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Three-steps iterative algorithms for mixed variational inequalities. (English) Zbl 1111.65058

Three-step iterative algorithms for mixed variational inequalities are presented. For finding the approximation solution of various types of variational inequalities and complementarity problems, projection and contraction methods are used. Due to the presence of a nonlinear term in the mixed variational inequality, the projection method and its variant forms can not be applied to suggest iterative algorithms for solving mixed variational inequalities. In the case that the nonlinear term is a proper, convex and lower semicontinuous function, there exists an equivalence between the mixed variational inequalities and the fixed point and the resolvent equations.

Main result: The new iterative method is obtained by using of three steps under suitable conditions. For the new method a proof of global convergence (requires only pseudomonotonicity) is proposed. Numerical experiments show that the authors’ method is more flexible and efficient to solve the traffic equilibrium problem.

MSC:
65K10Optimization techniques (numerical methods)
49J40Variational methods including variational inequalities
49M25Discrete approximations in calculus of variations