A New Popov’s Method for Solving Quasimonotone Variational Inequalities with Applications in Optimal Control Problems and Machine Learning Algorithms

This study presents an innovative Popov-type iterative method designed to tackle variational inequality problems (VIPs) with a quasimonotone operator in real Hilbert spaces. Unlike conventional techniques, this method eliminates the need for prior knowledge of the Lipschitz constant of the associated operator. Instead, it employs a self-adaptive step size with relaxed parameters at each iteration. Furthermore, the method integrates an inertial technique to boost the convergence rate of the algorithm. We establish the weak convergence of the proposed algorithm under certain mild conditions. The effectiveness and applicability of the method are assessed through preliminary numerical experiments conducted in both finite- and infinite-dimensional Hilbert spaces. Additionally, we extend our investigation to include an extreme learning machine (ELM) for data classification and optimal control problems as two practical applications. The numerical results from both classical and real-world applications validate the efficiency, accuracy, and applicability of the proposed algorithm.