Descent methods for mixed variational inequalities in a Hilbert space

In this paper, we propose two inertial projection methods to solve mixed variational inequalities without assuming monotonicity on the cost operator. Consequently, we show that the sequence of iterates generated by our proposed methods converge globally to a solution of the mixed variational inequality under the condition that the set of solutions of the dual mixed variational inequality is nonempty. Our convergence results are obtained without assuming any further stringent condition imposed in other related results in the literature. Moreover, our results improve on many important related results in the literature. Some numerical illustrations are given to show the efficiency of our proposed methods.

In this paper, a new projection method for mixed variational inequalities is introduced in Euclidean spaces. The Armijo-type linesearch is similar to that of He’s method for variational inequalities. Under some suitable assumptions, we prove that the sequence generated by the proposed method is globally convergent to a solution of the problem. If, in addition, a certain error bound holds, we analyze the convergence rate of the iterative sequence. The results presented in this paper generalize and improve some known results given in literatures.