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Nonlinear Optimization, Variational Inequalities and Equilibrium Problems
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Nonlinear Optimization, Variational Inequalities and Equilibrium Problems

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

Deadline for manuscript submissions: closed (28 February 2021) | Viewed by 34457

Special Issue Editor

Prof. Dr. Mihai Postolache

E-Mail Website
Guest Editor
1. Department of General Education, China Medical University, Taichung 40402, Taiwan
2. Department of Interior Design, Asia University, Taichung 41354, Taiwan
3. Gh. Mihoc-C. Iacob Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, 050711 Bucharest, Romania
4. Department of Mathematics and Informatics, University “Politehnica” of Bucharest, 060042 Bucharest, Romania
Interests: fixed point theory; continuous optimization; numerical algorithms
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The scope of our Special Issue, titled Nonlinear Optimization, Variational Inequalities, and Equilibrium Problems, is to bring together outstanding theoretical contributions in these active research areas, with real world applications.

As we know, optimization theory, either in a continuous or discrete version, plays an important role in economy, finance, and engineering, where studies on equilibria, control, and efficiency are strongly required. Specific research topics herein contain—among others—single or vector optimization, best approximation, optimal control, and algorithms design. Variational inequalities offer a suitable framework for specific problems of optimization, and have applications at least to signal processing and transportation networks. Last, but not least, the equilibrium problem is important for nonlinear analysis and applied sciences when studying models in mathematical biology, economics, and game theory.

It is the purpose of this Special Issue to report the hot and significant results in the directions specified above. We will select and accept only high-quality papers, impeccably written and organized, containing original research results, with illustrative applications, and a limited number of survey articles of exceptional merit.

The research topics include, but are not limited to, the following:

  • Discrete or continuous optimization;
  • General variational inequalities;
  • Best approximation problems;
  • Equations with nonlinear operators;
  • Dynamical systems and applications;
  • Mathematically-oriented control;
  • Numerical mathematics and analysis;
  • Algorithms for image encoding and recovery.

Prof. Dr. Mihai Postolache
Guest Editor

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Published Papers (14 papers)

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Research

13 pages, 263 KiB  
Article
Some Improvements of the Cauchy-Schwarz Inequality Using the Tapia Semi-Inner-Product
by Nicuşor Minculete and Hamid Reza Moradi
Mathematics 2020, 8(12), 2112; https://doi.org/10.3390/math8122112 - 26 Nov 2020
Cited by 6 | Viewed by 2492
Abstract
The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy–Schwarz inequality, which is improved by using the Tapia semi-inner-product. Finally, we obtain [...] Read more.
The aim of this article is to establish several estimates of the triangle inequality in a normed space over the field of real numbers. We obtain some improvements of the Cauchy–Schwarz inequality, which is improved by using the Tapia semi-inner-product. Finally, we obtain some new inequalities for the numerical radius and norm inequalities for Hilbert space operators. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
17 pages, 287 KiB  
Article
Primal Lower Nice Functions in Reflexive Smooth Banach Spaces
by Messaoud Bounkhel and Mostafa Bachar
Mathematics 2020, 8(11), 2066; https://doi.org/10.3390/math8112066 - 19 Nov 2020
Cited by 2 | Viewed by 1794
Abstract
In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces [...] Read more.
In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29]. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
29 pages, 901 KiB  
Article
A Generalized Viscosity Inertial Projection and Contraction Method for Pseudomonotone Variational Inequality and Fixed Point Problems
by Lateef Olakunle Jolaoso and Maggie Aphane
Mathematics 2020, 8(11), 2039; https://doi.org/10.3390/math8112039 - 16 Nov 2020
Cited by 2 | Viewed by 1911
Abstract
We introduce a new projection and contraction method with inertial and self-adaptive techniques for solving variational inequalities and split common fixed point problems in real Hilbert spaces. The stepsize of the algorithm is selected via a self-adaptive method and does not require prior [...] Read more.
We introduce a new projection and contraction method with inertial and self-adaptive techniques for solving variational inequalities and split common fixed point problems in real Hilbert spaces. The stepsize of the algorithm is selected via a self-adaptive method and does not require prior estimate of norm of the bounded linear operator. More so, the cost operator of the variational inequalities does not necessarily needs to satisfies Lipschitz condition. We prove a strong convergence result under some mild conditions and provide an application of our result to split common null point problems. Some numerical experiments are reported to illustrate the performance of the algorithm and compare with some existing methods. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
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19 pages, 10033 KiB  
Article
Relaxed Inertial Tseng’s Type Method for Solving the Inclusion Problem with Application to Image Restoration
by Jamilu Abubakar, Poom Kumam, Abdulkarim Hassan Ibrahim and Anantachai Padcharoen
Mathematics 2020, 8(5), 818; https://doi.org/10.3390/math8050818 - 18 May 2020
Cited by 29 | Viewed by 3014
Abstract
The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the [...] Read more.
The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
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11 pages, 266 KiB  
Article
Nondifferentiable Multiobjective Programming Problem under Strongly K-Gf-Pseudoinvexity Assumptions
by Ramu Dubey, Lakshmi Narayan Mishra, Luis Manuel Sánchez Ruiz and Deepak Umrao Sarwe
Mathematics 2020, 8(5), 738; https://doi.org/10.3390/math8050738 - 7 May 2020
Viewed by 1671
Abstract
In this paper we consider the introduction of the concept of (strongly) K- G f -pseudoinvex functions which enable to study a pair of nondifferentiable K-G- Mond-Weir type symmetric multiobjective programming model under such assumptions. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
15 pages, 804 KiB  
Article
Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators
by Yonghong Yao, Naseer Shahzad and Jen-Chih Yao
Mathematics 2020, 8(4), 461; https://doi.org/10.3390/math8040461 - 25 Mar 2020
Cited by 4 | Viewed by 1733
Abstract
The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding [...] Read more.
The projected subgradient algorithms can be considered as an improvement of the projected algorithms and the subgradient algorithms for the equilibrium problems of the class of monotone and Lipschitz continuous operators. In this paper, we present and analyze an iterative algorithm for finding a common element of the fixed point of pseudocontractive operators and the pseudomonotone equilibrium problem in Hilbert spaces. The suggested iterative algorithm is based on the projected method and subgradient method with a linearsearch technique. We show the strong convergence result for the iterative sequence generated by this algorithm. Some applications are also included. Our result improves and extends some existing results in the literature. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
31 pages, 9899 KiB  
Article
A Parallel-Viscosity-Type Subgradient Extragradient-Line Method for Finding the Common Solution of Variational Inequality Problems Applied to Image Restoration Problems
by Suthep Suantai, Pronpat Peeyada, Damrongsak Yambangwai and Watcharaporn Cholamjiak
Mathematics 2020, 8(2), 248; https://doi.org/10.3390/math8020248 - 14 Feb 2020
Cited by 14 | Viewed by 2763
Abstract
In this paper, we study a modified viscosity type subgradient extragradient-line method with a parallel monotone hybrid algorithm for approximating a common solution of variational inequality problems. Under suitable conditions in Hilbert spaces, the strong convergence theorem of the proposed algorithm to such [...] Read more.
In this paper, we study a modified viscosity type subgradient extragradient-line method with a parallel monotone hybrid algorithm for approximating a common solution of variational inequality problems. Under suitable conditions in Hilbert spaces, the strong convergence theorem of the proposed algorithm to such a common solution is proved. We then give numerical examples in both finite and infinite dimensional spaces to justify our main theorem. Finally, we can show that our proposed algorithm is flexible and has good quality for use with common types of blur effects in image recovery. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
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10 pages, 269 KiB  
Article
Results in wt-Distance over b-Metric Spaces
by Erdal Karapınar and Cristian Chifu
Mathematics 2020, 8(2), 220; https://doi.org/10.3390/math8020220 - 9 Feb 2020
Cited by 31 | Viewed by 2406
Abstract
In this manuscript, we introduce Meir-Keeler type contractions and Geraghty type contractions in the setting of the w t -distances over b-metric spaces. We examine the existence of a fixed point for such mappings. Under some additional assumption, we proved the uniqueness [...] Read more.
In this manuscript, we introduce Meir-Keeler type contractions and Geraghty type contractions in the setting of the w t -distances over b-metric spaces. We examine the existence of a fixed point for such mappings. Under some additional assumption, we proved the uniqueness of the found fixed point. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
14 pages, 271 KiB  
Article
Geometric Dynamics on Riemannian Manifolds
by Constantin Udriste and Ionel Tevy
Mathematics 2020, 8(1), 79; https://doi.org/10.3390/math8010079 - 3 Jan 2020
Cited by 8 | Viewed by 2802
Abstract
The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and [...] Read more.
The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
13 pages, 286 KiB  
Article
Iterative Algorithms for Pseudomonotone Variational Inequalities and Fixed Point Problems of Pseudocontractive Operators
by Yonghong Yao, Mihai Postolache and Jen-Chih Yao
Mathematics 2019, 7(12), 1189; https://doi.org/10.3390/math7121189 - 4 Dec 2019
Cited by 11 | Viewed by 2695
Abstract
In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. [...] Read more.
In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
16 pages, 391 KiB  
Article
Tseng Type Methods for Inclusion and Fixed Point Problems with Applications
by Raweerote Suparatulatorn and Anchalee Khemphet
Mathematics 2019, 7(12), 1175; https://doi.org/10.3390/math7121175 - 3 Dec 2019
Cited by 12 | Viewed by 2254
Abstract
An algorithm is introduced to find an answer to both inclusion problems and fixed point problems. This algorithm is a modification of Tseng type methods inspired by Mann’s type iteration and viscosity approximation methods. On certain conditions, the iteration obtained from the algorithm [...] Read more.
An algorithm is introduced to find an answer to both inclusion problems and fixed point problems. This algorithm is a modification of Tseng type methods inspired by Mann’s type iteration and viscosity approximation methods. On certain conditions, the iteration obtained from the algorithm converges strongly. Moreover, applications to the convex feasibility problem and the signal recovery in compressed sensing are considered. Especially, some numerical experiments of the algorithm are demonstrated. These results are compared to those of the previous algorithm. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
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12 pages, 230 KiB  
Article
Strongly Convex Functions of Higher Order Involving Bifunction
by Bandar B. Mohsen, Muhammad Aslam Noor, Khalida Inayat Noor and Mihai Postolache
Mathematics 2019, 7(11), 1028; https://doi.org/10.3390/math7111028 - 1 Nov 2019
Cited by 28 | Viewed by 2710
Abstract
Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for [...] Read more.
Some new concepts of the higher order strongly convex functions involving an arbitrary bifuction are considered in this paper. Some properties of the higher order strongly convex functions are investigated under suitable conditions. Some important special cases are discussed. The parallelogram laws for Banach spaces are obtained as applications of higher order strongly affine convex functions as novel applications. Results obtained in this paper can be viewed as refinement and improvement of previously known results. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
22 pages, 295 KiB  
Article
Weighted Method for Uncertain Nonlinear Variational Inequality Problems
by Cunlin Li, Mihai Postolache and Zhifu Jia
Mathematics 2019, 7(10), 974; https://doi.org/10.3390/math7100974 - 15 Oct 2019
Cited by 7 | Viewed by 1824
Abstract
A convex combined expectations regularized gap function with uncertain variable is presented to deal with uncertain nonlinear variational inequality problems (UNVIP). The UNVIP is transformed into a minimization problem through an uncertain weighted expected residual function. Moreover, the convergence of the global optimal [...] Read more.
A convex combined expectations regularized gap function with uncertain variable is presented to deal with uncertain nonlinear variational inequality problems (UNVIP). The UNVIP is transformed into a minimization problem through an uncertain weighted expected residual function. Moreover, the convergence of the global optimal solutions of the uncertain weighted expected residual minimization model is given through the integration by parts method under the compact space of the uncertain event. The limiting behaviors of the transformed model are analyzed. Furthermore, a compact approximation method is proposed in the unbounded uncertain event space. Through analysis of the convergence of UWERM model and reasonable hypothesis, the compact approximation method is verified under the circumstance of Holder continuity. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
21 pages, 2067 KiB  
Article
Inertial Method for Bilevel Variational Inequality Problems with Fixed Point and Minimizer Point Constraints
by Seifu Endris Yimer, Poom Kumam, Anteneh Getachew Gebrie and Rabian Wangkeeree
Mathematics 2019, 7(9), 841; https://doi.org/10.3390/math7090841 - 11 Sep 2019
Cited by 8 | Viewed by 3329
Abstract
In this paper, we introduce an iterative scheme with inertial effect using Mann iterative scheme and gradient-projection for solving the bilevel variational inequality problem over the intersection of the set of common fixed points of a finite number of nonexpansive mappings and the [...] Read more.
In this paper, we introduce an iterative scheme with inertial effect using Mann iterative scheme and gradient-projection for solving the bilevel variational inequality problem over the intersection of the set of common fixed points of a finite number of nonexpansive mappings and the set of solution points of the constrained optimization problem. Under some mild conditions we obtain strong convergence of the proposed algorithm. Two examples of the proposed bilevel variational inequality problem are also shown through numerical results. Full article
(This article belongs to the Special Issue Nonlinear Optimization, Variational Inequalities and Equilibrium Problems)
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