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Symmetry
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Symmetry in Optimization and Control with Real World Applications
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Symmetry in Optimization and Control with Real World Applications

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

Deadline for manuscript submissions: closed (26 March 2022) | Viewed by 40554

Special Issue Editors

Prof. Dr. Kok Lay Teo

E-Mail Website
Guest Editor
School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University of Technology, Perth, WA 6845, Australia
Interests: engineering mathematics; optimisation and control
Prof. Dr. Yonghong Wu

E-Mail Website
Co-Guest Editor
Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia
Interests: computational mathematics; applied mathematical modelling; differential equations and boundary value problems; fluid mechanics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Aviv Gibali

E-Mail Website
Co-Guest Editor
Faculty of Sciences, Applied Mathematics Department, Holon Institute of Technology, Holon, Israel
Interests: mathematical theory; algorithms systems of linear and nonlinear equations; inequalities optimization theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear colleagues,

Due to the advancement of technology, many challenging practically significant real world problems emerge. These problems arise in a wide range of disciplines, such as physics, chemistry, core engineering (civil, chemical, electrical and mechanical), biology, defence and computer sciences. They can often be formulated as optimization or control problems, and hence, methodologies and techniques in optimization and control are clearly indispensable for solving these problems. Due to the existence of symmetry in certain systems in nature and engineering, many mathematical models and optimization problems arising possess some form of symmetry. For example, in some constrained optimization problems, certain variables appear symmetrically in the objective and constraint functions. The purpose of this Special Issue is to publish a selection of original high-quality papers, presenting substantial novel latest research achievements in optimization and control with application to important real-world engineering problems involving complex systems. It is expected that the solution methods will be based on advanced theory, methodologies and techniques from optimization and control. Submitted papers are also expected to be within the general scope of the journal.

The focus of this Special Issue is on the advancement of mathematically rigorous computational techniques, engineering mathematics and real world engineering applications in areas listed below:

  • Dynamical systems and control
  • Optimization and optimal control
  • Inverse problems
  • Modelling and optimization of complex systems
Prof. Dr. Kok Lay Teo
Prof. Dr. Yonghong Wu
Prof. Dr. Aviv Gibali
Guest Editors

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

  • constrained optimization
  • control
  • dynamical systems
  • computational optimization

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Published Papers (15 papers)

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Editorial

Jump to: Research

4 pages, 171 KiB  
Editorial
Special Issue “Symmetry in Optimization and Control with Real-World Applications”
by Kok Lay Teo, Aviv Gibali and Yonghong Wu
Symmetry 2022, 14(8), 1529; https://doi.org/10.3390/sym14081529 - 26 Jul 2022
Viewed by 1057
Abstract
In the study of many real-world problems such as engineering design and industrial process control, one often needs to select certain elements/controls from a feasible set in order to optimize the design or system based on certain criteria [...] Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)

Research

Jump to: Editorial

25 pages, 599 KiB  
Article
Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem
by Kateryna Czerniachowska and Marcin Hernes
Symmetry 2021, 13(7), 1182; https://doi.org/10.3390/sym13071182 - 30 Jun 2021
Cited by 11 | Viewed by 2293
Abstract
The allocation of products on shelves is an important issue from the point of view of effective decision making by retailers. In this paper, we investigate a practical shelf space allocation model which takes into account the number of facings, capping, and nesting [...] Read more.
The allocation of products on shelves is an important issue from the point of view of effective decision making by retailers. In this paper, we investigate a practical shelf space allocation model which takes into account the number of facings, capping, and nesting of a product. We divide the shelf into the segments of variable size in which the products of the specific types could be placed. The interconnections between products are modelled with the help of categorizing the products into specific types as well as grouping some of them into clusters. This results in four groups of constraints—shelf constraints, shelf type constraints, product constraints, position allocation constraints—that are used in the model for aesthetic symmetry of a planogram. We propose a simulated annealing algorithm with improvement and reallocation procedures to solve the planogram profit maximization problem. Experiments are based on artificial data sets that have been generated according to real-world conditions. The efficiency of the designed algorithm has been estimated using the CPLEX solver. The computational tests demonstrate that the proposed algorithm gives valuable results in an acceptable time. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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24 pages, 5094 KiB  
Article
Improved Salp Swarm Algorithm with Simulated Annealing for Solving Engineering Optimization Problems
by Qing Duan, Lu Wang, Hongwei Kang, Yong Shen, Xingping Sun and Qingyi Chen
Symmetry 2021, 13(6), 1092; https://doi.org/10.3390/sym13061092 - 20 Jun 2021
Cited by 16 | Viewed by 3432
Abstract
Swarm-based algorithm can successfully avoid the local optimal constraints, thus achieving a smooth balance between exploration and exploitation. Salp swarm algorithm (SSA), as a swarm-based algorithm on account of the predation behavior of the salp, can solve complex daily life optimization problems in [...] Read more.
Swarm-based algorithm can successfully avoid the local optimal constraints, thus achieving a smooth balance between exploration and exploitation. Salp swarm algorithm (SSA), as a swarm-based algorithm on account of the predation behavior of the salp, can solve complex daily life optimization problems in nature. SSA also has the problems of local stagnation and slow convergence rate. This paper introduces an improved salp swarm algorithm, which improve the SSA by using the chaotic sequence initialization strategy and symmetric adaptive population division. Moreover, a simulated annealing mechanism based on symmetric perturbation is introduced to enhance the local jumping ability of the algorithm. The improved algorithm is referred to SASSA. The CEC standard benchmark functions are used to evaluate the efficiency of the SASSA and the results demonstrate that the SASSA has better global search capability. SASSA is also applied to solve engineering optimization problems. The experimental results demonstrate that the exploratory and exploitative proclivities of the proposed algorithm and its convergence patterns are vividly improved. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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17 pages, 3156 KiB  
Article
Hardware in the Loop Topology for an Omnidirectional Mobile Robot Using Matlab in a Robot Operating System Environment
by Constantin-Catalin Dosoftei, Alexandru-Tudor Popovici, Petru-Razvan Sacaleanu, Paul-Marcelin Gherghel and Cristina Budaciu
Symmetry 2021, 13(6), 969; https://doi.org/10.3390/sym13060969 - 30 May 2021
Cited by 13 | Viewed by 4721
Abstract
The symmetry of the omnidirectional robot motion abilities around its central vertical axis is an important advantage regarding its driveability for the flexible interoperation with fixed conveyor systems. The paper illustrates a Hardware in the Loop architectural approach for integrated development of an [...] Read more.
The symmetry of the omnidirectional robot motion abilities around its central vertical axis is an important advantage regarding its driveability for the flexible interoperation with fixed conveyor systems. The paper illustrates a Hardware in the Loop architectural approach for integrated development of an Ominidirectional Mobile Robot that is designed to serve in a dynamic logistic environment. Such logistic environments require complex algorithms for autonomous navigation between different warehouse locations, that can be efficiently developed using Robot Operating System nodes. Implementing path planning nodes benefits from using Matlab-Simulink, which provides a large selection of algorithms that are easily integrated and customized. The proposed solution is deployed for validation on a NVIDIA Jetson Nano, the embedded computer hosted locally on the robot, that runs the autonomous navigation software. The proposed solution permits the live connection to the omnidirectional prototype platform, allowing to deploy algorithms and acquire data for debugging the location, path planning and the mapping information during real time autonomous navigation experiments, very useful in validating different strategies. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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14 pages, 3333 KiB  
Article
Least Squares Support Vector Machine-Based Multivariate Generalized Predictive Control for Parabolic Distributed Parameter Systems with Control Constraints
by Ling Ai, Yang Xu, Liwei Deng and Kok Lay Teo
Symmetry 2021, 13(3), 453; https://doi.org/10.3390/sym13030453 - 10 Mar 2021
Cited by 5 | Viewed by 1789
Abstract
This manuscript addresses a new multivariate generalized predictive control strategy using the least squares support vector machine for parabolic distributed parameter systems. First, a set of proper orthogonal decomposition-based spatial basis functions constructed from a carefully selected set of data is used in [...] Read more.
This manuscript addresses a new multivariate generalized predictive control strategy using the least squares support vector machine for parabolic distributed parameter systems. First, a set of proper orthogonal decomposition-based spatial basis functions constructed from a carefully selected set of data is used in a Galerkin projection for the building of an approximate low-dimensional lumped parameter systems. Then, the temporal autoregressive exogenous model obtained by the least squares support vector machine is applied in the design of a multivariate generalized predictive control strategy. Finally, the effectiveness of the proposed multivariate generalized predictive control strategy is verified through a numerical simulation study on a typical diffusion-reaction process in radical symmetry. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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18 pages, 1692 KiB  
Article
A Heuristic Approach to Shelf Space Allocation Decision Support Including Facings, Capping, and Nesting
by Kateryna Czerniachowska and Marcin Hernes
Symmetry 2021, 13(2), 314; https://doi.org/10.3390/sym13020314 - 13 Feb 2021
Cited by 11 | Viewed by 2784
Abstract
Shelf space on which products are exhibited is a scarce resource in the retail environment. Retailers regularly make decisions related to allocating products to their outlets’ limited shelf space. The aim of the paper was to develop a practical shelf space allocation model [...] Read more.
Shelf space on which products are exhibited is a scarce resource in the retail environment. Retailers regularly make decisions related to allocating products to their outlets’ limited shelf space. The aim of the paper was to develop a practical shelf space allocation model offering the possibility of horizontal and vertical product grouping, representing an item (product) with facings, capping, and nesting, with the objective of maximizing the retailer’s profit. Because real category-management problems address a lot of retailer’s rules, we expanded the basic shelf space allocation model, using shelf constraints, product constraints, multi-shelves constraints, and category constraints. To solve the problem, we proposed two adjustable methods that allowed us to achieve good results within a short time interval. The validity of algorithms was estimated, using the CPLEX solver and illustrated with example problems. Experiments were performed on data generated on the basis of real retail values. To estimate the performance of the proposed approach, 45 cases were tested. Among them, the proposed approach found solutions in 34 cases, while CPLEX found solutions only in 23 cases. The profit ratio of the proposed approach is, on average, 94.57%, with minimal and maximal values of 86.80% and 99.84%, accordingly. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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19 pages, 323 KiB  
Article
Toward Applications of Linear Control Systems on the Real World and Theoretical Challenges
by Víctor Ayala, María Torreblanca and William Valdivia
Symmetry 2021, 13(2), 167; https://doi.org/10.3390/sym13020167 - 22 Jan 2021
Cited by 2 | Viewed by 2000
Abstract
Many significant real world challenges arise as optimization problems on different classes of control systems. In particular, ordinary differential equations with symmetries. The purpose of this review article is twofold. First, we give the information we have about the class of Linear Control [...] Read more.
Many significant real world challenges arise as optimization problems on different classes of control systems. In particular, ordinary differential equations with symmetries. The purpose of this review article is twofold. First, we give the information we have about the class of Linear Control Systems ΣG on a low dimension matrix Lie group G. Second, we invite the Mathematical community to consider possible applications through the Pontryagin Maximum Principle for ΣG. In addition, we challenge some theoretical open problems. The class ΣG is a perfect generalization of the classical Linear Control System on the Abelian group Rn. Let G be a Lie group of dimension two or three. Related to ΣG, this review describes the actual results about controllability, the time-optimal Hamiltonian equations and, the Pontryagin Maximum Principle. We show how to build ΣG, through several examples on low dimensional matrix groups. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
20 pages, 5366 KiB  
Article
Dynamic Cost Ant Colony Algorithm to Optimize Query for Distributed Database Based on Quantum-Inspired Approach
by Sayed A. Mohsin, Ahmed Younes and Saad M. Darwish
Symmetry 2021, 13(1), 70; https://doi.org/10.3390/sym13010070 - 2 Jan 2021
Cited by 13 | Viewed by 3532
Abstract
A distributed database model can be effectively optimized through using query optimization. In such a model, the optimizer attempts to identify the most efficient join order, which minimizes the overall cost of the query plan. Successful query processing largely relies on the methodology [...] Read more.
A distributed database model can be effectively optimized through using query optimization. In such a model, the optimizer attempts to identify the most efficient join order, which minimizes the overall cost of the query plan. Successful query processing largely relies on the methodology implemented by the query optimizer. Many researches are concerned with the fact that query processing is considered an NP-hard problem especially when the query becomes bigger. Regarding large queries, it has been found that heuristic methods cannot cover all search spaces and may lead to falling in a local minimum. This paper examines how quantum-inspired ant colony algorithm, a hybrid strategy of probabilistic algorithms, can be devised to improve the cost of query joins in distributed databases. Quantum computing has the ability to diversify and expand, and thus covering large query search spaces. This enables the selection of the best trails, which speeds up convergence and helps avoid falling into a local optimum. With such a strategy, the algorithm aims to identify an optimal join order to reduce the total execution time. Experimental results show that the proposed quantum-inspired ant colony offers a faster convergence with better outcome when compared with the classic model. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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16 pages, 4313 KiB  
Article
Investigation of Position and Velocity Stability of the Nanometer Resolution Linear Motor Stage with Air Bearings by Shaping of Controller Transfer Function
by Artur Piščalov, Edgaras Urbonas, Nikolaj Višniakov, Darius Zabulionis and Artūras Kilikevičius
Symmetry 2020, 12(12), 2062; https://doi.org/10.3390/sym12122062 - 11 Dec 2020
Cited by 2 | Viewed by 2289
Abstract
Modern industrial enterprises require high accuracy and precision feedback systems to fulfil cutting edge requirements of technological processes. As demand for a highly accurate system grows, a thin gap between throughput and quality exists. The conjunction of ultrafast lasers and modern control strategies [...] Read more.
Modern industrial enterprises require high accuracy and precision feedback systems to fulfil cutting edge requirements of technological processes. As demand for a highly accurate system grows, a thin gap between throughput and quality exists. The conjunction of ultrafast lasers and modern control strategies of mechatronic systems can be taken into account as an effective solution to reach both throughput and tolerances. In the present paper, the dynamic errors of the moving platform of the one degree of freedom stage, based on linear motor and air bearings, have been analyzed. A precision positioning system is investigated as a symmetric system which is based on symmetric linear motor. The goal of the present article is to investigate the controllers of the different architecture and to find the best controller that can ensure a stable and small dynamic error of the displacement of the stage platform at four different constant velocities of the moving platform. The relations between the controller order, velocity and the displacement dynamic error have been investigated. It is determined that higher-order controllers can reduce the dynamic error significantly at low velocities of the moving platforms: 1 and 5 mm/s. On the contrary, the low order controllers of 4th-degree polynomials of the transfer function can also provide small dynamic errors of the displacement of the platform. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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6 pages, 206 KiB  
Article
Generic Existence of Solutions of Symmetric Optimization Problems
by Alexander J. Zaslavski
Symmetry 2020, 12(12), 2004; https://doi.org/10.3390/sym12122004 - 4 Dec 2020
Cited by 4 | Viewed by 1558
Abstract
In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, [...] Read more.
In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
23 pages, 2392 KiB  
Article
A Fast Non-Linear Symmetry Approach for Guaranteed Consensus in Network of Multi-Agent Systems
by Rawad Abdulghafor and Sultan Almotairi
Symmetry 2020, 12(10), 1692; https://doi.org/10.3390/sym12101692 - 15 Oct 2020
Cited by 4 | Viewed by 1944
Abstract
There has been tremendous work on multi-agent systems (MAS) in recent years. MAS consist of multiple autonomous agents that interact with each order to solve a complex problem. Several applications of MAS can be found in computer networks, smart grids, and the modeling [...] Read more.
There has been tremendous work on multi-agent systems (MAS) in recent years. MAS consist of multiple autonomous agents that interact with each order to solve a complex problem. Several applications of MAS can be found in computer networks, smart grids, and the modeling of complex systems. Despite numerous benefits, a significant challenge for MAS is achieving a consensus among agents in a shared target task, which is difficult without applying certain mathematical equations. Non-linear models offer better possibility of resolving consensus for MAS; however, existing non-linear models are considerably complicated and present no guarantees for achieving consensus. This paper proposes a non-linear mathematical model of semi symmetry quadratic operator (SSQO) in order to resolve the issue of consensus in networks of MAS. The model is based on stochastic quadratic operator theory, with added new notations. An important feature for the proposed model is low complexity, fast consensus, and a guaranteed capability to reach a consensus. We present an evaluation of the proposed SSQO model and comparison with other existing models. We demonstrate that an average consensus can be achieved with our model in addition to the emulation effects for the MAS consensus. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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18 pages, 4681 KiB  
Article
The Application of a Hybrid Model Using Mathematical Optimization and Intelligent Algorithms for Improving the Talc Pellet Manufacturing Process
by Dussadee Buntam, Wachirapond Permpoonsinsup and Prayoon Surin
Symmetry 2020, 12(10), 1602; https://doi.org/10.3390/sym12101602 - 26 Sep 2020
Cited by 2 | Viewed by 2551
Abstract
Moisture is one of the most important factors impacting the talc pellet process. In this study, a hybrid model (HM) based on the combination of intelligent algorithms, self-organizing map (SOM), the adaptive neuron fuzzy inference system (ANFIS) and metaheuristic optimizations, genetic algorithm (GA) [...] Read more.
Moisture is one of the most important factors impacting the talc pellet process. In this study, a hybrid model (HM) based on the combination of intelligent algorithms, self-organizing map (SOM), the adaptive neuron fuzzy inference system (ANFIS) and metaheuristic optimizations, genetic algorithm (GA) and particle swarm optimization (PSO) is introduced, namely, HM-GA and HM-PSO. The main purpose is to predict the moisture in the talc pellet process related to symmetry in the aspect of real-world application problem. In the combination process, SOM classifies the suitable input data. The GA and PSO, as the training algorithms of ANFIS, are investigated to compare the prediction skill. Five factors, including talc powder, water, temperature, feed speed, and air flow of 52 experiment cases designed by central composite design (CCD), are the training set data. Three different measures evaluate the capacity of moisture prediction. The comparison results show that the HM-PSO can provide the smallest difference between train and test datasets under the condition of the moisture being less than 5%. As a result, the HM-PSO model achieves the best result in predicting the moisture for the talc pellet process with R = 0.9539, RMSE = 1.0693, and AAD = 0.393, compared to others. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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14 pages, 2246 KiB  
Article
Stability Analysis of Linear Feedback Systems in Control
by Mutti-Ur Rehman, Jehad Alzabut and Muhammad Fazeel Anwar
Symmetry 2020, 12(9), 1518; https://doi.org/10.3390/sym12091518 - 15 Sep 2020
Cited by 4 | Viewed by 2884
Abstract
This article presents a stability analysis of linear time invariant systems arising in system theory. The computation of upper bounds of structured singular values confer the stability analysis, robustness and performance of feedback systems in system theory. The computation of the bounds of [...] Read more.
This article presents a stability analysis of linear time invariant systems arising in system theory. The computation of upper bounds of structured singular values confer the stability analysis, robustness and performance of feedback systems in system theory. The computation of the bounds of structured singular values of Toeplitz and symmetric Toeplitz matrices for linear time invariant systems is presented by means of low rank ordinary differential equations (ODE’s) based methodology. The proposed methodology is based upon the inner-outer algorithm. The inner algorithm constructs and solves a gradient system of ODE’s while the outer algorithm adjusts the perturbation level with fast Newton’s iteration. The comparison of bounds of structured singular values approximated by low rank ODE’s based methodology results tighter bounds when compared with well-known MATLAB routine mussv, available in MATLAB control toolbox. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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16 pages, 301 KiB  
Article
A Symmetric FBF Method for Solving Monotone Inclusions
by Aviv Gibali and Yekini Shehu
Symmetry 2020, 12(9), 1456; https://doi.org/10.3390/sym12091456 - 4 Sep 2020
Cited by 4 | Viewed by 2651
Abstract
The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a forward–backward–forward splitting method with reflection steps (symmetric) in real Hilbert spaces. Weak and [...] Read more.
The forward–backward–forward (FBF) splitting method is a popular iterative procedure for finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. In this paper, we introduce a forward–backward–forward splitting method with reflection steps (symmetric) in real Hilbert spaces. Weak and strong convergence analyses of the proposed method are established under suitable assumptions. Moreover, a linear convergence rate of an inertial modified forward–backward–forward splitting method is also presented. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
23 pages, 1109 KiB  
Article
Convergence Analysis of Self-Adaptive Inertial Extra-Gradient Method for Solving a Family of Pseudomonotone Equilibrium Problems with Application
by Thanatporn Bantaojai, Nuttapol Pakkaranang, Habib ur Rehman, Poom Kumam and Wiyada Kumam
Symmetry 2020, 12(8), 1332; https://doi.org/10.3390/sym12081332 - 10 Aug 2020
Cited by 7 | Viewed by 2861
Abstract
In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous [...] Read more.
In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous iterations. The advantage of the method is that it operates without prior knowledge of Lipschitz-type constants and any line search method. The weak convergence of the method is established by taking mild conditions on a bifunction. In the context of an application, fixed-point theorems involving strict pseudo-contraction and results for pseudomonotone variational inequalities are considered. Many numerical results have been reported to explain the numerical behavior of the proposed method. Full article
(This article belongs to the Special Issue Symmetry in Optimization and Control with Real World Applications)
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