An efficient extragradient method for solving variational inequality problems with Pseudo-monotone operators

Abstract

Variational Inequality Problems (VIPs) provide a strong framework for exhibiting equilibrium problems in a variety of disciplines. Global Lipschitz continuity and strong monotonicity are two restrictive assumptions that are frequently used in traditional extragradient methods for solving VIPs, which limit their applicability to solve pseudo-monotone operators. This paper introduces a novel extragradient-type technique that eliminates the need for a global Lipschitz constant. A relaxation parameter that stabilizes the iterative process by taking a convex combination of the current point and a standard projection step is one of the two main innovations included in the new technique. The second innovation is an adaptive line search strategy that dynamically modifies the step size in response to local operator behaviour. We present a thorough convergence analysis that demonstrates the resulting sequence's weak convergence to a VIP solution. The suggested algorithm is more effective and reliable than previous approaches, especially for large-scale issues with sensitive initial circumstances, as shown by numerical experiments on well-known benchmark problems such as Sun's and Kojima-Shindo problems.