Explicit Extragradient-Like Method with Regularization for Variational Inequalities

In this paper, we introduce and analyze the convergence of a new algorithm for solving a monotone and Lipschitz variational inequality problem in a Hilbert space. The algorithm uses variable stepsizes which are generated over each iteration, based on some previous iterates, and by some cheap computations. Contrary to many known algorithms, the resulting algorithm can be easily implemented without the prior knowledge of Lipschitz contant of operator, and also without any linesearch procedure. Besides, the regularization technique is suitably incorporated in the algorithm to get further convergence. Theorem of strong convergence is established under mild conditions imposed on control parameters. Some experiments are provided to illustrate the numerical behavior of the algorithm in comparison with others.

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