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An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems

Okeke Chibueze C., Ugwunnadi Godwin C., Jolaoso Lateef O.

Demonstratio Mathematica

2022

Vol. 55, nr 1

506--527

The purpose of this article is to study and analyse a new extragradient-type algorithm with an inertial extrapolation step for solving split fixed-point problems for demicontractive mapping, equilibrium problem, and pseudomonotone variational inequality problem in real Hilbert spaces. One of the advantages of the proposed algorithm is that a strong convergence result is achieved without a prior estimate of the Lipschitz constant of the cost operator, which is very difficult to find. In addition, the stepsize is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the cost operator. Some numerical experiments are reported to show the performance and behaviour of the sequence generated by our algorithm. The obtained results in this article extend and improve many related recent results in this direction in the literature.

Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications

Pakkaranang Nuttapol, Rehman Habib ur, Kumam Wiyada

Demonstratio Mathematica

2021

Vol. 54, nr 1

280--298

The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.

A non-interior-point algorithm predictor-corrector algorithm for Variational Inequality Problem with equality linear constraints

Hu X., Huang C., Luo A., Chen H.

Przegląd Elektrotechniczny

2013

R. 89, nr 3b

101--103

We present a noninterior-point predictor-corrector algorithm for variational inequality Problem with equality linear constraints based on Chen-Kanzow-Smale smoothing techniques. This method is based upon a modified predictor-corrector interior-point algorithm. It is established the global linear convergence.

W artykule przedstawiono metodę punktów nie-wewnętrznych predykator-korektor, uwzględniający nierówności wariancyjne z liniowym ograniczeniem liniowości. Jego działanie oparto na technice Chen-Kanzow-Smale oraz zmodyfikowanej metodzie punktów wewnętrznych predykator-korektor. Analizie poddano liniową zbieżność algorytmu.

Strong convergence theorems for relatively quasi-nonexpansive mappings, variational inequality problems and systems of generalized mixed equilibrium problems

Shehu Y., Oguguo O.

Journal of Applied Analysis

2012

Vol. 18, nr 1

69-95

In this paper, we construct a new iterative scheme by hybrid methods to approximate a common element in the fixed points set of an infinite family of relatively quasi-nonexpansive mappings, the solutions set of a variational inequality problem and the solutions set of a system of generalized mixed equilibrium problems in a 2-uniformly convex real Banach space which is also uniformly smooth. Then, we prove strong convergence of the scheme to a common element of the three sets. We give several applications of our results in a Banach space. Our results extend many known recent results in the literature.