Modified resolvent algorithm for general mixed quasi-variational inequalities

It is well known [1] that the general mixed quasi-variational inequalities are equivalent to the implicit resolvent equations. In this paper, we use this alternative formulation to suggest and analyze a modified resolvent algorithm for solving general mixed quasi-variational inequalities. It is shown that the convergence of the new modified method only requires the pseudomonotonicity, which is a weaker condition than monotonicity. Since the general mixed quasi-variational inequalities include the general (mixed) variational inequalities and related optimization problems as special cases, results obtained in this paper continue to hold for these problems. Our results can be viewed as significant extensions of previously known results including those of Noor [1–3] and Solodov and Tseng [4] for various classes of variational inequalities.