On a unified implicit method for variational inequalities

摘要

It is well known that variational inequalities are equivalent to fixed point problems. We use this equivalent formulation to suggest and analyze a new unified implicit method of solving variational inequalities. This new method includes several known implicit and extragradient methods as special cases. We prove that the convergence of this new method requires only the pseudomonotonicity of the operator, which is a weaker condition than monotonicity. Several special cases are discussed. An example is given to illustrate the implementation of the proposed method. A comparison with other methods shows that the present method performs better.