao, Lei; Zhu, Daoli; Zhang, Shuzhong
• 署名单位: Shanghai Jiao Tong University; Shanghai Jiao Tong University; Shanghai Jiao Tong University; The Chinese University of Hong Kong, Shenzhen; Shenzhen Research Institute of Big Data; University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0167
• 发表日期: 2025
• 关键词: cutting plane method; 1st-order methods; equilibrium; CONVERGENCE; EXISTENCE

摘要

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 root ffififfi 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.