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15 pages, 294 KiB

Open AccessFeature PaperReview

Approximate Solutions of Variational Inequalities and the Ekeland Principle

by Raffaele Chiappinelli and David E. Edmunds

Mathematics 2025, 13(6), 1016; https://doi.org/10.3390/math13061016 - 20 Mar 2025

Viewed by 95

Abstract

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual

$X^{*}$ . We study the so-called variational inequality problem: given

$y \in X^{*,},$ does [...] Read more.

Let K be a closed convex subset of a real Banach space X, and let F be a map from X to its dual

$X^{*}$ . We study the so-called variational inequality problem: given

$y \in X^{*,},$ does there exist

$x_{0} \in K$ such that (in standard notation)

$⟨F (x_{0}) - y, x - x_{0}⟩ \geq 0$ for all

$x \in K ?$ After a short exposition of work in this area, we establish conditions on F sufficient to ensure a positive answer to the question of whether F is a gradient operator. A novel feature of the proof is the key role played by the well-known Ekeland variational principle. Full article

(This article belongs to the Special Issue Variational Problems and Applications, 3rd Edition)

13 pages, 258 KiB

Open AccessArticle

Proximal Point Methods for Solving Equilibrium Problems in Hadamard Spaces

by Behzad Djafari Rouhani and Vahid Mohebbi

Axioms 2025, 14(2), 127; https://doi.org/10.3390/axioms14020127 - 10 Feb 2025

Viewed by 435

Abstract

We investigate the

$Δ$ -convergence and strong convergence of a sequence generated by the proximal point method for pseudo-monotone equilibrium problems in Hadamard spaces. First, we show the

$Δ$ -convergence of the generated sequence to a solution of the equilibrium problem. Next, we [...] Read more.

We investigate the

$Δ$ -convergence and strong convergence of a sequence generated by the proximal point method for pseudo-monotone equilibrium problems in Hadamard spaces. First, we show the

$Δ$ -convergence of the generated sequence to a solution of the equilibrium problem. Next, we prove the strong convergence of the generated sequence with some additional conditions imposed on the bifunction. Finally, we prove the strong convergence of the generated sequence, by using Halpern’s regularization method, without any additional condition. Full article

13 pages, 278 KiB

Open AccessArticle

Korpelevich Method for Solving Bilevel Variational Inequalities on Riemannian Manifolds

by Jiagen Liao and Zhongping Wan

Axioms 2025, 14(2), 78; https://doi.org/10.3390/axioms14020078 - 22 Jan 2025

Viewed by 502

Abstract

The bilevel variational inequality on Riemannian manifolds refers to a mathematical problem involving the interaction between two levels of optimization, where one level is constrained by the other level. In this context, we present a variant of Korpelevich’s method specifically designed for solving [...] Read more.

The bilevel variational inequality on Riemannian manifolds refers to a mathematical problem involving the interaction between two levels of optimization, where one level is constrained by the other level. In this context, we present a variant of Korpelevich’s method specifically designed for solving bilevel variational inequalities on Riemannian manifolds with nonnegative sectional curvature and pseudomonotone vector fields. This variant aims to find a solution that satisfies certain conditions. Through our proposed algorithm, we are able to generate iteration sequences that converge to a solution, given mild conditions. Finally, we provide an example to demonstrate the effectiveness of our algorithm. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimization Algorithms and Its Applications)

32 pages, 1291 KiB

Open AccessArticle

A Subgradient Extragradient Framework Incorporating a Relaxation and Dual Inertial Technique for Variational Inequalities

by Habib ur Rehman, Kanokwan Sitthithakerngkiet and Thidaporn Seangwattana

Mathematics 2025, 13(1), 133; https://doi.org/10.3390/math13010133 - 31 Dec 2024

Viewed by 659

Abstract

This paper presents an enhanced algorithm designed to solve variational inequality problems that involve a pseudomonotone and Lipschitz continuous operator in real Hilbert spaces. The method integrates a dual inertial extrapolation step, a relaxation step, and the subgradient extragradient technique, resulting in faster [...] Read more.

This paper presents an enhanced algorithm designed to solve variational inequality problems that involve a pseudomonotone and Lipschitz continuous operator in real Hilbert spaces. The method integrates a dual inertial extrapolation step, a relaxation step, and the subgradient extragradient technique, resulting in faster convergence than existing inertia-based subgradient extragradient methods. A key feature of the algorithm is its ability to achieve weak convergence without needing a prior guess of the operator’s Lipschitz constant in the problem. Our method encompasses a range of subgradient extragradient techniques with inertial extrapolation steps as particular cases. Moreover, the inertia in our algorithm is more flexible, chosen from the interval

$[0, 1]$ . We establish R-linear convergence under the added hypothesis of strong pseudomonotonicity and Lipschitz continuity. Numerical findings are presented to showcase the algorithm’s effectiveness, highlighting its computational efficiency and practical relevance. A notable conclusion is that using double inertial extrapolation steps, as opposed to the single step commonly seen in the literature, provides substantial advantages for variational inequalities. Full article

(This article belongs to the Special Issue Current Topics in Optimization, Inequalities and Convex Function Theory)

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22 pages, 764 KiB

Open AccessArticle

An Inertial Subgradient Extragradient Method for Efficiently Solving Fixed-Point and Equilibrium Problems in Infinite Families of Demimetric Mappings

by Habib ur Rehman, Fouzia Amir, Jehad Alzabut and Mohammad Athar Azim

Mathematics 2025, 13(1), 20; https://doi.org/10.3390/math13010020 - 25 Dec 2024

Viewed by 520

Abstract

The primary objective of this article is to enhance the convergence rate of the extragradient method through the careful selection of inertial parameters and the design of a self-adaptive stepsize scheme. We propose an improved version of the extragradient method for approximating a [...] Read more.

The primary objective of this article is to enhance the convergence rate of the extragradient method through the careful selection of inertial parameters and the design of a self-adaptive stepsize scheme. We propose an improved version of the extragradient method for approximating a common solution to pseudomonotone equilibrium and fixed-point problems that involve an infinite family of demimetric mappings in real Hilbert spaces. We establish that the iterative sequences generated by our proposed algorithms converge strongly under suitable conditions. These results substantiate the effectiveness of our approach in achieving convergence, marking a significant advancement in the extragradient method. Furthermore, we present several numerical tests to illustrate the practical efficiency of the proposed method, comparing these results with those from established methods to demonstrate the improved convergence rates and solution accuracy achieved through our approach. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications: 2nd Edition)

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17 pages, 367 KiB

Open AccessArticle

Enhanced Projection Method for the Solution of the System of Nonlinear Equations Under a More General Assumption than Pseudo-Monotonicity and Lipschitz Continuity

by Kanikar Muangchoo and Auwal Bala Abubakar

Mathematics 2024, 12(23), 3734; https://doi.org/10.3390/math12233734 - 27 Nov 2024

Viewed by 655

Abstract

In this manuscript, we propose an efficient algorithm for solving a class of nonlinear operator equations. The algorithm is an improved version of previously established method. The algorithm’s features are as follows: (i) the search direction is bounded and satisfies the sufficient descent [...] Read more.

In this manuscript, we propose an efficient algorithm for solving a class of nonlinear operator equations. The algorithm is an improved version of previously established method. The algorithm’s features are as follows: (i) the search direction is bounded and satisfies the sufficient descent condition; (ii) the global convergence is achieved when the operator is continuous and satisfies a condition weaker than pseudo-monotonicity. Moreover, by comparing it with previously established method the algorithm’s efficiency was shown. The comparison was based on the iteration number required for each algorithm to solve a particular problem and the time taken. Some benchmark test problems, which included monotone and pseudo-monotone problems, were considered for the experiments. Lastly, the algorithm was utilized to solve the logistic regression (prediction) model. Full article

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19 pages, 312 KiB

Open AccessArticle

Modified Double Inertial Extragradient-like Approaches for Convex Bilevel Optimization Problems with VIP and CFPP Constraints

by Yue Zeng, Lu-Chuan Ceng, Liu-Fang Zheng and Xie Wang

Symmetry 2024, 16(10), 1324; https://doi.org/10.3390/sym16101324 - 8 Oct 2024

Viewed by 1035

Abstract

Convex bilevel optimization problems (CBOPs) exhibit a vital impact on the decision-making process under the hierarchical setting when image restoration plays a key role in signal processing and computer vision. In this paper, a modified double inertial extragradient-like approach with a line search [...] Read more.

Convex bilevel optimization problems (CBOPs) exhibit a vital impact on the decision-making process under the hierarchical setting when image restoration plays a key role in signal processing and computer vision. In this paper, a modified double inertial extragradient-like approach with a line search procedure is introduced to tackle the CBOP with constraints of the CFPP and VIP, where the CFPP and VIP represent a common fixed point problem and a variational inequality problem, respectively. The strong convergence analysis of the proposed algorithm is discussed under certain mild assumptions, where it constitutes both sections that possess a mutual symmetry structure to a certain extent. As an application, our proposed algorithm is exploited for treating the image restoration problem, i.e., the LASSO problem with the constraints of fractional programming and fixed-point problems. The illustrative instance highlights the specific advantages and potential infect of the our proposed algorithm over the existing algorithms in the literature, particularly in the domain of image restoration. Full article

(This article belongs to the Special Issue Model-Driven, Data-Driven and Symmetry Methods in Hyperspectral Image Processing)

27 pages, 398 KiB

Open AccessArticle

Mann-Type Inertial Accelerated Subgradient Extragradient Algorithm for Minimum-Norm Solution of Split Equilibrium Problems Induced by Fixed Point Problems in Hilbert Spaces

by Manatchanok Khonchaliew, Kunlanan Khamdam and Narin Petrot

Symmetry 2024, 16(9), 1099; https://doi.org/10.3390/sym16091099 - 23 Aug 2024

Cited by 1 | Viewed by 1496

Abstract

This paper presents the Mann-type inertial accelerated subgradient extragradient algorithm with non-monotonic step sizes for solving the split equilibrium and fixed point problems relating to pseudomonotone and Lipschitz-type continuous bifunctions and nonexpansive mappings in the framework of real Hilbert spaces. By sufficient conditions [...] Read more.

This paper presents the Mann-type inertial accelerated subgradient extragradient algorithm with non-monotonic step sizes for solving the split equilibrium and fixed point problems relating to pseudomonotone and Lipschitz-type continuous bifunctions and nonexpansive mappings in the framework of real Hilbert spaces. By sufficient conditions on the control sequences of the parameters of concern, the strong convergence theorem to support the proposed algorithm, which involves neither prior knowledge of the Lipschitz constants of bifunctions nor the operator norm of the bounded linear operator, is demonstrated. Some numerical experiments are performed to show the efficacy of the proposed algorithm. Full article

(This article belongs to the Section Mathematics)

18 pages, 392 KiB

Open AccessArticle

Method for Approximating Solutions to Equilibrium Problems and Fixed-Point Problems without Some Condition Using Extragradient Algorithm

by Anchalee Sripattanet and Atid Kangtunyakarn

Axioms 2024, 13(8), 525; https://doi.org/10.3390/axioms13080525 - 2 Aug 2024

Viewed by 751

Abstract

The objective of this research is to present a novel approach to enhance the extragradient algorithm’s efficiency for finding an element within a set of fixed points of nonexpansive mapping and the set of solutions for equilibrium problems. Specifically, we focus on applications [...] Read more.

The objective of this research is to present a novel approach to enhance the extragradient algorithm’s efficiency for finding an element within a set of fixed points of nonexpansive mapping and the set of solutions for equilibrium problems. Specifically, we focus on applications involving a pseudomonotone, Lipschitz-type continuous bifunction. Our main contribution lies in establishing a strong convergence theorem for this method, without relying on the assumption of

$lim_{n \to \infty} ∥ x_{n + 1} - x_{n} ∥ = 0$ . Moreover, the main theorem can be applied to effectively solve the combination of variational inequality problem (CVIP). In support of our main result, numerical examples are also presented. Full article

(This article belongs to the Section Mathematical Analysis)

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16 pages, 799 KiB

Open AccessArticle

A Method with Double Inertial Type and Golden Rule Line Search for Solving Variational Inequalities

by Uzoamaka Azuka Ezeafulukwe, Besheng George Akuchu, Godwin Chidi Ugwunnadi and Maggie Aphane

Mathematics 2024, 12(14), 2203; https://doi.org/10.3390/math12142203 - 13 Jul 2024

Viewed by 695

Abstract

In this work, we study a new line-search rule for solving the pseudomonotone variational inequality problem with non-Lipschitz mapping in real Hilbert spaces as well as provide a strong convergence analysis of the sequence generated by our suggested algorithm with double inertial extrapolation [...] Read more.

In this work, we study a new line-search rule for solving the pseudomonotone variational inequality problem with non-Lipschitz mapping in real Hilbert spaces as well as provide a strong convergence analysis of the sequence generated by our suggested algorithm with double inertial extrapolation steps. In order to speed up the convergence of projection and contraction methods with inertial steps for solving variational inequalities, we propose a new approach that combines double inertial extrapolation steps, the modified Mann-type projection and contraction method, and the line-search rule, which is based on the golden ratio

$(\sqrt{5} + 1) / 2$ . We demonstrate the efficiency, robustness, and stability of the suggested algorithm with numerical examples. Full article

(This article belongs to the Section E: Applied Mathematics)

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27 pages, 422 KiB

Open AccessArticle

Multivalued Variational Inequalities with Generalized Fractional Φ-Laplacians

by Vy Khoi Le

Fractal Fract. 2024, 8(6), 324; https://doi.org/10.3390/fractalfract8060324 - 29 May 2024

Viewed by 805

Abstract

In this article, we examine variational inequalities of the form

$\{\begin{matrix} ⟨ A (u), v - u ⟩ + ⟨ F (u), v - u ⟩ \geq 0, \forall v \in K \\ u \in K, \end{matrix}$ , [...] Read more.

In this article, we examine variational inequalities of the form

$\{\begin{matrix} ⟨ A (u), v - u ⟩ + ⟨ F (u), v - u ⟩ \geq 0, \forall v \in K \\ u \in K, \end{matrix}$ , where

$A$ is a generalized fractional

$Φ$ -Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and

$F$ is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term

$F$ such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions. Full article

(This article belongs to the Special Issue Feature Papers for Mathematical Physics Section)

20 pages, 363 KiB

Open AccessArticle

Modified Tseng Method for Solving Pseudomonotone Variational Inequality Problem in Banach Spaces

by Rose Maluleka, Godwin Chidi Ugwunnadi, Maggie Aphane, Hammed A. Abass and Abdul Rahim Khan

Symmetry 2024, 16(3), 363; https://doi.org/10.3390/sym16030363 - 18 Mar 2024

Cited by 1 | Viewed by 1257

Abstract

This article examines the process for solving the fixed-point problem of Bregman strongly nonexpansive mapping as well as the variational inequality problem of the pseudomonotone operator. Within the context of p-uniformly convex real Banach spaces that are also uniformly smooth, we introduce [...] Read more.

This article examines the process for solving the fixed-point problem of Bregman strongly nonexpansive mapping as well as the variational inequality problem of the pseudomonotone operator. Within the context of p-uniformly convex real Banach spaces that are also uniformly smooth, we introduce a modified Halpern iterative technique combined with an inertial approach and Tseng methods for finding a common solution of the fixed-point problem of Bregman strongly nonexpansive mapping and the pseudomonotone variational inequality problem. Using our iterative approach, we develop a strong convergence result for approximating the solution of the aforementioned problems. We also discuss some consequences of our major finding. The results presented in this paper complement and build upon many relevant discoveries in the literature. Full article

(This article belongs to the Section Mathematics)

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21 pages, 399 KiB

Open AccessArticle

Subgradient Extra-Gradient Algorithm for Pseudomonotone Equilibrium Problems and Fixed-Point Problems of Bregman Relatively Nonexpansive Mappings

by Roushanak Lotfikar, Gholamreza Zamani Eskandani, Jong-Kyu Kim and Michael Th. Rassias

Mathematics 2023, 11(23), 4821; https://doi.org/10.3390/math11234821 - 29 Nov 2023

Cited by 1 | Viewed by 1071

Abstract

In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive [...] Read more.

In this article, we introduce a new subgradient extra-gradient algorithm to find the common element of a set of fixed points of a Bregman relatively nonexpansive mapping and the solution set of an equilibrium problem involving a Pseudomonotone and Bregman–Lipschitz-type bifunction in reflexive Banach spaces. The advantage of the algorithm is that it is run without prior knowledge of the Bregman–Lipschitz coefficients. Finally, two numerical experiments are reported to illustrate the efficiency of the proposed algorithm. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

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17 pages, 521 KiB

Open AccessArticle

New Convergence Theorems for Pseudomonotone Variational Inequality on Hadamard Manifolds

by Zhaoli Ma and Lin Wang

Symmetry 2023, 15(11), 2085; https://doi.org/10.3390/sym15112085 - 19 Nov 2023

Viewed by 1253

Abstract

In this paper, we propose an efficient viscosity type subgradient extragradient algorithm for solving pseudomonotone variational inequality on Hadamard manifolds which is of symmetrical characteristic. Under suitable conditions, we obtain the convergence of the iteration sequence generated by the proposed algorithm to a [...] Read more.

In this paper, we propose an efficient viscosity type subgradient extragradient algorithm for solving pseudomonotone variational inequality on Hadamard manifolds which is of symmetrical characteristic. Under suitable conditions, we obtain the convergence of the iteration sequence generated by the proposed algorithm to a solution of a pseudomonotone variational inequality on Hadamard manifolds. We also employ our main result to solve a constrained convex minimization problem and present a numerical experiment to illustrate the asymptotic behavior of the algorithm. Our results develop and improve some recent results. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Nonlinear Analysis, Optimization and Related Topics)

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21 pages, 1237 KiB

Open AccessArticle

Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces

by Rose Maluleka, Godwin Chidi Ugwunnadi and Maggie Aphane

Axioms 2023, 12(10), 960; https://doi.org/10.3390/axioms12100960 - 11 Oct 2023

Cited by 1 | Viewed by 1285

Abstract

In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth [...] Read more.

In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efficiency of our proposed method. Full article

(This article belongs to the Special Issue Research on Fixed Point Theory and Application)

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