In this paper we consider a non-monotone (mixed) variational inequality 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, to be called ALAVI in the current paper, for solving a general constrained VI model. Under an assumption, to be 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 above mentioned coherence condition, thus are sufficient to ensure convergence of ALAVI. Under that assumption, we further show that ALAVI has in fact an $o(1/\sqrt{k})$ 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 non-monotone the local convergence rate of ALAVI improves to be linear. Numerical experiments on some randomly generated highly nonlinear and non-monotone VI problems show practical efficacy of the newly proposed method.

本文提出了一种名为ALAVI的增广Lagrangian原始-对偶方法，用于解决带有凸锥约束的非单调（混合）变分不等式模型，并证明了该方法的收敛性和全局收敛速率以及具有单调映射时的加速收敛速率，同时进行了大量的数值实验以证明方法的实用性。

锥约束非单调变分不等式问题的增广Lagrangian方法