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Strong convergence of three-step relaxed hybrid steepest-descent methods for variational inequalities. (English) Zbl 1158.65047

The paper shows strong convergence of a three-step relaxed hybrid steepest-descent method for solving variational inequalities defined by means of Lipschitzian and strongly monotone operators.

The convergence proof is slightly simpler that the one given by X. P. Ding, Y. C. Lin and J. C. Yao [Appl. Math. Mech., Engl. 28, 1029–1036 (2007; Zbl 1231.49004)] and is obtained under different assumptions on the parameters involved in the iterative procedure, than in the reference mentioned above.

The last part of the paper is devoted to the constrained pseudo-inverse problem which can be formulated as a variational inequality, as shown by H. K. Xu and T. H. Kim [J. Optimization Theory Appl. 119, 185–201 (2003; Zbl 1045.49018)].

Reviewer’s remark: I would like to point out that the authors probably meant the notation “Fix(T)=S b ” in the last section, instead of “F(T)=S b ”.

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