Mathematics

Editorial

Jump to: Research, Review

4 pages, 217 KiB

Open AccessEditorial

Variational Problems and Applications

by Savin Treanţă

Mathematics 2023, 11(1), 205; https://doi.org/10.3390/math11010205 - 30 Dec 2022

Viewed by 1157

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

Open AccessArticle

Variational Estimation Methods for Sturm–Liouville Problems

by Elena Corina Cipu and Cosmin Dănuţ Barbu

Mathematics 2022, 10(20), 3728; https://doi.org/10.3390/math10203728 - 11 Oct 2022

Cited by 3 | Viewed by 1278

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)

► Show Figures

Figure 1

16 pages, 519 KiB

Open AccessArticle

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://doi.org/10.3390/math10152763 - 4 Aug 2022

Cited by 4 | Viewed by 1333

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)

► Show Figures

Figure 1

7 pages, 354 KiB

Open AccessArticle

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://doi.org/10.3390/math10091429 - 23 Apr 2022

Cited by 2 | Viewed by 1483

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)

► Show Figures

Figure 1

18 pages, 765 KiB

Open AccessArticle

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://doi.org/10.3390/math10091411 - 22 Apr 2022

Cited by 18 | Viewed by 1795

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)

► Show Figures

Figure 1

25 pages, 803 KiB

Open AccessArticle

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://doi.org/10.3390/math10081312 - 14 Apr 2022

Cited by 17 | Viewed by 2782

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)

► Show Figures

Figure 1

15 pages, 586 KiB

Open AccessArticle

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://doi.org/10.3390/math10040611 - 16 Feb 2022

Cited by 9 | Viewed by 2093

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

Open AccessArticle

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://doi.org/10.3390/math10020264 - 16 Jan 2022

Cited by 14 | Viewed by 1820

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

Open AccessArticle

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://doi.org/10.3390/math10020204 - 10 Jan 2022

Cited by 28 | Viewed by 1887

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

Open AccessArticle

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://doi.org/10.3390/math9222979 - 22 Nov 2021

Cited by 19 | Viewed by 1965

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

Open AccessArticle

On Well-Posedness of Some Constrained Variational Problems

by Savin Treanţă

Mathematics 2021, 9(19), 2478; https://doi.org/10.3390/math9192478 - 4 Oct 2021

Cited by 6 | Viewed by 1527

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

Open AccessArticle

A Remark on the Change of Variable Theorem for the Riemann Integral

by Alexander Kuleshov

Mathematics 2021, 9(16), 1899; https://doi.org/10.3390/math9161899 - 10 Aug 2021

Cited by 3 | Viewed by 3166

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

Open AccessArticle

On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

by Savin Treanţă and Koushik Das

Mathematics 2021, 9(15), 1790; https://doi.org/10.3390/math9151790 - 28 Jul 2021

Cited by 10 | Viewed by 1554

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)}_{(\bar{b}, \bar{c})}

), which is much easier to study, and provide some characterization results of

(P)

and

{(P)}_{(\bar{b}, \bar{c})}

by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to

{(P)}_{(\bar{b}, \bar{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

Open AccessReview

Markov Moment Problem and Sandwich Conditions on Bounded Linear Operators in Terms of Quadratic Forms

by Octav Olteanu

Mathematics 2022, 10(18), 3288; https://doi.org/10.3390/math10183288 - 10 Sep 2022

Cited by 2 | Viewed by 1260

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

T_{1} \leq T \leq T_{2}

for the linear operator

T,

only in terms of quadratic forms, is deduced. Here,

T_{1}, T, and T_{2}

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

Open AccessFeature PaperReview

Recent Advances of Constrained Variational Problems Involving Second-Order Partial Derivatives: A Review

by Savin Treanţă

Mathematics 2022, 10(15), 2599; https://doi.org/10.3390/math10152599 - 26 Jul 2022

Cited by 3 | Viewed by 1098

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)

Journal Menu

Journal Browser

Variational Problems and Applications

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (15 papers)

Editorial

Research

Review

Further Information

Guidelines

MDPI Initiatives

Follow MDPI