Cost Bifunction Research Articles

Convergence Analysis of New Construction Explicit Methods for Solving Equilibrium Programming and Fixed Point Problems

In this paper, we present improved iterative methods for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. The method is built around two computing phases of a proximal-like mapping with inertial terms. Many such simpler step size rules that do not involve line search are examined, allowing the technique to be enforced more effectively without knowledge of the Lipschitz-type constant of the cost bifunction. When the control parameter conditions are properly defined, the iterative sequences converge weakly on a particular solution to the problem. We provide weak convergence theorems without knowing the Lipschitz-type bifunction constants. A few numerical tests were performed, and the results demonstrated the appropriateness and rapid convergence of the new methods over traditional ones.

Inertial Modification Using Self-Adaptive Subgradient Extragradient Techniques for Equilibrium Programming Applied to Variational Inequalities and Fixed-Point Problems

Equilibrium problems are articulated in a variety of mathematical computing applications, including minimax and numerical programming, saddle-point problems, fixed-point problems, and variational inequalities. In this paper, we introduce improved iterative techniques for evaluating the numerical solution of an equilibrium problem in a Hilbert space with a pseudomonotone and a Lipschitz-type bifunction. These techniques are based on two computing steps of a proximal-like mapping with inertial terms. We investigated two simplified stepsize rules that do not require a line search, allowing the technique to be carried out more successfully without knowledge of the Lipschitz-type constant of the cost bifunction. Once control parameter constraints are put in place, the iterative sequences converge on a particular solution to the problem. We prove strong convergence theorems without knowing the Lipschitz-type bifunction constants. A sequence of numerical tests was performed, and the results confirmed the correctness and speedy convergence of the new techniques over the traditional ones.

Modified proximal-like extragradient methods for two classes of equilibrium problems in Hilbert spaces with applications

The objective of this study is to introduce two new extragradient methods to solve equilibrium problems involving two distinct classes of bifunction. Based on the pseudomonotonicity hypothesis and the specific Lipschitz-type cost bifunction condition, we have shown a weak convergence theorem for the first proposed method. The bifunction is strongly pseudomonotone in the second method, but the step-size rule does not depend on the strongly pseudomonotone constant and Lipschitz-type constants. In contrast, the first convergence result, we prove a strong convergence theorem with the use of a particular class of variable step-size rule sequences. To confirm the validity of the proposed convergence results, we examine two well-known Nash–Cournot equilibrium models for the numerical experiment and show that the competitive advantage of our proposed methods is based on time of execution and number of iterations.

An Accelerated Popov’s Subgradient Extragradient Method for Strongly Pseudomonotone Equilibrium Problems in a Real Hilbert Space With Applications

In this paper, we introduce a subgradient extragradient method to find the numerical solution of strongly pseudomonotone equilibrium problems with the Lipschitz-type condition on a bifunction in a real Hilbert space. The strong convergence theorem for the proposed method is precisely established on the basis of the standard cost bifunction assumptions. The application of our convergence results is also considered in the context of variational inequalities. For numerical analysis, we consider the well-known Nash-Cournot oligopolistic equilibrium model to support our well-established convergence results.

Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results.

Inertial Iterative Self-Adaptive Step Size Extragradient-Like Method for Solving Equilibrium Problems in Real Hilbert Space with Applications

A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this paper, we introduce a modified iterative method to solve equilibrium problems in real Hilbert space. This method can be seen as a modification of the paper titled “A new two-step proximal algorithm of solving the problem of equilibrium programming” by Lyashko et al. (Optimization and its applications in control and data sciences, Springer book pp. 315–325, 2016). A weak convergence result has been proven by considering the mild conditions on the cost bifunction. We have given the application of our results to solve variational inequality problems. A detailed numerical study on the Nash–Cournot electricity equilibrium model and other test problems is considered to verify the convergence result and its performance.

A modified self‐adaptive extragradient method for pseudomonotone equilibrium problem in a real Hilbert space with applications

In this paper, we consider an improvement of the extragradient method to figure out the numerical solution for pseudomonotone equilibrium problems in arbitrary real Hilbert space. A new method is proposed with an inertial scheme and a self adaptive step size rule that is revised on each iteration based on the previous three iterations. The weak convergence of the method is proved by assuming standard cost bifunction assumptions. We also consider the application of our results to solve different kinds of variational inequality problems and a particular class of fixed point problems. For a numerical part, we study the well‐known Nash‐Cournot equilibrium model and other test problems to support our well‐established convergence results and to ensure that our proposed method has a competitive edge over CPU time and a number of iterations.

New extragradient methods for solving equilibrium problems in Banach spaces

In this paper, three new algorithms are proposed for solving a pseudomonotone equilibrium problem with a Lipschitz-type condition in a 2-uniformly convex and uniformly smooth Banach space. The algorithms are constructed around the $$\phi $$ -proximal mapping associated with cost bifunction. The first algorithm is designed with the prior knowledge of the Lipschitz-type constant of bifunction. This means that the Lipschitz-type constant is an input parameter of the algorithm while the next two algorithms are modified such that they can work without any information of the Lipschitz-type constant, and then they can be implemented more easily. Some convergence theorems are proved under mild conditions. Our results extend and enrich existing algorithms for solving equilibrium problem in Banach spaces. The numerical behavior of the new algorithms is also illustrated via several experiments.

A new Popov's subgradient extragradient method for two classes of equilibrium programming in a real Hilbert space

In this paper, we proposed two different methods for solving pseudomonotone and strongly pseudomonotone equilibrium problems. We can examine these methods as an extension and improvement of the Popov's extragradient method. We replaced the second minimization problem onto a closed convex set in the Popov's extragradient method, with a half-space minimization problem that is updated on each iteration and also formulates a useful method for determining the appropriate stepsize on each iteration. The weak convergence theorem of the first method and strong convergence theorem for the second method is well-established based on a standard assumption on a cost bifunction. We also consider various numerical examples to support our well-established convergence results, and we can see that the proposed methods depict a significant improvement in terms of the number of iterations and execution time.

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

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.

A Self-Adaptive Extra-Gradient Methods for a Family of Pseudomonotone Equilibrium Programming with Application in Different Classes of Variational Inequality Problems

The main objective of this article is to propose a new method that would extend Popov’s extragradient method by changing two natural projections with two convex optimization problems. We also show the weak convergence of our designed method by taking mild assumptions on a cost bifunction. The method is evaluating only one value of the bifunction per iteration and it is uses an explicit formula for identifying the appropriate stepsize parameter for each iteration. The variable stepsize is going to be effective for enhancing iterative algorithm performance. The variable stepsize is updating for each iteration based on the previous iterations. After numerical examples, we conclude that the effect of the inertial term and variable stepsize has a significant improvement over the processing time and number of iterations.

Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem

In this paper, we propose a new method, which is set up by incorporating an inertial step with the extragradient method for solving a strongly pseudomonotone equilibrium problems. This method had to comply with a strongly pseudomonotone property and a certain Lipschitz-type condition of a bifunction. A strong convergence result is provided under some mild conditions, and an iterative sequence is accomplished without previous knowledge of the Lipschitz-type constants of a cost bifunction. A sufficient explanation is that the method operates with a slow-moving stepsize sequence that converges to zero and non-summable. For numerical explanations, we analyze a well-known equilibrium model to support our well-established convergence result, and we can see that the proposed method seems to have a significant consistent improvement over the performance of the existing methods.

The Ishikawa Subgradient Extragradient Method for Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

In this paper, we introduce a novel algorithm for finding a common point of the solution set of a class of equilibrium problems involving pseudo-monotone bifunctions and satisfying the Lipschitz-type condition and the set of fixed points of a quasi-nonexpansive in a real Hilbert space. This algorithm can be considered as a combination of the subgradient extragradient method for equilibrium problems and the Ishikawa method for fixed point problems. The strong convergence of the iterates generated by the proposed method is obtained under the main assumptions that the fixed-point mapping is demiclosed at 0 and Lipschitz-type constant of the cost bifunction is unknown. Some numerical examples are implemented to show the computational efficiency of the proposed algorithm.

Modified Popov's explicit iterative algorithms for solving pseudomonotone equilibrium problems

This paper proposes two algorithms that are based on a subgradient and an inertial scheme with the explicit iterative method for solving pseudomonotone equilibrium problems. The weak convergence of both algorithms is well-established under standard assumptions on the cost bifunction. The advantage of these algorithms is that they did not require any line search procedure or any knowledge about bifunction Lipschitz-type constants for step-size evaluation. A practical explanation of this is that they use a sequence of step-size that are revised at each iteration based on some previous iteration. For a numerical experiment, we consider a well-known Nash-Cournot equilibrium model of electricity markets and also other test problems to assist the well-established convergence results and be able to see that our proposed algorithms have a competitive advantage over the time of execution and the number of iterations.

The extragradient algorithm with inertial effects extended to equilibrium problems

In this paper, two algorithms are proposed for a class of pseudomonotone and strongly pseudomonotone equilibrium problems. These algorithms can be viewed as a extension of the paper title, the extragradient algorithm with inertial effects for solving the variational inequality proposed by Dong et al. (Optimization 65:2217–2226, 2016. https://doi.org/10.1080/02331934.2016.1239266). The weak convergence of the first algorithm is well established based on the standard assumption imposed on the cost bifunction. We provide a strong convergence for the second algorithm without knowing the strongly pseudomonoton and the Lipschitz-type constants of cost bifunction. The practical interpretation of a second algorithm is that the algorithm uses a sequence of step sizes that is converging to zero and non-summable. Numerical examples are used to assist the well-established convergence result, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.

Modified two-step extragradient method for solving the pseudomonotone equilibrium programming in a real Hibert space

The purpose of this paper is to come up with an inertial extragradient method for dealing with a class of pseudomonotone equilibrium problems. This method can be a view as an extension of the paper title “A new twostep proximal algorithm of solving the problem of equilibrium programming” by Lyashko and Semenov et al. (Optimization and Its Applications in Control and Data Sciences: 315—325, 2016). The theorem of weak convergence for solutions of the pseudomonotone equilibrium problems is well-established under standard assumptions placed on cost bifunction in the structure of a real Hilbert spaces. For a numerical experiment, we take up a well-known Nash Cournot equilibrium model of electricity markets to support the well-established convergence results and be adequate to see that our proposed algorithms have a competitive superiority over the time of execution and the number of iterations.

Weak convergence of explicit extragradient algorithms for solving equilibirum problems

This paper aims to propose two new algorithms that are developed by implementing inertial and subgradient techniques to solve the problem of pseudomonotone equilibrium problems. The weak convergence of these algorithms is well established based on standard assumptions of a cost bi-function. The advantage of these algorithms was that they did not need a line search procedure or any information on Lipschitz-type bifunction constants for step-size evaluation. A practical explanation for this is that they use a sequence of step-sizes that are updated at each iteration based on some previous iterations. For numerical examples, we discuss two well-known equilibrium models that assist our well-established convergence results, and we see that the suggested algorithm has a competitive advantage over time of execution and the number of iterations.

Strong convergence of inertial algorithms for solving equilibrium problems

In this paper, we introduce several inertial-like algorithms for solving equilibrium problems (EP) in real Hilbert spaces. The algorithms are constructed using the resolvent of the EP associated bifunction and combines the inertial and the Mann-type technique. Under mild and standard conditions imposed on the cost bifunction and control parameters strong convergence of the algorithms is established. We present several numerical examples to illustrate the behavior of our schemes and emphasize their convergence advantages compared with some related methods.

New inertial algorithm for a class of equilibrium problems

The article introduces a new algorithm for solving a class ofequilibrium problems involving strongly pseudomonotone bifunctions with Lipschitz-type condition. We describe how to incorporate the proximal-like regularized technique with inertial effects. The main novelty of the algorithm is that it can be done without previously knowing the information on the strongly pseudomonotone and Lipschitz-type constants of cost bifunction. A reasonable explain for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. Theorem of strong convergence is proved. In the case, when the information on the modulus of strong pseudomonotonicity and Lispchitz-type constant is known,the rate of linear convergence of the algorithm has been established. Several of numerical experiments are performed to illustrate the convergence of the algorithm and also compare it with other algorithms.

The Bruck's ergodic iteration method for the Ky Fan inequality over the fixed point set

ABSTRACTWe introduce a new iteration algorithm for solving the Ky Fan inequality over the fixed point set of a nonexpansive mapping, where the cost bifunction is monotone without Lipschitz-type continuity. The algorithm is based on the idea of the ergodic iteration method for solving multi-valued variational inequality which is proposed by Bruck [On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), pp. 159–164] and the auxiliary problem principle for equilibrium problems P.N. Anh, T.N. Hai, and P.M. Tuan. [On ergodic algorithms for equilibrium problems, J. Glob. Optim. 64 (2016), pp. 179–195]. By choosing suitable regularization parameters, we also present the convergence analysis in detail for the algorithm and give some illustrative examples.