Extragradient Sliding for Composite Non-Monotone Variational Inequalities

Variational inequalities offer a versatile and straightforward approach to analyzing a broad range of equilibrium problems in both theoretical and practical fields. In this paper, we consider a composite generally non-monotone variational inequality represented as a sum of Lq-Lipschitz monotone and Lp-Lipschitz generally non-monotone operators. We applied a special sliding version of the classical Extragradient method to this problem and obtain better convergence results. In particular, to achieve varepsilon-accuracy of the solution, the oracle complexity of the non-monotone operator Q for our algorithm is Oleft(Lp2/varepsilon2right) in contrast to the basic Extragradient algorithm with Oleft((Lp+Lq)2/varepsilon2right). The results of numerical experiments confirm the theoretical findings and show the superiority of the proposed method.