Fractal and Fractional

Editorial

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5 pages, 210 KiB

Open AccessEditorial

Advances in Optimization and Nonlinear Analysis

by Savin Treanţă

Fractal Fract. 2022, 6(7), 364; https://doi.org/10.3390/fractalfract6070364 - 30 Jun 2022

Viewed by 1198

Abstract

There are many applications of optimization and nonlinear analysis in various fields of basic science, engineering, and natural phenomena [...] Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

Research

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46 pages, 30367 KiB

Open AccessArticle

Optimal Design of TD-TI Controller for LFC Considering Renewables Penetration by an Improved Chaos Game Optimizer

by Ahmed H. A. Elkasem, Mohamed Khamies, Mohamed H. Hassan, Ahmed M. Agwa and Salah Kamel

Fractal Fract. 2022, 6(4), 220; https://doi.org/10.3390/fractalfract6040220 - 13 Apr 2022

Cited by 36 | Viewed by 3154

Abstract

This study presents an innovative strategy for load frequency control (LFC) using a combination structure of tilt-derivative and tilt-integral gains to form a TD-TI controller. Furthermore, a new improved optimization technique, namely the quantum chaos game optimizer (QCGO) is applied to tune the [...] Read more.

This study presents an innovative strategy for load frequency control (LFC) using a combination structure of tilt-derivative and tilt-integral gains to form a TD-TI controller. Furthermore, a new improved optimization technique, namely the quantum chaos game optimizer (QCGO) is applied to tune the gains of the proposed combination TD-TI controller in two-area interconnected hybrid power systems, while the effectiveness of the proposed QCGO is validated via a comparison of its performance with the traditional CGO and other optimizers when considering 23 bench functions. Correspondingly, the effectiveness of the proposed controller is validated by comparing its performance with other controllers, such as the proportional-integral-derivative (PID) controller based on different optimizers, the tilt-integral-derivative (TID) controller based on a CGO algorithm, and the TID controller based on a QCGO algorithm, where the effectiveness of the proposed TD-TI controller based on the QCGO algorithm is ensured using different load patterns (i.e., step load perturbation (SLP), series SLP, and random load variation (RLV)). Furthermore, the challenges of renewable energy penetration and communication time delay are considered to test the robustness of the proposed controller in achieving more system stability. In addition, the integration of electric vehicles as dispersed energy storage units in both areas has been considered to test their effectiveness in achieving power grid stability. The simulation results elucidate that the proposed TD-TI controller based on the QCGO controller can achieve more system stability under the different aforementioned challenges. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

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38 pages, 4909 KiB

Open AccessArticle

Non-Dominated Sorting Manta Ray Foraging Optimization for Multi-Objective Optimal Power Flow with Wind/Solar/Small- Hydro Energy Sources

by Fatima Daqaq, Salah Kamel, Mohammed Ouassaid, Rachid Ellaia and Ahmed M. Agwa

Fractal Fract. 2022, 6(4), 194; https://doi.org/10.3390/fractalfract6040194 - 31 Mar 2022

Cited by 14 | Viewed by 3145

Abstract

This present study describes a novel manta ray foraging optimization approach based non-dominated sorting strategy, namely (NSMRFO), for solving the multi-objective optimization problems (MOPs). The proposed powerful optimizer can efficiently achieve good convergence and distribution in both the search and objective spaces. In [...] Read more.

This present study describes a novel manta ray foraging optimization approach based non-dominated sorting strategy, namely (NSMRFO), for solving the multi-objective optimization problems (MOPs). The proposed powerful optimizer can efficiently achieve good convergence and distribution in both the search and objective spaces. In the NSMRFO algorithm, the elitist non-dominated sorting mechanism is followed. Afterwards, a crowding distance with a non-dominated ranking method is integrated for the purpose of archiving the Pareto front and improving the optimal solutions coverage. To judge the NSMRFO performances, a bunch of test functions are carried out including classical unconstrained and constrained functions, a recent benchmark suite known as the completions on evolutionary computation 2020 (CEC2020) that contains twenty-four multimodal optimization problems (MMOPs), some engineering design problems, and also the modified real-world issue known as IEEE 30-bus optimal power flow involving the wind/solar/small-hydro power generations. Comparison findings with multimodal multi-objective evolutionary algorithms (MMMOEAs) and other existing multi-objective approaches with respect to performance indicators reveal the NSMRFO ability to balance between the coverage and convergence towards the true Pareto front (PF) and Pareto optimal sets (PSs). Thus, the competing algorithms fail in providing better solutions while the proposed NSMRFO optimizer is able to attain almost all the Pareto optimal solutions. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

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15 pages, 806 KiB

Open AccessArticle

Hermite-Hadamard Inequalities in Fractional Calculus for Left and Right Harmonically Convex Functions via Interval-Valued Settings

by Muhammad Bilal Khan, Jorge E. Macías-Díaz, Savin Treanțǎ, Mohammed S. Soliman and Hatim Ghazi Zaini

Fractal Fract. 2022, 6(4), 178; https://doi.org/10.3390/fractalfract6040178 - 23 Mar 2022

Cited by 36 | Viewed by 1999

Abstract

The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR-𝓗-convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions ( [...] Read more.

The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR-𝓗-convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions (IV-Fs) linked to Hermite–Hadamard (H-H) and Hermite–Hadamard–Fejér (H-H-Fejér) type inequalities via interval-valued Riemann–Liouville fractional integrals (IV-RL-fractional integrals). We also attain some related inequalities for the product of two LR-𝓗-convex IV-Fs. These findings enable us to identify a new class of inclusions that may be seen as significant generalizations of results proved by Iscan and Chen. Some examples are included in our findings that may be used to determine the validity of the results. The findings in this work can be seen as a considerable advance over previously published findings. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

15 pages, 3939 KiB

Open AccessArticle

Multistability of the Vibrating System of a Micro Resonator

by Yijun Zhu and Huilin Shang

Fractal Fract. 2022, 6(3), 141; https://doi.org/10.3390/fractalfract6030141 - 2 Mar 2022

Cited by 7 | Viewed by 2127

Abstract

Multiple attractors and their fractal basins of attraction can lead to the loss of global stability and integrity of Micro Electro Mechanical Systems (MEMS). In this paper, multistability of a class of electrostatic bilateral capacitive micro-resonator is researched in detail. First, the dynamical [...] Read more.

Multiple attractors and their fractal basins of attraction can lead to the loss of global stability and integrity of Micro Electro Mechanical Systems (MEMS). In this paper, multistability of a class of electrostatic bilateral capacitive micro-resonator is researched in detail. First, the dynamical model is established and made dimensionless. Second, via the perturbating method and the numerical description of basins of attraction, the multiple periodic motions under primary resonance are discussed. It is found that the variation of AC voltage can induce safe jump of the micro resonator. In addition, with the increase of the amplitude of AC voltage, hidden attractors and chaos appear. The results may have some potential value in the design of MEMS devices. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

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7 pages, 277 KiB

Open AccessArticle

The Method of Fundamental Solutions for the 3D Laplace Inverse Geometric Problem on an Annular Domain

by Mojtaba Sajjadmanesh, Hassen Aydi, Eskandar Ameer and Choonkil Park

Fractal Fract. 2022, 6(2), 66; https://doi.org/10.3390/fractalfract6020066 - 27 Jan 2022

Cited by 1 | Viewed by 1995

Abstract

In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in [...] Read more.

In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in a least-square sense and minimisation of the objective function. This approach can also be considered with noisy boundary Cauchy data. The simplicity and efficiency of this method is illustrated in several numerical examples. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

22 pages, 4157 KiB

Open AccessArticle

The Role of the Discount Policy of Prepayment on Environmentally Friendly Inventory Management

by Shirin Sultana, Abu Hashan Md Mashud, Yosef Daryanto, Sujan Miah, Adel Alrasheedi and Ibrahim M. Hezam

Fractal Fract. 2022, 6(1), 26; https://doi.org/10.3390/fractalfract6010026 - 2 Jan 2022

Cited by 11 | Viewed by 2560

Abstract

Nowadays, more and more consumers consider environmentally friendly products in their purchasing decisions. Companies need to adapt to these changes while paying attention to standard business systems such as payment terms. The purpose of this study is to optimize the entire profit function [...] Read more.

Nowadays, more and more consumers consider environmentally friendly products in their purchasing decisions. Companies need to adapt to these changes while paying attention to standard business systems such as payment terms. The purpose of this study is to optimize the entire profit function of a retailer and to find the optimal selling price and replenishment cycle when the demand rate depends on the price and carbon emission reduction level. This study investigates an economic order quantity model that has a demand function with a positive impact of carbon emission reduction besides the selling price. In this model, the supplier requests payment in advance on the purchased cost while offering a discount according to the payment in the advanced decision. Three different types of payment-in-advance cases are applied: (1) payment in advance with equal numbers of instalments, (2) payment in advance with a single instalment, and (3) the absence of payment in advance. Numerical examples and sensitivity analysis illustrate the proposed model. Here, the total profit increases for all three cases with higher values of carbon emission reduction level. Further, the study finds that the profit becomes maximum for case 2, whereas the selling price and cycle length become minimum. This study considers the sustainable inventory model with payment-in-advance settings when the demand rate depends on the price and carbon emission reduction level. From the literature review, no researcher has undergone this kind of study in the authors’ knowledge. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

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12 pages, 300 KiB

Open AccessArticle

Application of the Pick Function in the Lieb Concavity Theorem for Deformed Exponentials

by Guozeng Yang, Yonggang Li, Jing Wang and Huafei Sun

Fractal Fract. 2022, 6(1), 20; https://doi.org/10.3390/fractalfract6010020 - 31 Dec 2021

Cited by 1 | Viewed by 1377

Abstract

The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity [...] Read more.

The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity theorem for deformed exponentials. In this paper, the Pick function is used to obtain a generalization of the Lieb concavity theorem for deformed exponentials, and some corollaries associated with exterior algebra are obtained. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

16 pages, 695 KiB

Open AccessArticle

Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions

by Muhammad Bilal Khan, Savin Treanțǎ, Mohamed S. Soliman, Kamsing Nonlaopon and Hatim Ghazi Zaini

Fractal Fract. 2022, 6(1), 6; https://doi.org/10.3390/fractalfract6010006 - 23 Dec 2021

Cited by 33 | Viewed by 2539

Abstract

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation (

\leq_{p}

). This order relation is defined on interval space. We have proved [...] Read more.

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation (

\leq_{p}

). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “

\subseteq

” coincident to pseudo-order relation “

\leq_{p}

” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

19 pages, 361 KiB

Open AccessArticle

Semidefinite Multiobjective Mathematical Programming Problems with Vanishing Constraints Using Convexificators

by Kin Keung Lai, Mohd Hassan, Sanjeev Kumar Singh, Jitendra Kumar Maurya and Shashi Kant Mishra

Fractal Fract. 2022, 6(1), 3; https://doi.org/10.3390/fractalfract6010003 - 22 Dec 2021

Cited by 8 | Viewed by 2701

Abstract

In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong

S —

stationary conditions from Fritz John [...] Read more.

In this paper, we establish Fritz John stationary conditions for nonsmooth, nonlinear, semidefinite, multiobjective programs with vanishing constraints in terms of convexificator and introduce generalized Cottle type and generalized Guignard type constraints qualification to achieve strong

S —

stationary conditions from Fritz John stationary conditions. Further, we establish strong

S —

stationary necessary and sufficient conditions, independently from Fritz John conditions. The optimality results for multiobjective semidefinite optimization problem in this paper is related to two recent articles by Treanta in 2021. Treanta in 2021 discussed duality theorems for special class of quasiinvex multiobjective optimization problems for interval-valued components. The study in our article can also be seen and extended for the interval-valued optimization motivated by Treanta (2021). Some examples are provided to validate our established results. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

17 pages, 344 KiB

Open AccessArticle

Hermite–Jensen–Mercer-Type Inequalities via Caputo–Fabrizio Fractional Integral for h-Convex Function

by Miguel Vivas-Cortez, Muhammad Shoaib Saleem, Sana Sajid, Muhammad Sajid Zahoor and Artion Kashuri

Fractal Fract. 2021, 5(4), 269; https://doi.org/10.3390/fractalfract5040269 - 10 Dec 2021

Cited by 15 | Viewed by 2542

Abstract

Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h-convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. [...] Read more.

Integral inequalities involving many fractional integral operators are used to solve various fractional differential equations. In the present paper, we will generalize the Hermite–Jensen–Mercer-type inequalities for an h-convex function via a Caputo–Fabrizio fractional integral. We develop some novel Caputo–Fabrizio fractional integral inequalities. We also present Caputo–Fabrizio fractional integral identities for differentiable mapping, and these will be used to give estimates for some fractional Hermite–Jensen–Mercer-type inequalities. Some familiar results are recaptured as special cases of our results. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

18 pages, 1084 KiB

Open AccessArticle

Stability of Parametric Intuitionistic Fuzzy Multi-Objective Fractional Transportation Problem

by Mohamed A. El Sayed, Mohamed A. El-Shorbagy, Farahat A. Farahat, Aisha F. Fareed and Mohamed A. Elsisy

Fractal Fract. 2021, 5(4), 233; https://doi.org/10.3390/fractalfract5040233 - 19 Nov 2021

Cited by 10 | Viewed by 2309

Abstract

In this study, a parametric intuitionistic fuzzy multi-objective fractional transportation problem (PIF-MOFTP) is proposed. The current PIF-MOFTP has a single-scalar parameter in the objective functions and an intuitionistic fuzzy supply and demand. Based on the

(α, β)

-cut concept a [...] Read more.

In this study, a parametric intuitionistic fuzzy multi-objective fractional transportation problem (PIF-MOFTP) is proposed. The current PIF-MOFTP has a single-scalar parameter in the objective functions and an intuitionistic fuzzy supply and demand. Based on the

(α, β)

-cut concept a parametric

(α, β)

-MOFTP is established. Then, a fuzzy goal programming (FGP) approach is utilized to obtain

(α, β)

-Pareto optimal solution. We investigated the stability set of the first kind (SSFK) corresponding to the solution by extending the Kuhn-Tucker optimality conditions of multi-objective programming problems. An algorithm to crystalize the progressing SSFK for PIF-MOFTP as well as an illustrative numerical example is presented. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

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17 pages, 323 KiB

Open AccessArticle

Existence of Solutions to a Class of Nonlinear Arbitrary Order Differential Equations Subject to Integral Boundary Conditions

by Ananta Thakur, Javid Ali and Rosana Rodríguez-López

Fractal Fract. 2021, 5(4), 220; https://doi.org/10.3390/fractalfract5040220 - 15 Nov 2021

Cited by 4 | Viewed by 2044

Abstract

We investigate the existence of positive solutions for a class of fractional differential equations of arbitrary order

δ > 2

, subject to boundary conditions that include an integral operator of the fractional type. The consideration of this type of boundary conditions allows [...] Read more.

We investigate the existence of positive solutions for a class of fractional differential equations of arbitrary order

δ > 2

, subject to boundary conditions that include an integral operator of the fractional type. The consideration of this type of boundary conditions allows us to consider heterogeneity on the dependence specified by the restriction added to the equation as a relevant issue for applications. An existence result is obtained for the sublinear and superlinear case by using the Guo–Krasnosel’skii fixed point theorem through the definition of adequate conical shells that allow us to localize the solution. As additional tools in our procedure, we obtain the explicit expression of Green’s function associated to an auxiliary linear fractional boundary value problem, and we study some of its properties, such as the sign and some useful upper and lower estimates. Finally, an example is given to illustrate the results. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

17 pages, 308 KiB

Open AccessArticle

Well Posedness of New Optimization Problems with Variational Inequality Constraints

by Savin Treanţă

Fractal Fract. 2021, 5(3), 123; https://doi.org/10.3390/fractalfract5030123 - 15 Sep 2021

Cited by 8 | Viewed by 2088

Abstract

In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by [...] Read more.

In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by introducing the set of approximating solutions for the considered class of constrained optimization problems, we established some characterization results on well posedness. Furthermore, to illustrate the theoretical developments included in this paper, we present some examples. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

Review

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31 pages, 1308 KiB

Open AccessReview

Recent Advances and Applications of Spiral Dynamics Optimization Algorithm: A Review

by Madiah Binti Omar, Kishore Bingi, B Rajanarayan Prusty and Rosdiazli Ibrahim

Fractal Fract. 2022, 6(1), 27; https://doi.org/10.3390/fractalfract6010027 - 2 Jan 2022

Cited by 17 | Viewed by 4349

Abstract

This paper comprehensively reviews the spiral dynamics optimization (SDO) algorithm and investigates its characteristics. SDO algorithm is one of the most straightforward physics-based optimization algorithms and is successfully applied in various broad fields. This paper describes the recent advances of the SDO algorithm, [...] Read more.

This paper comprehensively reviews the spiral dynamics optimization (SDO) algorithm and investigates its characteristics. SDO algorithm is one of the most straightforward physics-based optimization algorithms and is successfully applied in various broad fields. This paper describes the recent advances of the SDO algorithm, including its adaptive, improved, and hybrid approaches. The growth of the SDO algorithm and its application in various areas, theoretical analysis, and comparison with its preceding and other algorithms are also described in detail. A detailed description of different spiral paths, their characteristics, and the application of these spiral approaches in developing and improving other optimization algorithms are comprehensively presented. The review concludes the current works on the SDO algorithm, highlighting its shortcomings and suggesting possible future research perspectives. Full article

(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)

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