Novel Proximal Point Iterative Algorithms for Generalized Variational Inequalities: Convergence Analysis and Numerical Experiments

In this paper, we examine aspects of general variational inequalities (GVIs) that are similar to fixed-point problems. Special cases of the proposed unique iterative methods are introduced, including the implicit, explicit, and extra-gradient methods of the proximal point approach. The convergence of the derived expression is analyzed under appropriate assumptions to demonstrate the effectiveness of the proposed method. Numerical results are also presented to evaluate the algorithms’ performance in practice. In these experiments, matrix-scaling tests were conducted, and parameter sensitivity analyses were performed; the convergence speed, computational efficiency, robustness, and stability of the experiments were evaluated. From the results, the implicit proximal method seems to give very good results, and convergence is achieved after fewer iterations than other algorithms. Additionally, we examine the non-convergent behavior of other algorithms under varying parameters. Overall, our study validates the theoretical findings and highlights the effectiveness of advanced proximal methods for large-scale GVI problems, while also suggesting directions for future research in this area.