On gradient projection methods for strongly pseudomonotone variational inequalities without Lipschitz continuity

Abstract

In this paper, we propose two new self-adaptive algorithms for solving strongly pseudomonotone variational inequalities. Our algorithms use dynamic step-sizes, chosen based on information of the previous step and their strong convergence is proved without the Lipschitz continuity of the underlying mappings. In contrast to the existing self-adaptive algorithms in Bello Cruz et al. (Comput Optim Appl 46:247–263, 2010), Santos et al. (Comput Appl Math 30:91–107, 2011), which use fast diminishing step-sizes, the new algorithms use arbitrarily slowly diminishing step sizes. This feature helps to speed up our algorithms. Moreover, we provide the worst case complexity bound of the proposed algorithms, which have not been done in the previous works. Some preliminary numerical experiences and comparisons are also reported.