Pham Ngoc Anh

A hybrid extragradient method extended to fixed point problems and equilibrium problems

In this article, we present a new hybrid extragradient iteration method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of equilibrium problems for a pseudomonotone and Lipschitz-type continuous bifunction. We obtain strongly convergent theorems for the sequences generated by these processes in a real Hilbert space.

Strong Convergence Theorems for Nonexpansive Mappings and Ky Fan Inequalities

We introduce a new iteration method and prove strong convergence theorems for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of monotone and Lipschitz-type continuous Ky Fan inequality. Under certain conditions on parameters, we show that the iteration sequences generated by this method converge strongly to the common element in a real Hilbert space. Some preliminary computational experiences are reported.

On the Contraction and Nonexpansiveness Properties of the Marginal Mappings in Generalized Variational Inequalities Involving Co-Coercive Operators

We investigate the contraction and nonexpansiveness properties of the marginal mappings for gap functions in generalized variational inequalities dealing with strongly monotone and co-coercive operators in a real Hilbert space. We show that one can choose regularization operators such that the solution of a strongly monotone variational inequality can be obtained as the fixed point of a certain contractive mapping. Moreover a solution of a co-coercive variational inequality can be computed by finding a fixed point of a certain nonexpansive mapping. The results give a further analysis for some methods based on the auxiliary problem principle. They also lead to new algorithms for solving generalized variational inequalities involving co-coercive operators. By the Banach contraction mapping principle the convergence rate can be easily established.

On ergodic algorithms for equilibrium problems

In this paper, we present a new iteration method for solving monotone equilibrium problems. This new method is based on the ergodic iteration method Ronald and Bruck in (J Math Anal Appl 61:159–164, 1977) and the auxiliary problem principle Noor in (J Optim Theory Appl 122:371–386, 2004), but it includes the usage of symmetric and positive definite matrices. The proposed algorithm is very simple. Moreover, it simplifies the assumptions necessary in order to converge to the solution. Specifically, whereas previous methods require strong monotonicity and Lipschitz-type continuous conditions, our proposed method only requires weak monotonicity conditions. Applications to the generalized variational inequality problem and some numerical results are reported.

The subgradient extragradient method extended to equilibrium problems

A globally convergent algorithm for equilibrium problems with pseudomonotone bifunctions is proposed. The algorithm is based on the idea of the subgradient extragradient method for solving variational inequalities proposed by Censor et al. [Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148 (2011), 318–335.] and Armijo linesearch techniques. In addition, we give a modified version of our algorithm for finding a common point of the solution set of equilibrium problems and the fixed point set of a nonexpansive mapping. We also analyse the weak convergence of both algorithms in a real Hilbert space.

A Cutting Hyperplane Method for Generalized Monotone Nonlipschitzian Multivalued Variational Inequalities

We present a new method for solving multivalued variational inequalities, where the underlying function is upper semicontinuous and satisfies a certain generalized monotone assumption. First, we construct an appropriate hyperplane which separates the current iterative point from the solution set. Then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. We also analyze the global convergence of the algorithm under minimal assumptions.

A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems

We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions.

Contraction mapping fixed point algorithms for solving multivalued mixed variational inequalities

We show how to choose regularization parameters such that the solution of a multivalued strongly monotone mixed variational inequality can be obtained by computing the fixed point of a certain multivalued mapping having a contraction selection. Moreover a solution of a multivalued cocoercive variational inequality can be computed by finding a fixed point of a certain mapping having nonexpansive selection. By the Banach contraction mapping principle it is easy to establish the convergence rate.

Fixed point methods for pseudomonotone variational inequalities involving strict pseudocontractions

We introduce a new iteration method for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of strict pseudocontractions in a real Hilbert space. The weak convergence of the iterative sequences generated by the method is obtained thanks to improve and extend some recent results under the assumptions that the cost mapping associated with the variational inequality problem only is pseudomonotone and not necessarily inverse strongly monotone. Finally, we present some numerical examples to illustrate the behaviour of the proposed algorithm.

FIXED POINT SOLUTION METHODS FOR SOLVING EQUILIBRIUM PROBLEMS

In this paper, we propose new iteration methods for find- ing a common point of the solution set of a pseudomonotone equilibrium problem and the solution set of a monotone equilibrium problem. The methods are based on both the extragradient-type method and the vis- cosity approximation method. We obtain weak convergence theorems for the sequences generated by these methods in a real Hilbert space.