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Mathematics
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Nonlinear Analysis and Optimization
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Nonlinear Analysis and Optimization

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 14596

Special Issue Editors

Prof. Dr. Ching-Feng Wen

E-Mail Website
Guest Editor
Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
Interests: fixed point theory; theory and algorithms on variational inequalities; set-valued and variational analysis; nonlinear analysis; optimization; well-posedness and optimal control
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jen-Chih Yao

E-Mail Website
Guest Editor
1. Research Center for Interneural Computing, China Medical University Hospital, Taichung City 404332, Taiwan
2. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan
Interests: vector optimization; fixed point theory; variational inequalities; complementarity problems; variational analysis; equilibrium problems; optimal control; generalized convexity and generalized monotonicity
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yeong-Cheng Liou

E-Mail Website
Guest Editor
Department of Healthcare Administration and Medical Informatics, and Research Center of Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 807, Taiwan
Interests: optimization and decision analysis; approximations and fixed point theory; content and knowledge management systems; intelligent system and cryptography; supply chain and logistics management; game theory and economic computing; optimal portfolio and finance engineering; algorithms and software engineering

Special Issue Information

Dear Colleagues,

Nonlinear analysis is a rapidly growing area of mathematics, with numerous applications in optimization, control theory, economics, engineering, and other disciplines. Recently, with the tools of nonlinear analysis, various reformulations for optimization problems and techniques in analyzing the convergence of algorithms have explored new and rich directions. The goals of this Special Issue are to stimulate further research and to highlight recent advances in these fields as well as to promote, encourage, and bring together researchers in the fields of nonlinear analysis and optimization. This Issue is devoted to the publication of original articles of current interest on every theoretical, computational, and application aspect of nonlinear analysis, variational analysis, convex analysis, fixed point theory, and optimization techniques, as well as their applications to science, engineering, and other disciplines.

Prof. Dr. Ching-Feng Wen
Prof. Dr. Jen-Chih Yao
Prof. Dr. Yeong-Cheng Liou
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • Nonlinear analysis
  • Optimization
  • Optimal control
  • Variational analysis
  • Set-valued optimization

Published Papers (7 papers)

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Research

17 pages, 290 KiB  
Article
Weak Measurable Optimal Controls for the Problems of Bolza
by Gerardo Sánchez Licea
Mathematics 2021, 9(2), 191; https://0-doi-org.brum.beds.ac.uk/10.3390/math9020191 - 19 Jan 2021
Cited by 1 | Viewed by 1378
Abstract
Two sufficiency theorems for parametric and a nonparametric problems of Bolza in optimal control are derived. The dynamics of the problems are nonlinear, the initial and final states are free, and the main results can be applied when nonlinear mixed time-state-control inequality and [...] Read more.
Two sufficiency theorems for parametric and a nonparametric problems of Bolza in optimal control are derived. The dynamics of the problems are nonlinear, the initial and final states are free, and the main results can be applied when nonlinear mixed time-state-control inequality and equality constraints are presented. The deviation between admissible costs and optimal costs around the optimal control is estimated by functionals playing the role of the square of some norms. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
9 pages, 270 KiB  
Article
Krasnoselskii–Mann Viscosity Approximation Method for Nonexpansive Mappings
by Najla Altwaijry, Tahani Aldhaban, Souhail Chebbi and Hong-Kun Xu
Mathematics 2020, 8(7), 1153; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071153 - 14 Jul 2020
Cited by 2 | Viewed by 1884
Abstract
We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., [...] Read more.
We show that the viscosity approximation method coupled with the Krasnoselskii–Mann iteration generates a sequence that strongly converges to a fixed point of a given nonexpansive mapping in the setting of uniformly smooth Banach spaces. Our result shows that the geometric property (i.e., uniform smoothness) of the underlying space plays a role in relaxing the conditions on the choice of regularization parameters and step sizes in iterative methods. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
11 pages, 467 KiB  
Article
Strong Convergence of Modified Inertial Mann Algorithms for Nonexpansive Mappings
by Bing Tan, Zheng Zhou and Songxiao Li
Mathematics 2020, 8(4), 462; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040462 - 25 Mar 2020
Cited by 12 | Viewed by 2118
Abstract
We investigated two new modified inertial Mann Halpern and inertial Mann viscosity algorithms for solving fixed point problems. Strong convergence theorems under some fewer restricted conditions are established in the framework of infinite dimensional Hilbert spaces. Finally, some numerical examples are provided to [...] Read more.
We investigated two new modified inertial Mann Halpern and inertial Mann viscosity algorithms for solving fixed point problems. Strong convergence theorems under some fewer restricted conditions are established in the framework of infinite dimensional Hilbert spaces. Finally, some numerical examples are provided to support our main results. The algorithms and results presented in this paper can generalize and extend corresponding results previously known in the literature. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
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16 pages, 635 KiB  
Article
A Hybrid Forward–Backward Algorithm and Its Optimization Application
by Liya Liu, Xiaolong Qin and Jen-Chih Yao
Mathematics 2020, 8(3), 447; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030447 - 19 Mar 2020
Cited by 1 | Viewed by 3006
Abstract
In this paper, we study a hybrid forward–backward algorithm for sparse reconstruction. Our algorithm involves descent, splitting and inertial ideas. Under suitable conditions on the algorithm parameters, we establish a strong convergence solution theorem in the framework of Hilbert spaces. Numerical experiments are [...] Read more.
In this paper, we study a hybrid forward–backward algorithm for sparse reconstruction. Our algorithm involves descent, splitting and inertial ideas. Under suitable conditions on the algorithm parameters, we establish a strong convergence solution theorem in the framework of Hilbert spaces. Numerical experiments are also provided to illustrate the application in the field of signal processing. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
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13 pages, 290 KiB  
Article
On Null-Continuity of Monotone Measures
by Jun Li
Mathematics 2020, 8(2), 205; https://0-doi-org.brum.beds.ac.uk/10.3390/math8020205 - 06 Feb 2020
Cited by 3 | Viewed by 1718
Abstract
The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by [...] Read more.
The null-continuity of monotone measures is a weaker condition than continuity from below and possesses many special properties. This paper further studies this structure characteristic of monotone measures. Some basic properties of null-continuity are shown and the characteristic of null-continuity is described by using convergence of sequence of measurable functions. It is shown that the null-continuity is a necessary condition that the classical Riesz’s theorem remains valid for monotone measures. When considered measurable space ( X , A ) is S-compact, the null-continuity condition is also sufficient for Riesz’s theorem. By means of the equivalence of null-continuity and property (S) of monotone measures, a version of Egoroff’s theorem for monotone measures on S-compact spaces is also presented. We also study the Sugeno integral and the Choquet integral by using null-continuity and generalize some previous results. We show that the monotone measures defined by the Sugeno integral (or the Choquet integral) preserve structural characteristic of null-continuity of the original monotone measures. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
15 pages, 265 KiB  
Article
System of Multi-Valued Mixed Variational Inclusions with XOR-Operation in Real Ordered Uniformly Smooth Banach Spaces
by Rais Ahmad, Imran Ali, Xiao-Bing Li, Mohd. Ishtyak and Ching-Feng Wen
Mathematics 2019, 7(11), 1027; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111027 - 01 Nov 2019
Cited by 2 | Viewed by 1351
Abstract
In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation ⊕ in real ordered uniformly smooth Banach spaces. This system consists of bimappings, multi-valued mappings and Cayley operators. An iterative algorithm is suggested to find the solution [...] Read more.
In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation ⊕ in real ordered uniformly smooth Banach spaces. This system consists of bimappings, multi-valued mappings and Cayley operators. An iterative algorithm is suggested to find the solution to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ and consequently an existence and convergence result is proved. In support of our main result, an example is constructed. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
17 pages, 858 KiB  
Article
Modified Proximal Algorithms for Finding Solutions of the Split Variational Inclusions
by Suthep Suantai, Suparat Kesornprom and Prasit Cholamjiak
Mathematics 2019, 7(8), 708; https://0-doi-org.brum.beds.ac.uk/10.3390/math7080708 - 06 Aug 2019
Cited by 19 | Viewed by 2267
Abstract
We investigate the split variational inclusion problem in Hilbert spaces. We propose efficient algorithms in which, in each iteration, the stepsize is chosen self-adaptive, and proves weak and strong convergence theorems. We provide numerical experiments to validate the theoretical results for solving the [...] Read more.
We investigate the split variational inclusion problem in Hilbert spaces. We propose efficient algorithms in which, in each iteration, the stepsize is chosen self-adaptive, and proves weak and strong convergence theorems. We provide numerical experiments to validate the theoretical results for solving the split variational inclusion problem as well as the comparison to algorithms defined by Byrne et al. and Chuang, respectively. It is shown that the proposed algorithms outrun other algorithms via numerical experiments. As applications, we apply our method to compressed sensing in signal recovery. The proposed methods have as a main advantage that the computation of the Lipschitz constants for the gradient of functions is dropped in generating the sequences. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization)
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