Extragradient Sliding for Composite Non-Monotone Variational Inequalities

Abstract: 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 $L_q$-Lipschitz monotone and $L_p$-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 $O\left(L_p^2/\varepsilon^2\right)$ in contrast to the basic Extragradient algorithm with $O\left((L_p+L_q)^2/\varepsilon^2\right)$. The results of numerical experiments confirm the theoretical findings and show the superiority of the proposed method.