Low Computational Cost Algorithms for Solving Variational Inequalities over the Fixed Point Set

Abstract

The variational inequality problem has many important applications in the fields of signal processing, image processing, optimal control, and many others. In this paper, we discuss several extragradient-like algorithms for solving variational inequalities over the fixed point set of a nonexpansive mapping. The considered methods are based on some existing ones. Our algorithms use dynamic step-sizes, chosen based on information of previous steps and under the assumptions that the involving mapping is pseudomonotone and Lipschitz continuous, the sequence generated by our algorithms converges to the desired solution. Compared with the original extragradient algorithm, the new ones have an advantage: they do not require to compute any projection onto the feasible set. This feature helps to reduce the computational cost of our methods. Moreover, to implement the new algorithms, we do not need to know the Lipschitz constant of the involving mapping. Also, we present some numerical experiments to verify the efficiency of the new algorithms.