A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming

Lyashko, S. I.; Семенов, В. В.

doi:10.1007/978-3-319-42056-1_10

Cited by 63 publications

(48 citation statements)

References 33 publications

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“…Problem (1.1) is also well known as the Ky Fan inequality early studied in [9]. In recent years, this problem has received a lot of attention by many authors, for instance, see [1,2,6,10,11,12,13,14,15,17,23,26,27,28,32]. This interest can be that, as be seen, it unifies all the aforementioned particular problems in a convenient way.…”

Section: Introductionmentioning

confidence: 95%

“…Note that if A : H → H is a Lipschitz continuous operator, i.e., there exists L > 0 such that ||Ax−Ay|| ≤ L||x−y|| for all x, y ∈ H, then the bifunction f (x, y) = Ax, y − x satisfies the Lipschitz-type condition with c 1 = L 2µ and c 2 = Lµ 2 for any µ > 0. The Lipschitz-type condition is often used in establishing the convergence of extragradient-like methods (two-step proximal-like methods) for EPs (see, e.g., [11,13,15,23,32,34]). Let U : H → H be a mapping with the fixed point set F ix(U ).…”

Section: Preliminariesmentioning

confidence: 99%

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Strong Convergence of a New Hybrid Algorithm for Fixed Point Problems and Equilibrium Problems

Hieu

2018

Mathematical Modelling and Analysis

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The paper considers the problem of finding a common solution of a pseudomonotone and Lipschitz-type equilibrium problem and a fixed point problem for a quasi nonexpansive mapping in a Hilbert space. A new hybrid algorithm is introduced for approximating a solution of this problem. The presented algorithm can be considered as a combination of the extragradient method (two-step proximal-like method) and a modified version of the normal Mann iteration. It is well known that the normal Mann iteration has the weak convergence, but in this paper we has obtained the strong convergence of the new algorithm under some mild conditions on parameters. Several numerical experiments are reported to illustrate the convergence of the algorithm and also to show the advantages of it over existing methods.

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Section: Introductionmentioning

confidence: 95%

Section: Preliminariesmentioning

confidence: 99%

Strong Convergence of a New Hybrid Algorithm for Fixed Point Problems and Equilibrium Problems

Hieu

2018

Mathematical Modelling and Analysis

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“…In 2016, Lyashko et al [33] developed an extragradient method for solving pseudomonotone equilibrium problems in a real Hilbert space. It is required to solve two optimization problems on a closed convex set for each next iteration, with a reasonable fixed stepsize depends upon on the Lipschitz-type constants.…”

Section: Introductionmentioning

confidence: 99%

“…It is required to solve two optimization problems on a closed convex set for each next iteration, with a reasonable fixed stepsize depends upon on the Lipschitz-type constants. The superiority of the Lyashko et al [33] method compared to the Tran et al [20] extragradient method is that the value of the bifunction f is to determine only once for each iteration. Inertial-type methods are based on the discrete variant of a second-order dissipative dynamical system.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

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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.

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

Rehman

Dong

Cho

2020

Math Methods in App Sciences

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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.

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A New Two-Step Proximal Algorithm of Solving the Problem of Equilibrium Programming

Cited by 63 publications

References 33 publications

Strong Convergence of a New Hybrid Algorithm for Fixed Point Problems and Equilibrium Problems

Strong Convergence of a New Hybrid Algorithm for Fixed Point Problems and Equilibrium Problems

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

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

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