The common solutions to pseudomonotone equilibrium problems

In this paper, we propose two iterative methods for finding a common solution of a finite family of equilibrium problems for pseudomonotone bifunctions. The first is a parallel hybrid extragradient-cutting algorithm which is extended from the previously known one for variational inequalities to equilibrium problems. The second is a new cyclic hybrid extragradient-cutting algorithm. In the cyclic algorithm, using the known techniques, we can perform and develop practical numerical experiments.Comment: 12 pages, accepted for publication in Bull. Iranian Math. Soc. (2015

A shorter proof on recent iterative algorithms constructed by the relaxed (u,v)(u, v)-cocoercive mappings and a similar case for inverse-strongly monotone mappings

In this short note, using the class of the relaxed $(u, v)$-cocoercive mappings and $\alpha$-inverse strongly monotone mappings, we prove that if an important condition holds then we can prove the convergence of the proposed algorithm, more shorter than the original proof.Comment: 8 page

A New Iterative Projection Method for Approximating Fixed Point Problems and Variational Inequality Problems

In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this result, we provide some applications of the considered problem not just giving a pure extension of existing mathematical problems.Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1403.320

Hybrid algorithms without the extra-steps for equilibrium problems

In this paper, we introduce some new hybrid algorithms for finding a solution of a system of equilibrium problems. In these algorithms, by constructing specially cutting-halfspaces, we avoid using the extra-steps as in the extragradient method and the Armijo linesearch method which are inherently costly when the feasible set has a complex structure. The strong convergence of the algorithms is established.Comment: 11 pages (Submitted

A modified subgradient extragradient method for solving the variational inequality problem

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. \cite{CGR}, replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant

A hybrid method without extrapolation step for solving variational inequality problems

In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on two well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in hybrid method. We prove a strong convergence of the sequences generated by our method

Strong convergence with a modified iterative projection method for hierarchical fixed point problems and variational inequalities

This paper deals with a modified iterative projection method for approximating a solution of the hierarchical fixed point problem for a sequene of nearly nonexpansive mappings with respect to a nonexpansive mapping. It is shown that under certain approximate assumptions on the operators and parameters, the modified iterative sequence converges strongly to a common element of the set of the common fixed points of nearly nonexpansive mappings.Also, this point solves some variational inequality. As a special case, this projection method can be used to find the minimum norm solution of the given variational inequality. The results here improve and extend some recent corresponding results of other authors.Comment: 11 page

Golden ratio algorithms with new stepsize rules for variational inequalities

In this paper, we introduce two golden ratio algorithms with new stepsize rules for solving pseudomonotone and Lipschitz variational inequalities in finite dimensional Hilbert spaces. The presented stepsize rules allow the resulting algorithms to work without the prior knowledge of the Lipschitz constant of operator. The first algorithm uses a sequence of stepsizes which is previously chosen, diminishing and non-summable. While the stepsizes in the second one are updated at each iteration and by a simple computation. A special point is that the sequence of stepsizes generated by the second algorithm is separated from zero. The convergence as well as the convergence rate of the proposed algorithms are established under some standard conditions. Also, we give several numerical results to show the behavior of the algorithms in comparisons with other algorithms.Comment: 19 pages, 4 figures (Accepted for publication on April 16, 2019

System of split variational inequality problems

In this paper, we introduce a system of split variational inequality problems in real Hilbert spaces. Using projection method, we propose an iterative algorithm for the system of split variational inequality problems. Further, we prove that the sequence generated by the iterative algorithm converges strongly to a solution of the system of split variational inequality problems. Furthermore, we discuss some consequences of the main result. The iterative algorithms and results presented in this paper generalize, unify and improve the previously known results of this area.Comment: 14page

An explicit iterative method to solve generalized mixed equilibrium problem, variational inequality problem and hierarchical fixed point problem for a nearly nonexpansive mapping

In this paper, we introduce a new iterative method to find a common solution of a generalized mixed equilibrium problem, a variational inequality problem and a hierarchical fixed point problem for a demicontinuous nearly nonexpansive mapping. We prove that the proposed method converges strongly to a common solution of above problems under the suitable conditions. It is also noted that the main theorem is proved without usual demiclosedness condition. Also, under the appropriate assumptions on the control sequences and operators, our iterative method can be reduced to recent methods. So, the results here improve and extend some recent corresponding results given by many other authors.Comment: arXiv admin note: text overlap with arXiv:1403.360