MDPI - Publisher of Open Access Journals

14 pages, 265 KiB

Open AccessArticle

Existence Results for Some Classes of Weighted Equilibrium Problems

by Miruna-Mihaela Beldiman and Andrei-Dan Halanay

Axioms 2025, 14(4), 316; https://doi.org/10.3390/axioms14040316 - 21 Apr 2025

Viewed by 330

Starting from some systems of vector equilibrium problems, we obtain the existence of the solution for a class of weighted equilibrium problems, under different types of generalized pseudo-monotonicity assumptions. We present both new and previous results, making a connection between them and giving [...] Read more.

Starting from some systems of vector equilibrium problems, we obtain the existence of the solution for a class of weighted equilibrium problems, under different types of generalized pseudo-monotonicity assumptions. We present both new and previous results, making a connection between them and giving a few examples. Using the main theorem, we derive the solution existence for the initial systems and discuss a corresponding set-valued problem. Finally, we consider the case of a real normed space. We extend some previously obtained results from the literature about weighted variational inequalities, and we also give proofs for some results we previously announced. We give some relevant examples for our notions. 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 646

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 858

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)

► Show Figures

Figure 1

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 1142

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)

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 874

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)

► Show Figures

Figure 1

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 796

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)

► Show Figures

Figure 1

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 1372

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)

► Show Figures

Figure 1

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 1388

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)

► Show Figures

Figure 1

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 1423

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)

► Show Figures

Figure 1

27 pages, 400 KiB

Open AccessArticle

Parallel Subgradient-like Extragradient Approaches for Variational Inequality and Fixed-Point Problems with Bregman Relatively Asymptotical Nonexpansivity

by Lu-Chuan Ceng, Yun-Ling Cui, Sheng-Long Cao, Bing Li, Cong-Shan Wang and Hui-Ying Hu

Symmetry 2023, 15(9), 1749; https://doi.org/10.3390/sym15091749 - 12 Sep 2023

Viewed by 1362

Abstract

In a uniformly smooth and p-uniformly convex Banach space, let the pair of variational inequality and fixed-point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed-point problems implicating two uniformly continuous and [...] Read more.

In a uniformly smooth and p-uniformly convex Banach space, let the pair of variational inequality and fixed-point problems (VIFPPs) consist of two variational inequality problems (VIPs) involving two uniformly continuous and pseudomonotone mappings and two fixed-point problems implicating two uniformly continuous and Bregman relatively asymptotically nonexpansive mappings. This article designs two parallel subgradient-like extragradient algorithms with an inertial effect for solving this pair of VIFPPs, where each algorithm consists of two parts which are of a mutually symmetric structure. With the help of suitable registrations, it is proven that the sequences generated by the suggested algorithms converge weakly and strongly to a solution of this pair of VIFPPs, respectively. Lastly, an illustrative instance is presented to verify the implementability and applicability of the suggested approaches. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)

27 pages, 401 KiB

Open AccessArticle

Modified Inertial-Type Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Finite Bregman Relatively Nonexpansive and Demicontractive Mappings

by Cong-Shan Wang, Lu-Chuan Ceng, Bing Li, Sheng-Long Cao, Hui-Ying Hu and Yun-Shui Liang

Axioms 2023, 12(9), 832; https://doi.org/10.3390/axioms12090832 - 28 Aug 2023

Cited by 3 | Viewed by 1199

Abstract

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in [...] Read more.

In this paper, we design two inertial-type subgradient extragradient algorithms with the linear-search process for resolving the two pseudomonotone variational inequality problems (VIPs) of and the common fixed point problem (CFPP) of finite Bregman relatively nonexpansive operators and Bregman relatively demicontractive operators in Banach spaces of both p-uniform convexity and uniform smoothness, which are more general than Hilbert ones. By the aid of suitable restrictions, it is shown that the sequences fabricated by the suggested schemes converge weakly and strongly to a solution of a pair of VIPs with a CFPP constraint, respectively. Additionally, the illustrative instance is furnished to back up the practicability and implementability of the suggested methods. This paper reveals the competitive advantage of the proposed algorithms over the existing algorithms; that is, the existing hybrid projection method for a single VIP with an FPP constraint is extended to develop the modified inertial-type subgradient extragradient method for a pair of VIPs with an CFPP constraint. Full article

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

13 pages, 322 KiB

Open AccessArticle

An Alternated Inertial Projection Algorithm for Multi-Valued Variational Inequality and Fixed Point Problems

by Huan Zhang, Xiaolan Liu, Yan Sun and Ju Hu

Mathematics 2023, 11(8), 1850; https://doi.org/10.3390/math11081850 - 13 Apr 2023

Viewed by 1423

Abstract

In this paper, we propose an alternated inertial projection algorithm for solving multi-valued variational inequality problem and fixed point problem of demi-contractive mapping. On one hand, this algorithm only requires the mapping is pseudo-monotone. On the other hand, this algorithm is combined with [...] Read more.

In this paper, we propose an alternated inertial projection algorithm for solving multi-valued variational inequality problem and fixed point problem of demi-contractive mapping. On one hand, this algorithm only requires the mapping is pseudo-monotone. On the other hand, this algorithm is combined with the alternated inertial method to accelerate the convergence speed. The global convergence of the algorithm can be obtained under mild conditions. Preliminary numerical results show that the convergence speed of our algorithm is faster than some existing algorithms. Full article

(This article belongs to the Special Issue Advances on Nonlinear Functional Analysis)

► Show Figures

Figure 1

26 pages, 591 KiB

Open AccessArticle

A Relaxed Inertial Tseng’s Extragradient Method for Solving Split Variational Inequalities with Multiple Output Sets

by Timilehin Opeyemi Alakoya and Oluwatosin Temitope Mewomo

Mathematics 2023, 11(2), 386; https://doi.org/10.3390/math11020386 - 11 Jan 2023

Cited by 7 | Viewed by 2075

Abstract

Recently, the split inverse problem has received great research attention due to its several applications in diverse fields. In this paper, we study a new class of split inverse problems called the split variational inequality problem with multiple output sets. We propose a [...] Read more.

Recently, the split inverse problem has received great research attention due to its several applications in diverse fields. In this paper, we study a new class of split inverse problems called the split variational inequality problem with multiple output sets. We propose a new Tseng extragradient method, which uses self-adaptive step sizes for approximating the solution to the problem when the cost operators are pseudomonotone and non-Lipschitz in the framework of Hilbert spaces. We point out that while the cost operators are non-Lipschitz, our proposed method does not involve any linesearch procedure for its implementation. Instead, we employ a more efficient self-adaptive step size technique with known parameters. In addition, we employ the relaxation method and the inertial technique to improve the convergence properties of the algorithm. Moreover, under some mild conditions on the control parameters and without the knowledge of the operators’ norm, we prove that the sequence generated by our proposed method converges strongly to a minimum-norm solution to the problem. Finally, we apply our result to study certain classes of optimization problems, and we present several numerical experiments to demonstrate the applicability of our proposed method. Several of the existing results in the literature in this direction could be viewed as special cases of our results in this study. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory and Its Applications)

► Show Figures

Figure 1

15 pages, 300 KiB

Open AccessArticle

Well-Posedness Results of Certain Variational Inequalities

by Savin Treanţă

Mathematics 2022, 10(20), 3809; https://doi.org/10.3390/math10203809 - 15 Oct 2022

Viewed by 2114

Abstract

Well-posedness and generalized well-posedness results are examined for a class of commanded variational inequality problems. In this regard, by using the concepts of hemicontinuity, monotonicity, and pseudomonotonicity of the considered functional, and by introducing the set of approximating solutions of the considered commanded [...] Read more.

Well-posedness and generalized well-posedness results are examined for a class of commanded variational inequality problems. In this regard, by using the concepts of hemicontinuity, monotonicity, and pseudomonotonicity of the considered functional, and by introducing the set of approximating solutions of the considered commanded variational inequality problems, we establish several well-posedness and generalized well-posedness results. Moreover, some illustrative examples are provided to highlight the effectiveness of the results obtained in the paper. Full article

(This article belongs to the Topic Mathematical Modeling)

29 pages, 940 KiB

Open AccessArticle

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

by Habib ur Rehman, Wiyada Kumam and Kamonrat Sombut

Mathematics 2022, 10(10), 1751; https://doi.org/10.3390/math10101751 - 20 May 2022

Cited by 10 | Viewed by 1849

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications)

► Show Figures

Figure 1

Search Results (47)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (47)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI