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Variational Problems and Applications
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Variational Problems and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Dynamical Systems".

Deadline for manuscript submissions: closed (31 July 2022) | Viewed by 22899

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Prof. Dr. Savin Treanta

E-Mail Website
Guest Editor
Department of Applied Mathematics, Faculty of Applied Sciences, University Politehnica of Bucharest, 060042 Bucharest, Romania
Interests: variational and functional analysis; optimization methods; control theory; differential equations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Over time, many researchers have been interested in obtaining solution procedures in variational (interval/fuzzy) analysis and robust control. In order to formulate necessary and sufficient optimality/efficiency conditions and duality theorems for different classes of robust and interval-valued/fuzzy variational problems, various approaches have been proposed. The current Special Issue is situated around studies of uncertain variational problems. This Special Issue aims to develop the research in this field by formulating and demonstrating some characterization results of well-posedness and robust efficient solutions in new classes of (multiobjective) variational (control) problems governed by multiple and/or path-independent curvilinear integral cost functionals and robust mixed and/or isoperimetric constraints involving first- and second-order partial differential equations. Therefore, I cordially invite you to publish your results on related subjects (variational inequalities, evolutionary problems, and so on) in this Special Issue. 

Prof. Dr. Savin Treanta
Guest Editor

Manuscript Submission Information

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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.

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Keywords

  • optimization problems
  • optimal control
  • variational problems
  • well-posedness
  • partial differential equations
  • generalized convexity
  • dynamical systems

Published Papers (15 papers)

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Editorial

Jump to: Research, Review

4 pages, 217 KiB  
Editorial
Variational Problems and Applications
by Savin Treanţă
Mathematics 2023, 11(1), 205; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010205 - 30 Dec 2022
Viewed by 827
Abstract
Over the years, many researchers have been interested in obtaining solution procedures in variational (interval/fuzzy) analysis and robust control [...] Full article
(This article belongs to the Special Issue Variational Problems and Applications)

Research

Jump to: Editorial, Review

18 pages, 398 KiB  
Article
Variational Estimation Methods for Sturm–Liouville Problems
by Elena Corina Cipu and Cosmin Dănuţ Barbu
Mathematics 2022, 10(20), 3728; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203728 - 11 Oct 2022
Cited by 2 | Viewed by 859
Abstract
In this paper, we are concerned with approach solutions for Sturm–Liouville problems (SLP) using variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also set forth, and the connection between the solution of each formulation is then proved. Variational estimations [...] Read more.
In this paper, we are concerned with approach solutions for Sturm–Liouville problems (SLP) using variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also set forth, and the connection between the solution of each formulation is then proved. Variational estimations (the variational equation associated through the Euler–Lagrange variational principle and Nehari’s method, shooting method and bisection method) and iterative variational methods (He’s method and HPM) for regular RSL are unitary presented in final part of the paper, which ends with applications. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
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16 pages, 519 KiB  
Article
Stationary Conditions and Characterizations of Solution Sets for Interval-Valued Tightened Nonlinear Problems
by Kin Keung Lai, Shashi Kant Mishra, Sanjeev Kumar Singh and Mohd Hassan
Mathematics 2022, 10(15), 2763; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152763 - 04 Aug 2022
Cited by 4 | Viewed by 1032
Abstract
In this paper, we obtain characterizations of solution sets of the interval-valued mathematical programming problems with switching constraints. Stationary conditions which are weaker than the standard Karush–Kuhn–Tucker conditions need to be discussed in order to find the necessary optimality conditions. We introduce corresponding [...] Read more.
In this paper, we obtain characterizations of solution sets of the interval-valued mathematical programming problems with switching constraints. Stationary conditions which are weaker than the standard Karush–Kuhn–Tucker conditions need to be discussed in order to find the necessary optimality conditions. We introduce corresponding weak, Mordukhovich, and strong stationary conditions for the corresponding interval-valued mathematical programming problems with switching constraints (IVPSC) and interval-valued tightened nonlinear problems (IVTNP), because the W-stationary condition of IVPSC is equivalent to Karush–Kuhn–Tucker conditions of the IVTNP. Furthermore, we use strong stationary conditions to characterize the several solutions sets for IVTNP, in which the last ones are particular solutions sets for IVPSC at the same time, because the feasible set of tightened nonlinear problems (IVTNP) is a subset of the feasible set of the mathematical programs with switching constraints (IVPSC). Full article
(This article belongs to the Special Issue Variational Problems and Applications)
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7 pages, 354 KiB  
Article
Interaction Behaviours between Soliton and Cnoidal Periodic Waves for Nonlocal Complex Modified Korteweg–de Vries Equation
by Junda Peng, Bo Ren, Shoufeng Shen and Guofang Wang
Mathematics 2022, 10(9), 1429; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091429 - 23 Apr 2022
Cited by 1 | Viewed by 1159
Abstract
The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other [...] Read more.
The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other different nonlinear excitations are constructed via the nonauto-Bäcklund transformation theorem. By selecting cnoidal periodic waves, the interaction between one kink soliton and the cnoidal periodic waves is derived. The specific Jacobi function-type solution and graphs of its analysis are provided in this paper. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
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18 pages, 765 KiB  
Article
Chaos Embed Marine Predator (CMPA) Algorithm for Feature Selection
by Adel Fahad Alrasheedi, Khalid Abdulaziz Alnowibet, Akash Saxena, Karam M. Sallam and Ali Wagdy Mohamed
Mathematics 2022, 10(9), 1411; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091411 - 22 Apr 2022
Cited by 14 | Viewed by 1441
Abstract
Data mining applications are growing with the availability of large data; sometimes, handling large data is also a typical task. Segregation of the data for extracting useful information is inevitable for designing modern technologies. Considering this fact, the work proposes a chaos embed [...] Read more.
Data mining applications are growing with the availability of large data; sometimes, handling large data is also a typical task. Segregation of the data for extracting useful information is inevitable for designing modern technologies. Considering this fact, the work proposes a chaos embed marine predator algorithm (CMPA) for feature selection. The optimization routine is designed with the aim of maximizing the classification accuracy with the optimal number of features selected. The well-known benchmark data sets have been chosen for validating the performance of the proposed algorithm. A comparative analysis of the performance with some well-known algorithms advocates the applicability of the proposed algorithm. Further, the analysis has been extended to some of the well-known chaotic algorithms; first, the binary versions of these algorithms are developed and then the comparative analysis of the performance has been conducted on the basis of mean features selected, classification accuracy obtained and fitness function values. Statistical significance tests have also been conducted to establish the significance of the proposed algorithm. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
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25 pages, 803 KiB  
Article
Guided Hybrid Modified Simulated Annealing Algorithm for Solving Constrained Global Optimization Problems
by Khalid Abdulaziz Alnowibet, Salem Mahdi, Mahmoud El-Alem, Mohamed Abdelawwad and Ali Wagdy Mohamed
Mathematics 2022, 10(8), 1312; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081312 - 14 Apr 2022
Cited by 14 | Viewed by 2127
Abstract
In this paper, a hybrid gradient simulated annealing algorithm is guided to solve the constrained optimization problem. In trying to solve constrained optimization problems using deterministic, stochastic optimization methods or hybridization between them, penalty function methods are the most popular approach due to [...] Read more.
In this paper, a hybrid gradient simulated annealing algorithm is guided to solve the constrained optimization problem. In trying to solve constrained optimization problems using deterministic, stochastic optimization methods or hybridization between them, penalty function methods are the most popular approach due to their simplicity and ease of implementation. There are many approaches to handling the existence of the constraints in the constrained problem. The simulated-annealing algorithm (SA) is one of the most successful meta-heuristic strategies. On the other hand, the gradient method is the most inexpensive method among the deterministic methods. In previous literature, the hybrid gradient simulated annealing algorithm (GLMSA) has demonstrated efficiency and effectiveness to solve unconstrained optimization problems. In this paper, therefore, the GLMSA algorithm is generalized to solve the constrained optimization problems. Hence, a new approach penalty function is proposed to handle the existence of the constraints. The proposed approach penalty function is used to guide the hybrid gradient simulated annealing algorithm (GLMSA) to obtain a new algorithm (GHMSA) that finds the constrained optimization problem. The performance of the proposed algorithm is tested on several benchmark optimization test problems and some well-known engineering design problems with varying dimensions. Comprehensive comparisons against other methods in the literature are also presented. The results indicate that the proposed method is promising and competitive. The comparison results between the GHMSA and the other four state-Meta-heuristic algorithms indicate that the proposed GHMSA algorithm is competitive with, and in some cases superior to, other existing algorithms in terms of the quality, efficiency, convergence rate, and robustness of the final result. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
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15 pages, 586 KiB  
Article
Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings
by Muhammad Bilal Khan, Savin Treanțǎ, Mohamed S. Soliman, Kamsing Nonlaopon and Hatim Ghazi Zaini
Mathematics 2022, 10(4), 611; https://0-doi-org.brum.beds.ac.uk/10.3390/math10040611 - 16 Feb 2022
Cited by 9 | Viewed by 1719
Abstract
The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the [...] Read more.
The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions (IV-Fs) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs). Then, we obtain Hermite–Hadamard (𝓗-𝓗) and Hermite–Hadamard–Fejér (𝓗-𝓗-Fejér) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
16 pages, 304 KiB  
Article
Hermite-Hadamard-Type Fractional Inclusions for Interval-Valued Preinvex Functions
by Kin Keung Lai, Jaya Bisht, Nidhi Sharma and Shashi Kant Mishra
Mathematics 2022, 10(2), 264; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020264 - 16 Jan 2022
Cited by 14 | Viewed by 1361
Abstract
We introduce a new class of interval-valued preinvex functions termed as harmonically h-preinvex interval-valued functions. We establish new inclusion of Hermite–Hadamard for harmonically h-preinvex interval-valued function via interval-valued Riemann–Liouville fractional integrals. Further, we prove fractional Hermite–Hadamard-type inclusions for the product of [...] Read more.
We introduce a new class of interval-valued preinvex functions termed as harmonically h-preinvex interval-valued functions. We establish new inclusion of Hermite–Hadamard for harmonically h-preinvex interval-valued function via interval-valued Riemann–Liouville fractional integrals. Further, we prove fractional Hermite–Hadamard-type inclusions for the product of two harmonically h-preinvex interval-valued functions. In this way, these findings include several well-known results and newly obtained results of the existing literature as special cases. Moreover, applications of the main results are demonstrated by presenting some examples. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
17 pages, 347 KiB  
Article
Riemann–Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation
by Muhammad Bilal Khan, Hatim Ghazi Zaini, Savin Treanțǎ, Mohamed S. Soliman and Kamsing Nonlaopon
Mathematics 2022, 10(2), 204; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020204 - 10 Jan 2022
Cited by 26 | Viewed by 1535
Abstract
The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two [...] Read more.
The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In this paper, we study a new version of interval-valued functions (I-V·Fs), known as left and right χ-pre-invex interval-valued functions (LR-χ-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, we derive numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way. In addition, instructive instances are provided to aid our conclusions. Meanwhile, we’ll discuss a few specific examples that may be extrapolated from our primary findings. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
14 pages, 311 KiB  
Article
Optimality Conditions and Duality for a Class of Generalized Convex Interval-Valued Optimization Problems
by Yating Guo, Guoju Ye, Wei Liu, Dafang Zhao and Savin Treanţǎ
Mathematics 2021, 9(22), 2979; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222979 - 22 Nov 2021
Cited by 16 | Viewed by 1528
Abstract
This paper is devoted to derive optimality conditions and duality theorems for interval-valued optimization problems based on gH-symmetrically derivative. Further, the concepts of symmetric pseudo-convexity and symmetric quasi-convexity for interval-valued functions are proposed to extend above optimization conditions. Examples are also presented to [...] Read more.
This paper is devoted to derive optimality conditions and duality theorems for interval-valued optimization problems based on gH-symmetrically derivative. Further, the concepts of symmetric pseudo-convexity and symmetric quasi-convexity for interval-valued functions are proposed to extend above optimization conditions. Examples are also presented to illustrate corresponding results. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
12 pages, 269 KiB  
Article
On Well-Posedness of Some Constrained Variational Problems
by Savin Treanţă
Mathematics 2021, 9(19), 2478; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192478 - 04 Oct 2021
Cited by 4 | Viewed by 1213
Abstract
By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by [...] Read more.
By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by defining the set of approximating solutions for the class of variational problems under study, we establish several results on well-posedness. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
4 pages, 223 KiB  
Article
A Remark on the Change of Variable Theorem for the Riemann Integral
by Alexander Kuleshov
Mathematics 2021, 9(16), 1899; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161899 - 10 Aug 2021
Cited by 3 | Viewed by 2728
Abstract
In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), [...] Read more.
In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function f on its entire domain) can be omitted while the change of variable formula still holds. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
13 pages, 284 KiB  
Article
On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals
by Savin Treanţă and Koushik Das
Mathematics 2021, 9(15), 1790; https://0-doi-org.brum.beds.ac.uk/10.3390/math9151790 - 28 Jul 2021
Cited by 9 | Viewed by 1252
Abstract
In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named [...] Read more.
In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form. Full article
(This article belongs to the Special Issue Variational Problems and Applications)

Review

Jump to: Editorial, Research

16 pages, 359 KiB  
Review
Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms
by Octav Olteanu
Mathematics 2022, 10(18), 3288; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183288 - 10 Sep 2022
Cited by 2 | Viewed by 997
Abstract
As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by [...] Read more.
As is well-known, unlike the one-dimensional case, there exist nonnegative polynomials in several real variables that are not sums of squares. First, we briefly review a method of approximating any real-valued nonnegative continuous compactly supported function defined on a closed unbounded subset by dominating special polynomials that are sums of squares. This also works in several-dimensional cases. To perform this, a Hahn–Banach-type theorem (Kantorovich theorem on an extension of positive linear operators), a Haviland theorem, and the notion of a moment-determinate measure are applied. Second, completions and other results on solving full Markov moment problems in terms of quadratic forms are proposed based on polynomial approximation. The existence and uniqueness of the solution are discussed. Third, the characterization of the constraints T1≤T≤T2 for the linear operator T, only in terms of quadratic forms, is deduced. Here, T1, T,and T2 are bounded linear operators. Concrete spaces, operators, and functionals are involved in our corollaries or examples. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
13 pages, 320 KiB  
Review
Recent Advances of Constrained Variational Problems Involving Second-Order Partial Derivatives: A Review
by Savin Treanţă
Mathematics 2022, 10(15), 2599; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152599 - 26 Jul 2022
Cited by 2 | Viewed by 847
Abstract
This paper comprehensively reviews the nonlinear dynamics given by some classes of constrained control problems which involve second-order partial derivatives. Specifically, necessary optimality conditions are formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and [...] Read more.
This paper comprehensively reviews the nonlinear dynamics given by some classes of constrained control problems which involve second-order partial derivatives. Specifically, necessary optimality conditions are formulated and proved for the considered variational control problems governed by integral functionals. In addition, the well-posedness and the associated variational inequalities are considered in the present review paper. Full article
(This article belongs to the Special Issue Variational Problems and Applications)
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