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Symmetry
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Symmetry in Nonlinear Functional Analysis and Optimization Theory
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Symmetry in Nonlinear Functional Analysis and Optimization Theory

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2021) | Viewed by 25669

Special Issue Editor

Prof. Dr. Sun Young Cho

E-Mail Website
Guest Editor
Department of Mathematics, Gyeongsang National University, Jinju 660-701, Korea
Interests: nonlinear functional analysis; optimization theory; complementary problems; differential equation; equilibrium problems; monotone operators; convex feasibility problems; split feasibility problems; machine learning

Special Issue Information

Dear Colleagues,

Nonlinear Functional Analysis and Optimization Theory are two closed related two research fields in applied mathematics. A lot of problems such as differential equations and integral equations in nonlinear analysis, can be solved via optimization methods. In particular, fixed/zero-point problems nonlinear operators are under the spotlight of mathematicians working on optimization theory. Recently, a number of optimization methods, such as, projection-like methods, have been investigated for solving various nonlinear equations. Many important applications have been carried out in engineering fields, such as, transportation, economics, medicine, and machine learning.

In this Special Issue, we will focus on high-quality research on nonlinear functional analysis and optimization theory, in particular complementary problems, differential equation, integral equations, equilibrium problems, monotone operators, fixed/zero points, convex feasibility problems, split feasibility problems and their applications to the real world.

Please note that all submitted papers must be within the general scope of the Symmetry journal.

Prof. Sun Young Cho
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • complementary problems
  • differential equation
  • equilibrium problems
  • monotone operators
  • convex feasibility problems
  • split feasibility problems
  • machine learning

Published Papers (13 papers)

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Research

10 pages, 1654 KiB  
Article
An Application of the Kalman Filter Recursive Algorithm to Estimate the Gaussian Errors by Minimizing the Symmetric Loss Function
by Cristian Busu and Mihail Busu
Symmetry 2021, 13(2), 240; https://doi.org/10.3390/sym13020240 - 31 Jan 2021
Cited by 3 | Viewed by 2292
Abstract
Kalman filtering is a linear quadratic estimation (LQE) algorithm that uses a time series of observed data to produce estimations of unknown variables. The Kalman filter (KF) concept is widely used in applied mathematics and signal processing. In this study, we developed a [...] Read more.
Kalman filtering is a linear quadratic estimation (LQE) algorithm that uses a time series of observed data to produce estimations of unknown variables. The Kalman filter (KF) concept is widely used in applied mathematics and signal processing. In this study, we developed a methodology for estimating Gaussian errors by minimizing the symmetric loss function. Relevant applications of the kinetic models are described at the end of the manuscript. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
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23 pages, 807 KiB  
Article
Ostrowski Type Inequalities Involving Harmonically Convex Functions and Applications
by Nousheen Akhtar, Muhammad Uzair Awan, Muhammad Zakria Javed, Michael Th. Rassias, Marcela V. Mihai, Muhammad Aslam Noor and Khalida Inayat Noor
Symmetry 2021, 13(2), 201; https://doi.org/10.3390/sym13020201 - 27 Jan 2021
Cited by 7 | Viewed by 1610
Abstract
The main objective of this paper is to derive some new generalizations of Ostrowski type inequalities for the functions whose first derivatives absolute value are harmonically convex. We also discuss some special cases of the obtained results. In the last section, we present [...] Read more.
The main objective of this paper is to derive some new generalizations of Ostrowski type inequalities for the functions whose first derivatives absolute value are harmonically convex. We also discuss some special cases of the obtained results. In the last section, we present some applications of the obtained results. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
16 pages, 310 KiB  
Article
Split Common Coincidence Point Problem: A Formulation Applicable to (Bio)Physically-Based Inverse Planning Optimization
by Charles E. Chidume and Lois C. Okereke
Symmetry 2020, 12(12), 2086; https://doi.org/10.3390/sym12122086 - 15 Dec 2020
Cited by 1 | Viewed by 1789
Abstract
Inverse planning is a method of radiotherapy treatment planning where the care team begins with the desired dose distribution satisfying prescribed clinical objectives, and then determines the treatment parameters that will achieve it. The variety in symmetry, form, and characteristics of the objective [...] Read more.
Inverse planning is a method of radiotherapy treatment planning where the care team begins with the desired dose distribution satisfying prescribed clinical objectives, and then determines the treatment parameters that will achieve it. The variety in symmetry, form, and characteristics of the objective functions describing clinical criteria requires a flexible optimization approach in order to obtain optimized treatment plans. Therefore, we introduce and discuss a nonlinear optimization formulation called the split common coincidence point problem (SCCPP). We show that the SCCPP is a suitable formulation for the inverse planning optimization problem with the flexibility of accommodating several biological and/or physical clinical objectives. Also, we propose an iterative algorithm for approximating the solution of the SCCPP, and using Bregman techniques, we establish that the proposed algorithm converges to a solution of the SCCPP and to an extremum of the inverse planning optimization problem. We end with a note on useful insights on implementing the algorithm in a clinical setting. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
22 pages, 419 KiB  
Article
An Efficient Parallel Extragradient Method for Systems of Variational Inequalities Involving Fixed Points of Demicontractive Mappings
by Lateef Olakunle Jolaoso and Maggie Aphane
Symmetry 2020, 12(11), 1915; https://doi.org/10.3390/sym12111915 - 20 Nov 2020
Cited by 4 | Viewed by 1416
Abstract
Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which [...] Read more.
Herein, we present a new parallel extragradient method for solving systems of variational inequalities and common fixed point problems for demicontractive mappings in real Hilbert spaces. The algorithm determines the next iterate by computing a computationally inexpensive projection onto a sub-level set which is constructed using a convex combination of finite functions and an Armijo line-search procedure. A strong convergence result is proved without the need for the assumption of Lipschitz continuity on the cost operators of the variational inequalities. Finally, some numerical experiments are performed to illustrate the performance of the proposed method. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
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14 pages, 326 KiB  
Article
The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality
by Temadher A. Almaadeed, Akram Taati, Maziar Salahi and Abdelouahed Hamdi
Symmetry 2020, 12(8), 1369; https://doi.org/10.3390/sym12081369 - 17 Aug 2020
Cited by 2 | Viewed by 1641
Abstract
In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different [...] Read more.
In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
13 pages, 286 KiB  
Article
Implicit Extragradient-Like Method for Fixed Point Problems and Variational Inclusion Problems in a Banach Space
by Sun Young Cho
Symmetry 2020, 12(6), 998; https://doi.org/10.3390/sym12060998 - 11 Jun 2020
Cited by 16 | Viewed by 1689
Abstract
In a uniformly convex and q-uniformly smooth Banach space with q ∈ ( 1 , 2 ] , one use VIP to indicate a variational inclusion problem involving two accretive mappings and CFPP to denote the common fixed-point problem of an infinite [...] Read more.
In a uniformly convex and q-uniformly smooth Banach space with q ∈ ( 1 , 2 ] , one use VIP to indicate a variational inclusion problem involving two accretive mappings and CFPP to denote the common fixed-point problem of an infinite family of strict pseudocontractions of order q. In this paper, we introduce a composite extragradient implicit method for solving a general symmetric system of variational inclusions (GSVI) with certain VI and CFPP. We then investigate its convergence analysis under some weak conditions. Finally, we consider the celebrated LASSO problem in Hilbert spaces. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
25 pages, 1165 KiB  
Article
The Inertial Sub-Gradient Extra-Gradient Method for a Class of Pseudo-Monotone Equilibrium Problems
by Habib ur Rehman, Poom Kumam, Wiyada Kumam, Meshal Shutaywi and Wachirapong Jirakitpuwapat
Symmetry 2020, 12(3), 463; https://doi.org/10.3390/sym12030463 - 15 Mar 2020
Cited by 44 | Viewed by 3265
Abstract
In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We [...] Read more.
In this article, we focus on improving the sub-gradient extra-gradient method to find a solution to the problems of pseudo-monotone equilibrium in a real Hilbert space. The weak convergence of our method is well-established based on the standard assumptions on a bifunction. We also present the application of our results that enable to solve numerically the pseudo-monotone and monotone variational inequality problems, in addition to the particular presumptions required by the operator. We have used various numerical examples to support our well-proved convergence results, and we can show that the proposed method involves a considerable influence over-running time and the total number of iterations. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
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16 pages, 305 KiB  
Article
A Fixed-Point Subgradient Splitting Method for Solving Constrained Convex Optimization Problems
by Nimit Nimana
Symmetry 2020, 12(3), 377; https://doi.org/10.3390/sym12030377 - 03 Mar 2020
Cited by 3 | Viewed by 2073
Abstract
In this work, we consider a bilevel optimization problem consisting of the minimizing sum of two convex functions in which one of them is a composition of a convex function and a nonzero linear transformation subject to the set of all feasible points [...] Read more.
In this work, we consider a bilevel optimization problem consisting of the minimizing sum of two convex functions in which one of them is a composition of a convex function and a nonzero linear transformation subject to the set of all feasible points represented in the form of common fixed-point sets of nonlinear operators. To find an optimal solution to the problem, we present a fixed-point subgradient splitting method and analyze convergence properties of the proposed method provided that some additional assumptions are imposed. We investigate the solving of some well known problems by using the proposed method. Finally, we present some numerical experiments for showing the effectiveness of the obtained theoretical result. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
13 pages, 275 KiB  
Article
Proximally Compatible Mappings and Common Best Proximity Points
by V. Pragadeeswarar, R. Gopi, M. De la Sen and Stojan Radenović
Symmetry 2020, 12(3), 353; https://doi.org/10.3390/sym12030353 - 01 Mar 2020
Cited by 4 | Viewed by 1943
Abstract
The purpose of this paper is to introduce and analyze a new idea of proximally compatible mappings and we extend some results of Jungck via proximally compatible mappings. Furthermore, we obtain common best proximity point theorems for proximally compatible mappings through two different [...] Read more.
The purpose of this paper is to introduce and analyze a new idea of proximally compatible mappings and we extend some results of Jungck via proximally compatible mappings. Furthermore, we obtain common best proximity point theorems for proximally compatible mappings through two different ways of construction of sequences. In addition, we provide an example to support our main result. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
14 pages, 322 KiB  
Article
Sensitivity Analysis of Mixed Cayley Inclusion Problem with XOR-Operation
by Imran Ali, Mohd. Ishtyak, Rais Ahmad and Ching-Feng Wen
Symmetry 2020, 12(2), 220; https://doi.org/10.3390/sym12020220 - 02 Feb 2020
Cited by 1 | Viewed by 1811
Abstract
In this paper, we consider the parametric mixed Cayley inclusion problem with Exclusive or (XOR)-operation and show its equivalence with the parametric resolvent equation problem with XOR-operation. Since the sensitivity analysis, Cayley operator, inclusion problems, and XOR-operation are all applicable for solving many [...] Read more.
In this paper, we consider the parametric mixed Cayley inclusion problem with Exclusive or (XOR)-operation and show its equivalence with the parametric resolvent equation problem with XOR-operation. Since the sensitivity analysis, Cayley operator, inclusion problems, and XOR-operation are all applicable for solving many problems occurring in basic and applied sciences, such as financial modeling, climate models in geography, analyzing “Black Box processes”, computer programming, economics, and engineering, etc., we study the sensitivity analysis of the parametric mixed Cayley inclusion problem with XOR-operation. For this purpose, we use the equivalence of the parametric mixed Cayley inclusion problem with XOR-operation and the parametric resolvent equation problem with XOR-operation, which is an alternative approach to study the sensitivity analysis. In support of some of the concepts used in this paper, an example is provided. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
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17 pages, 285 KiB  
Article
On a General Extragradient Implicit Method and Its Applications to Optimization
by Lu-Chuan Ceng and Qing Yuan
Symmetry 2020, 12(1), 124; https://doi.org/10.3390/sym12010124 - 08 Jan 2020
Viewed by 1472
Abstract
Let X be a Banach space with both q-uniformly smooth and uniformly convex structures. This article introduces and considers a general extragradient implicit method for solving a general system of variational inequalities (GSVI) with the constraints of a common fixed point problem [...] Read more.
Let X be a Banach space with both q-uniformly smooth and uniformly convex structures. This article introduces and considers a general extragradient implicit method for solving a general system of variational inequalities (GSVI) with the constraints of a common fixed point problem (CFPP) of a countable family of nonlinear mappings { S n } n = 0 ∞ and a monotone variational inclusion, zero-point, problem. Here, the constraints are symmetrical and the general extragradient implicit method is based on Korpelevich’s extragradient method, implicit viscosity approximation method, Mann’s iteration method, and the W-mappings constructed by { S n } n = 0 ∞ . Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
16 pages, 286 KiB  
Article
Parallel Tseng’s Extragradient Methods for Solving Systems of Variational Inequalities on Hadamard Manifolds
by Lu-Chuan Ceng, Yekini Shehu and Yuanheng Wang
Symmetry 2020, 12(1), 43; https://doi.org/10.3390/sym12010043 - 24 Dec 2019
Cited by 4 | Viewed by 2010
Abstract
The aim of this article is to study two efficient parallel algorithms for obtaining a solution to a system of monotone variational inequalities (SVI) on Hadamard manifolds. The parallel algorithms are inspired by Tseng’s extragradient techniques with new step sizes, which are established [...] Read more.
The aim of this article is to study two efficient parallel algorithms for obtaining a solution to a system of monotone variational inequalities (SVI) on Hadamard manifolds. The parallel algorithms are inspired by Tseng’s extragradient techniques with new step sizes, which are established without the knowledge of the Lipschitz constants of the operators and line-search. Under the monotonicity assumptions regarding the underlying vector fields, one proves that the sequences generated by the methods converge to a solution of the monotone SVI whenever it exists. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
13 pages, 266 KiB  
Article
Hybrid Algorithms for Variational Inequalities Involving a Strict Pseudocontraction
by Sun Young Cho
Symmetry 2019, 11(12), 1502; https://doi.org/10.3390/sym11121502 - 11 Dec 2019
Cited by 2 | Viewed by 1820
Abstract
In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a [...] Read more.
In a real Hilbert space, we investigate the Tseng’s extragradient algorithms with hybrid adaptive step-sizes for treating a Lipschitzian pseudomonotone variational inequality problem and a strict pseudocontraction fixed-point problem, which are symmetry. By imposing some appropriate weak assumptions on parameters, we obtain a norm solution of the problems, which solves a certain hierarchical variational inequality. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory)
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