scholarly journals Convergence Analysis of Self-Adaptive Inertial Extra-Gradient Method for Solving a Family of Pseudomonotone Equilibrium Problems with Application

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1332
Author(s):  
Thanatporn Bantaojai ◽  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Poom Kumam ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Space ◽  
Line Search ◽  
Fixed Point Theorems ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Search Method ◽  
Type Condition ◽  
Variable Stepsize ◽  
Mild Conditions ◽  
Line Search Method

In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous iterations. The advantage of the method is that it operates without prior knowledge of Lipschitz-type constants and any line search method. The weak convergence of the method is established by taking mild conditions on a bifunction. In the context of an application, fixed-point theorems involving strict pseudo-contraction and results for pseudomonotone variational inequalities are considered. Many numerical results have been reported to explain the numerical behavior of the proposed method.

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

2021 ◽  
Vol 54 (1) ◽  
pp. 280-298
Author(s):  
Nuttapol Pakkaranang ◽  
Habib ur Rehman ◽  
Wiyada Kumam
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Type Condition ◽  
Step Size ◽  
Extragradient Methods ◽  
Iterative Schemes ◽  
Mild Conditions

Abstract 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.

scholarly journals A General Inertial Projection-Type Algorithm for Solving Equilibrium Problem in Hilbert Spaces with Applications in Fixed-Point Problems

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 101 ◽  
Author(s):  
Nopparat Wairojjana ◽  
Habib ur Rehman ◽  
Manuel De la Sen ◽  
Nuttapol Pakkaranang
Keyword(s):  
Fixed Point ◽  
Mathematical Programming ◽  
Hilbert Spaces ◽  
Fixed Point Theorems ◽  
Equilibrium Problems ◽  
Type Condition ◽  
Variable Stepsize ◽  
Mild Conditions ◽  
Strict Pseudocontraction ◽  
Type Algorithm

A plethora of applications from mathematical programming, such as minimax, and mathematical programming, penalization, fixed point to mention a few can be framed as equilibrium problems. Most of the techniques for solving such problems involve iterative methods that is why, in this paper, we introduced a new extragradient-like method to solve equilibrium problems in real Hilbert spaces with a Lipschitz-type condition on a bifunction. The advantage of a method is a variable stepsize formula that is updated on each iteration based on the previous iterations. The method also operates without the previous information of the Lipschitz-type constants. The weak convergence of the method is established by taking mild conditions on a bifunction. For application, fixed-point theorems that involve strict pseudocontraction and results for pseudomonotone variational inequalities are studied. We have reported various numerical results to show the numerical behaviour of the proposed method and correlate it with existing ones.

scholarly journals A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1165 ◽  
Author(s):  
Pasakorn Yordsorn ◽  
Poom Kumam ◽  
Habib ur Rehman ◽  
Abdulkarim Hassan Ibrahim
Keyword(s):  
Hilbert Space ◽  
Weak Convergence ◽  
Line Search ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Adaptive Method ◽  
Type Condition ◽  
Computational Performance ◽  
Weak Convergence Theorem

In this paper, we presented a modification of the extragradient method to solve pseudomonotone equilibrium problems involving the Lipschitz-type condition in a real Hilbert space. The method uses an inertial effect and a formula for stepsize evaluation, that is updated for each iteration based on some previous iterations. The key advantage of the algorithm is that it is achieved without previous knowledge of the Lipschitz-type constants and also without any line search procedure. A weak convergence theorem for the proposed method is well established by assuming mild cost bifunction conditions. Many numerical experiments are presented to explain the computational performance of the method and to equate them with others.

scholarly journals Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 76
Author(s):  
Chainarong Khanpanuk ◽  
Nuttapol Pakkaranang ◽  
Nopparat Wairojjana ◽  
Nattawut Pholasa
Keyword(s):  
Hilbert Space ◽  
Iterative Method ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Inertial Term ◽  
Mild Conditions ◽  
Lipschitz Constants

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature.

scholarly journals A randomized adaptive trust region line search method

2020 ◽  
Vol 10 (2) ◽  
pp. 259-263
Author(s):  
Saman Babaie-Kafaki ◽  
Saeed Rezaee
Keyword(s):  
Simulated Annealing ◽  
Numerical Experiments ◽  
Line Search ◽  
Optimization Problems ◽  
Trust Region ◽  
Search Method ◽  
Test Problems ◽  
Unconstrained Optimization Problems ◽  
Line Search Method ◽  
Region Line

Hybridizing the trust region, line search and simulated annealing methods, we develop a heuristic algorithm for solving unconstrained optimization problems. We make some numerical experiments on a set of CUTEr test problems to investigate efficiency of the suggested algorithm. The results show that the algorithm is practically promising.

scholarly journals Strong Convergence Theorems for a Pair of Strictly Pseudononspreading Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bin-Chao Deng ◽  
Tong Chen
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Real Hilbert Space ◽  
Convergence Theorems ◽  
Mild Conditions ◽  
Strongly Monotone

LetHbe a real Hilbert space. LetT1,T2:H→Hbek1-,k2-strictly pseudononspreading mappings; letαnandβnbe two real sequences in (0,1). For givenx0∈H, the sequencexnis generated iteratively byxn+1=βnxn+1-βnTw1αnγfxn+I-μαnBTw2xn,∀n∈N, whereTwi=1−wiI+wiTiwithi=1,2andB:H→His strongly monotone and Lipschitzian. Under some mild conditions on parametersαnandβn, we prove that the sequencexnconverges strongly to the setFixT1∩FixT2of fixed points of a pair of strictly pseudononspreading mappingsT1andT2.

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