Article
Operations Research & Management Science
Qamrul Hasan Ansari, Feeroz Babu
Summary: This paper discusses the proximal point algorithm for finding a singularity of the sum of a single-valued vector field and a set-valued vector field on Hadamard manifolds, with a focus on the convergence analysis of the proposed algorithm. Applications to composite minimization problems and variational inequality problems are also presented.
OPTIMIZATION LETTERS
(2021)
Article
Mathematics, Applied
Shih-Sen Chang, Jen-Chih Yao, M. Liu, L. C. Zhao
Summary: This paper considers the inertial proximal point algorithm for finding zero points of variational inclusions on Hadamard manifolds. It is proved that the sequence generated by the algorithm converges to an element of the zero points set under suitable conditions. As applications, the results are utilized to study the minimization problem and saddle point problem in the setting of Hadamard manifolds.
APPLICABLE ANALYSIS
(2023)
Article
Operations Research & Management Science
Konrawut Khammahawong, Poom Kumam, Parin Chaipunya, Juan Martinez-Moreno
Summary: This article presents two Tseng's methods for finding singularity points of an inclusion problem on a Hadamard manifold, proving convergence to the singularity point for any sequence generated by the methods when it exists. Applications to convex minimization problems and variational inequality problems are also provided.
Article
Mathematics, Applied
Parin Chaipunya, Konrawut Khammahawong, Poom Kumam
Summary: This paper introduces a new iterative algorithm to solve inclusion problems in Hadamard manifolds and studies its applications to convex minimization problems and variational inequality problems. A numerical example is presented to support the main theorem.
JOURNAL OF INEQUALITIES AND APPLICATIONS
(2021)
Article
Operations Research & Management Science
Shahroud Azami, Ali Barani, Morteza Oveisiha
Summary: This paper solves quasiconvex multiobjective optimization problems on Hadamard manifolds using inexact scalarization proximal methods, and the convergence to Pareto critical points is proved. In the convex case, the convergence to weak Pareto solutions is established.
Article
Multidisciplinary Sciences
Jinhua Zhu, Jinfang Tang, Shih-sen Chang, Min Liu, Liangcai Zhao
Summary: This paper introduces an iterative algorithm for finding a common solution to a finite family of problems on Hadamard manifolds, proving some strong convergence theorems. The results extend recent findings in literature.
Article
Operations Research & Management Science
Joao S. Andrade, Jurandir de O. Lopes, Joao Carlos de O. Souza
Summary: We propose an inertial proximal point method for variational inclusion involving difference of two maximal monotone vector fields in Hadamard manifolds and present some conditions for boundedness and full convergence of the sequence. The efficiency of the method is verified through numerical experiments comparing its performance with classical versions of the method for monotone and non-monotone problems.
JOURNAL OF GLOBAL OPTIMIZATION
(2023)
Article
Mathematics
Shih-sen Chang, Jinfang Tang, Chingfeng Wen
Summary: The article introduces a new algorithm that proves the convergence of a sequence to a common element in the set of fixed points for quasi-pseudo-contractive mappings and demi-contraction mappings, as well as the set of zeros of monotone inclusion problems on Hadamard manifolds. The results are then applied to study minimization problems and equilibrium problems in Hadamard manifolds.
ACTA MATHEMATICA SCIENTIA
(2021)
Article
Computer Science, Software Engineering
Anatoli Juditsky, Arkadi Nemirovski
Summary: This work aims to extend the scope of CVX to handle convex-concave saddle point problems and variational inequalities with monotone operators. By introducing the concept of cones family K, these problems can be transformed into conic problems on a cone from K and solved by corresponding solvers. The K-representations of convex-concave functions and monotone vector fields allow for algorithmic calculus to recognize and convert saddle point problems or variational inequalities into conic problems.
OPTIMIZATION METHODS & SOFTWARE
(2022)
Article
Operations Research & Management Science
Sani Salisu, Poom Kumam, Songpon Sriwongsa
Summary: In this article, a viscosity-type scheme is proposed for approximating a common solution of various mathematical problems in the framework of Hadamard spaces. The paper provides a strong convergence theorem for the generated sequence to a solution of the problem. The results are then applied to solve different problems and compare with existing methods, extending and complementing recent findings in the literature.
Article
Engineering, Electrical & Electronic
Peipei Tang, Chengjing Wang, Bo Jiang
Summary: In this paper, we introduce a PPMM algorithm for nonconvex rank regression problems. The algorithm applies the proximal majorization-minimization algorithm to solve the nonconvex problem, with the inner subproblems solved by a sparse semismooth Newton method based proximal point algorithm. Numerical experiments demonstrate that our proposed algorithm outperforms the existing state-of-the-art algorithms.
IEEE TRANSACTIONS ON SIGNAL PROCESSING
(2023)
Article
Mathematics, Applied
Shih-sen Chang, Jen-Chih Yao, L. Yang, Ching-Feng Wen, D. P. Wu
Summary: This paper introduces a new splitting iterative algorithm for finding a common solution in Hadamard manifolds, and proves its convergence under mild conditions. The algorithm is applied to study minimization problems and saddle point problems in Hadamard manifolds as well.
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Olaniyi S. Iyiola, Yekini Shehu
Summary: This paper proposes a two-point inertial proximal point algorithm for finding zero of maximal monotone operators in Hilbert spaces. The weak convergence results and non-asymptotic O(1/n) convergence rate of the proposed algorithm in a non-ergodic sense are obtained. Applications of the results to various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, are given. Numerical results are provided to demonstrate the accelerating behaviors of the proposed method compared to other related methods in the literature.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Radu Ioan Bot, Axel Boehm
Summary: This article introduces the latest research progress on minimax problems, particularly on proving the global convergence rate of alternating gradient descent algorithm in non-convex-concave settings.
SIAM JOURNAL ON OPTIMIZATION
(2023)
Article
Operations Research & Management Science
Pedro Jorge S. Santos, Joao Carlos de O. Souza
Summary: This paper studies the convergence of a proximal point method for solving quasi-equilibrium problems in Hilbert spaces. The method proposed by Moudafi and Iusem and Sosa is extended to the more general context of quasi-equilibrium problems. In this method, a quasi-equilibrium problem is solved by computing a solution of an equilibrium problem at each iteration. Weak convergence of the sequence to a solution of the QEP is obtained under some mild assumptions. Encouraging numerical experiments are presented to demonstrate the performance of the method.
Article
Operations Research & Management Science
Qamrul Hasan Ansari, Feeroz Babu
Summary: This paper discusses the proximal point algorithm for finding a singularity of the sum of a single-valued vector field and a set-valued vector field on Hadamard manifolds, with a focus on the convergence analysis of the proposed algorithm. Applications to composite minimization problems and variational inequality problems are also presented.
OPTIMIZATION LETTERS
(2021)
Article
Operations Research & Management Science
Donghui Fang, Qamrul Hasan Ansari, Jen-Chih Yao
Summary: This study considers a DC composite optimization problem and introduces some new regularity conditions, obtaining complete characterizations for certain duality properties of the problem.
Article
Mathematics, Applied
Doaa Filali, Mohammad Dilshad, Mohammad Akram, Feeroz Babu, Izhar Ahmad
Summary: This article introduces and analyzes the viscosity method for solving hierarchical variational inequalities on Hadamard manifolds involving a phi-contraction mapping and fixed points of a nonexpansive mapping. The results generalize and extend existing ones from Hilbert/Banach spaces and Hadamard manifolds. Additionally, an application to nonsmooth optimization problems is presented along with convergence analysis clarified through computational numerical experiments.
JOURNAL OF INEQUALITIES AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Qamrul Hasan Ansari, Feeroz Babu, Mohd Zeeshan
Summary: A new path incremental quasi-subgradient method is introduced on a Riemannian manifold with nonnegative sectional curvature to find the optimal solution of a sum of geodesic quasi-convex functions. The convergence analysis of the algorithm is studied, and its practical applicability is demonstrated in solving geodesic quasi-convex feasibility problems and sum of ratio problems on Riemannian manifolds.
NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION
(2021)
Article
Mathematics
Mohammad Akram, Mohammad Dilshad, Arvind Kumar Rajpoot, Feeroz Babu, Rais Ahmad, Jen-Chih Yao
Summary: In this paper, we present an improved iterative method proposed by Wang and apply it to find the common solution for the fixed point problem (FPP) and the split variational inclusion problem (SpVIP) in Hilbert space. We discuss the weak convergence of SpVIP and the strong convergence of the common solution for SpVIP and FPP under appropriate assumptions. We also compare our iterative schemes with existing related schemes.
Article
Mathematics, Applied
Feeroz Babu, Akram Ali, Ali H. Alkhaldi
Summary: In this paper, a non-monotone equilibrium problem on Hadamard manifolds is considered, and an Armijo's type extragradient algorithm is defined for this problem. The algorithm introduced does not require monotonicity of the objective bifunction and nonemptiness of the solution set. A convergence result of the algorithm is presented under very mild assumptions. Moreover, the applications of the established results to non-monotone set-valued variational inequalities and generalized Nash equilibrium problems are investigated.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics
Qamrul Hasan Ansari, Feeroz Babu, Moin Uddin
Summary: This article proposes two regularized iterative algorithms for solving variational inequality problems on Hadamard manifolds. By regularizing the variational inclusion problem and proving the convergence of the solution, the algorithms are shown to provide a solution for the hierarchical variational inequality problem. A computational experiment validates the effectiveness of the proposed algorithms.
ARABIAN JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics, Applied
Shamshad Husain, Mohd Asad, Feeroz Babu
Summary: The primary objective of this study is to approximate a nontrivial solution to a significant extension of the hierarchical fixed point problem. We present an accelerated simultaneous Halpern-type iterative algorithm and analyze its convergence. Numerical experiments are provided to demonstrate the rationality and superiority of our algorithm. This paper improves, generalizes, and extends some notable recent findings in this field.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
D. R. Sahu, Babu Feeroz, Sharma Shikher
Summary: The objective of this work is to design a new iterative method for solving the inclusion problem (A + B)(-1)(0) based on Armijo's type-modified extragradient method. The proposed method reduces the computational cost and improves the convergence rate by requiring one projection at each iteration. A convergence theorem is established, significantly improving existing results. Concrete examples on Hadamard manifolds and numerical confirmations are provided. Moreover, convergence results for variational inequality problems without vector field monotonicity are demonstrated.
NUMERICAL ALGORITHMS
(2023)
Article
Operations Research & Management Science
Qamrul Hasan Ansari, Feeroz Babu, Muzaffar Sarkar Raju
Summary: In this paper, a proximal point method using Bregman distance is proposed to solve quasiconvex pseudomonotone equilibrium problems. Under certain assumptions, it is proven that the algorithm is well defined and the generated sequence converges to a solution of the equilibrium problem when the bifunction is strongly quasiconvex in its second argument. This method extends the validity of the convergence analysis of proximal point methods for equilibrium problems beyond the usual assumption of the bifunction's convexity in the second argument. Numerical examples are provided to illustrate the algorithm and convergence result.
Article
Operations Research & Management Science
Feeroz Babu, Akram Ali, Ali H. Alkhaldi
Summary: This paper examines the necessary and sufficient optimality conditions, known as the Karush-Kuhn-Tucker (KKT) conditions, for non-smooth quasi-convex optimization problems on Riemannian manifolds. The authors introduce and establish the existence of the quasi-subdifferential on Riemannian manifolds. They provide auxiliary results and formulate numerical examples to verify the proposed outcomes. Moreover, the study also presents original results in Euclidean spaces, which differ from previous findings.
Article
Mathematics, Applied
Qamrul Hasan Ansari, Feeroz Babu, Mohd. Zeeshan
Summary: In this paper, the authors investigate a hierarchical variational inequality problem on Hadamard manifolds using set-valued monotone vector fields. They develop implicit and explicit viscosity methods for solving weak contraction mappings and study an inexact version of the explicit viscosity method. Two examples and computational experiments are provided to illustrate the proposed methods.
FIXED POINT THEORY
(2023)
Article
Mathematics, Applied
Qamrul Hasan Ansari, Feeroz Babu, Mohd Zeeshan
Summary: This paper discusses the solution methods for hierarchical variational inequality problems in the framework of Hadamard manifolds. Bilevel variational inequality problems and bilevel optimization problems are considered as special cases. Implicit and explicit viscosity methods are developed to solve the problem for weakly contraction mappings. An inexact version of the explicit viscosity method is also studied. Finally, two examples and computational experiments are provided to illustrate the implicit and explicit viscosity methods.
FIXED POINT THEORY
(2022)
Article
Mathematics, Applied
Qamrul Hasan Ansari, Feeroz Babu, Mohd. Zeeshan
Summary: The present paper deals with the coercivity conditions to establish necessary and sufficient condition for a solutions of the variational inequality problem in the setting of Hadamard manifolds. As an application of our results, the existence of solutions of the minimization problems in the setting of Hadamard spaces is derived.
JOURNAL OF NONLINEAR AND CONVEX ANALYSIS
(2022)
Article
Mathematics
Qamrul Hasan Ansari, Feeroz Babu, D. R. Sahu
Summary: This paper considers variational inclusion systems and its special cases in the setting of Hadamard manifolds. An iterative algorithm for solving the system and its convergence analysis are proposed. The application to constraint minimization problems on Hadamard manifolds is presented. A numerical example is used to illustrate the proposed algorithms and convergence analysis. The algorithms and convergence results in this paper improve or extend known algorithms from linear structure to Hadamard manifolds.
ACTA MATHEMATICA SCIENTIA
(2022)
