Extension of the subgradient extragradient algorithm for solving variational inequalities without monotonicity

Two improved subgradient extragradient algorithms are proposed for solving nonmonotone variational inequalities under the nonempty assumption of the solution set of the dual variational inequalities. First, when the mapping is Lipschitz continuous, we propose an improved subgradient extragradient algorithm with self-adaptive step-size (ISEGS for short). In ISEGS, the next iteration point is obtained by projecting sequentially the current iteration point onto two different half-spaces, and only one projection onto the feasible set is required in the process of constructing the half-spaces per iteration. The self-adaptive technique allows us to determine the step-size without using the Lipschitz constant. Second, we extend our algorithm into the case where the mapping is merely continuous. The Armijo line search approach is used to handle the non-Lipschitz continuity of the mapping. The global convergence of both algorithms is established without monotonicity assumption of the mapping. The computational complexity of the two proposed algorithms is analyzed. Some numerical examples are given to show the efficiency of the new algorithms.