Article
Mathematics, Applied
Bui Van Dinh, Hy Duc Manh, Tran Thi Huyen Thanh
Summary: In this paper, a modified projection algorithm is proposed for solving variational inequality problems without the requirement of pseudomonotonicity in the cost mapping. The algorithm does not use embedded projection methods and the linesearch procedure is unnecessary when the cost mapping is Lipschitz. Several numerical examples are provided to illustrate the efficiency of the proposed algorithms.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Lanmei Deng, Rong Hu, Yaping Fang
Summary: Two projection extragradient algorithms are proposed for solving equilibrium problems without monotonicity and Lipschitz-type property in Hilbert spaces. The strategy of embedding a subgradient projection step in the extragradient algorithm and employing an Armijo-linesearch guarantees weak and strong convergence of the generated sequences to a solution of the equilibrium problem, respectively. Numerical experiments demonstrate the efficiency of the algorithms.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Alfredo N. Iusem, Vahid Mohebbi
Summary: This article investigates the vector equilibrium problem in Hadamard manifolds and proposes an extragradient method to solve this problem. Under suitable assumptions, it is proven that the generated sequence converges to a solution of the problem. Moreover, examples of Hadamard manifolds and vector equilibrium problems to which the main result can be applied are provided, along with some numerical experiments.
JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS
(2021)
Article
Mathematics, Applied
Pham Ngoc Anh, Nguyen Van Hong, Aviv Gibali
Summary: In this study, we address a bilevel equilibrium problem defined over the intersection of the fixed points set of demicontractive mappings. We propose an inexact simultaneous projection method for solving the problem and establish its strong convergence under mild and standard conditions. Primary numerical experiments in both finite and infinite dimensional spaces are conducted, comparing the algorithm's performance with related results in existing literature, highlighting its computational and theoretical advantages.
FIXED POINT THEORY
(2023)
Article
Mathematics, Applied
Suthep Suantai, Suparat Kesornprom, Nattawut Pholasa, Yeol Je Cho, Prasit Cholamjiak
Summary: The study introduces a new relaxed projection algorithm for solving the split feasibility problem and proves some weak convergence theorems under certain conditions in the framework of Hilbert spaces. Through numerical examples in signal processing to verify the theoretical analysis results, improvements were made to the corresponding results in the literature.
Article
Operations Research & Management Science
T. O. Alakoya, L. O. Jolaoso, O. T. Mewomo
Summary: This paper studies a classical variational inequality and fixed point problems on a level set of a convex function, proposing a modified algorithm with self-adaptive stepsize, two projections onto half-spaces, strong convergence, and inertial technique for faster convergence. Numerical experiments demonstrate the efficiency of this algorithm compared to existing ones in literature.
Article
Mathematics
Yanlai Song, Omar Bazighifan
Summary: This paper introduces a new iterative algorithm for solving variational inequality problems over the set of solutions to the generalized equilibrium problems in a Hilbert space. The method combines the Tseng's extragradient method, inertial idea, and iterative regularization, with a non-monotonic stepsize rule without line search.
Article
Mathematics, Applied
Vahid Mohebbi
Summary: In this paper, we study the extragradient method for solving vector quasi-equilibrium problems and propose a regularization procedure. We prove the strong convergence of the generated sequences and provide examples and numerical experiments.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Operations Research & Management Science
Kanyanee Saechou, Atid Kangtunyakarn
Summary: This paper investigates the properties of the modification of equilibrium problem (MEP) and the new subgradient extragradient algorithm. The convergence of the algorithm is proved under certain conditions, and it is applied to solve various equilibrium problems.
Article
Mathematics
Thidaporn Seangwattana
Summary: The purpose of this work is to modify an Extragradient method to find a common solution of fixed point, variational inequality, and equilibrium problems in a Hilbert space without assuming the monotonicity of a bifunction. A weak convergence theorem is presented by the proposed method, which can find solutions of various problems without the monotonicity of a bifunction when reducing some mappings in the method.
THAI JOURNAL OF MATHEMATICS
(2021)
Article
Operations Research & Management Science
Huimin He, Jigen Peng, Qinwei Fan
Summary: This paper discusses the split common fixed point problem for demicontractive operators and introduces an iterative viscosity approximation method (VAM) for solving SCFPP. It is shown that under certain conditions, the sequence generated by VAM strongly converges to a solution of SCFPP, which is identified as the unique solution of a variational inequality. The main result of this paper extends and improves upon previous results by Yao et al., Boikanyo, and Cui-Wang.
Article
Mathematics
Victor Amarachi Uzor, Timilehin Opeyemi Alakoya, Oluwatosin Temitope Mewomo
Summary: This paper studies the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We propose a new inertial iterative scheme that combines Tseng's extragradient method with the viscosity method and adaptive step size technique to find a common solution of the investigated problem. We prove strong convergence of our algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, numerical experiments are provided to demonstrate the efficiency of our method compared to existing methods in the literature.
Article
Mathematics, Applied
Qingjie Hu, Liping Zhu, Cuili Chang, Wenqi Zhang
Summary: In this paper, a truncated three-term conjugate gradient method is proposed for nonconvex unconstrained optimization. The method is proven to possess global convergence and efficiency through complexity analysis and numerical results.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Stanislaw Migorski, Changjie Fang, Shengda Zeng
Summary: This paper introduces a new algorithm for solving variational inequalities in Hilbert spaces, proving the strong convergence of the algorithm without the need for the Lipschitz constant of the operator. Additionally, several numerical experiments for the proposed algorithm are presented.
APPLICABLE ANALYSIS
(2021)
Article
Engineering, Multidisciplinary
Grace Nnennaya Ogwo, Timilehin Opeyemi Alakoya, Oluwatosin Temitope Mewomo
Summary: In this paper, the pseudomonotone variational inequality problems with non-Lipschitz operators are studied. An inertial subgradient extragradient method with Halpern technique and Armijo type step size is proposed to approximate the solution in the framework of 2-uniformly convex real Banach spaces. It is proven that the sequence generated by the proposed method converges strongly to the solution under some mild conditions and without the weak sequential continuity condition often assumed by authors. Numerical experiments are provided to compare the proposed method with other existing methods in the literature. The result extends and improves several existing results in this direction.
JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION
(2023)
Article
Operations Research & Management Science
Phan Tu Vuong
Summary: We revisit a dynamical system for solving variational inequalities and prove the global exponential stability of the trajectories under certain assumptions on the considered operator. Numerical examples are provided to confirm the theoretical results. The stability result obtained in this paper improves and complements some recent works.
NETWORKS & SPATIAL ECONOMICS
(2022)
Article
Computer Science, Software Engineering
Masoud Ahookhosh, Ronan M. T. Fleming, Phan T. Vuong
Summary: This paper introduces two globally convergent Levenberg-Marquardt methods for finding zeros of Holder metrically subregular mappings that may have non-isolated zeros. The global convergence and worst-case global complexity of these methods are proved and studied. Encouraging numerical results derived from real-world biological data are reported.
OPTIMIZATION METHODS & SOFTWARE
(2022)
Article
Mathematics, Applied
Duong Viet Thong, Phan Tu Vuong
Summary: The work investigates pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces and proposes two new methods which do not require knowledge of the Lipschitz constant associated with the variational inequality mapping. Under suitable conditions, the proposed algorithms demonstrate weak and strong convergence, with linear convergence achieved under strong pseudomonotonicity and Lipschitz continuity assumptions. Numerical examples in fractional programming and optimal control problems show the potential of the algorithms and compare their performances to related results.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Duong Viet Thong, Aviv Gibali, Phan Tu Vuong
Summary: This paper introduces an explicit proximal method for the generalized variational inequality problem in real Hilbert spaces, with strong convergence and an adaptive step-size rule that eliminates the need for prior knowledge of the Lipschitz constant of the mapping involved. Intensive numerical experiments validate the advantages and applicability of the method.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Operations Research & Management Science
Phan Tu Vuong, Xiaozheng He, Duong Viet Thong
Summary: This study investigates the convergence properties of a projected neural network for solving inverse variational inequalities, demonstrating exponential stability and linear convergence under standard assumptions. By considering applications in the road pricing problem, the effectiveness of the proposed method is illustrated, providing a positive answer to an open question and improving upon recent results in the literature.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
(2021)
Article
Operations Research & Management Science
Duong Viet Thong, Aviv Gibali, Mathias Staudigl, Phan Tu Vuong
Summary: This paper introduces the concept of dynamic user equilibrium (DUE) and related algorithms and toolboxes, proposes new strongly convergent algorithms for computing DUE in the infinite-dimensional space of path flows, and compares them with the numerical solution strategy employed by Friesz and Han on standard test instances.
NETWORKS & SPATIAL ECONOMICS
(2021)
Article
Mathematics, Applied
Le Van Vinh, Van Nam Tran, Phan Tu Vuong
Summary: This paper considers a second-order dynamical system for solving equilibrium problems in Hilbert spaces and proves the existence and uniqueness of strong global solutions under suitable conditions. The exponential convergence of trajectories is established under strong pseudo-monotonicity and Lipschitz-type conditions. Furthermore, a discrete version of the second-order dynamical system is investigated, which leads to a fixed point-type algorithm with inertial effect and relaxation. The linear convergence of this algorithm is proven under suitable parameter conditions. Numerical experiments are conducted to confirm the theoretical results.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Duong Viet Thong, PhanTu Vuong
Summary: This paper investigates extragradient-type algorithms with inertial effect for solving strongly pseudo-monotone variational inequalities and provides a R-linear convergence analysis. The results show that the algorithms can achieve linear convergence without the prior knowledge and with the stepsize bounded from below by a positive number.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Operations Research & Management Science
Shisheng Cui, Uday Shanbhag, Mathias Staudigl, Phan Vuong
Summary: In this paper, we study monotone inclusions on a Hilbert space, where the operator is the sum of a maximal monotone operator and a single-valued monotone, Lipschitz continuous, and expectation-valued operator. Motivated by previous work on relaxed inertial methods, we propose a stochastic extension of the relaxed inertial forward-backward-forward method and show that it produces a sequence that weakly converges to the solution set. We also analyze the convergence rate and complexity of the algorithm under strong monotonicity conditions. Numerical experiments on two-stage games and an overlapping group Lasso problem demonstrate the advantages of our method compared to competitors.
COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
(2022)
Article
Operations Research & Management Science
Duong Viet Thong, Phan Tu Vuong, Pham Ky Anh, Le Dung Muu
Summary: This paper proposes a new projection-type method with inertial extrapolation for solving pseudo-monotone and Lipschitz continuous variational inequalities in Hilbert spaces. The method does not require the knowledge of the Lipschitz constant or the sequential weak continuity of the operator. A self-adaptive procedure is introduced to generate dynamic step-sizes converging to a positive constant. The convergence of the sequence generated by the method to a solution of the considered variational inequality is proved with a nonasymptotic O(1/n) convergence rate.
NETWORKS & SPATIAL ECONOMICS
(2022)
Article
Mathematics, Applied
F. J. Aragon-Artacho, R. Campoy, P. T. Vuong
Summary: The article introduces an extension of the Boosted Difference of Convex functions Algorithm (BDCA) that can be applied to difference of convex functions programs with linear constraints. The study proves that every cluster point of the sequence generated by this algorithm is a Karush-Kuhn-Tucker point of the problem if the feasible set has a Slater point. Additionally, when the objective function is quadratic, the study proves that any sequence generated by the algorithm is bounded and R-linearly (geometrically) convergent. Numerical experiments show that this new extension of BDCA outperforms DCA.
SET-VALUED AND VARIATIONAL ANALYSIS
(2022)
Article
Mathematics, Applied
Thanh Quoc Trinh, Le Van Vinh, Phan Tu Vuong
Summary: We prove the finite convergence of sequences generated by some extragradient-type methods solving variational inequalities under the weakly sharp condition of the solution set. Additionally, we provide estimations for the number of iterations needed to guarantee convergence to a point in the solution set, and prove the optimality of these estimations. Numerical examples are provided to illustrate the theoretical results.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Operations Research & Management Science
Le Thi Thanh Hai, Thanh Quoc Trinh, Phan Tu Vuong
Summary: In this paper, we refine the convergence analysis of Popov's projection algorithm for solving pseudo-monotone variational inequalities in Hilbert spaces. By using a new Lyapunov function, our analysis results in a larger range of stepsize. Moreover, we establish the linear convergence of the sequence generated by Popov's algorithm when the operator is strongly pseudo-monotone and Lipschitz continuous. As a by-product, we extend the range of stepsize in the projected reflected gradient algorithm for solving unconstrained pseudo-monotone variational inequalities.
Article
Mathematics, Applied
Pham Duy Khanh, Le Van Vinh, Phan Tu Vuong
Summary: Under the assumption of error bound, the research establishes the linear convergence rate of a gradient projection method for solving co-coercive variational inequalities. The results unify and improve various findings in variational inequalities, fixed point problems, and convex feasible problems, with numerical experiments conducted to illustrate the theoretical results.
JOURNAL OF NONLINEAR AND VARIATIONAL ANALYSIS
(2021)
Article
Automation & Control Systems
Phan Tu Vuong
Summary: This study considers a second-order dynamical system for solving variational inequalities in Hilbert spaces, proving the existence and uniqueness of strong global solutions under standard conditions. Exponential convergence of trajectories is demonstrated under assumptions of strong pseudo-monotonicity and Lipschitz continuity. A discrete version of the system leads to a relaxed inertial projection algorithm with proven linear convergence under suitable parameter conditions, with potential extension to general monotone inclusion problems discussed, and numerical experiments confirming theoretical results.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
