An Augmented Lagrangian Approach to Conically Constrained Nonmonotone Variational Inequality Problems

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 [Formula: see text] global rate of convergence where k is the iteration count. By introducing a new gap function, this rate further improves to be [Formula: see text] 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. Funding: L. Zhao and D. Zhu were partially supported by the Major Project of the National Natural Science Foundation of China [Grant 72293582], the National Key R&D Program of China [Grant 2023YFA0915202], and the Fundamental Research Funds for the Central Universities (the Interdisciplinary Program of Shanghai Jiao Tong University) [Grant YG2024QNA36]. L. Zhao was partially supported by the Startup Fund for Young Faculty at SJTU (SFYF at SJTU) [Grant 22X010503839].