Editorial

3 pages, 174 KiB

Open AccessEditorial

Preface to the Special Issue on “Optimization Theory and Applications”

by Árpád Bűrmen and Tadej Tuma

Mathematics 2022, 10(24), 4790; https://doi.org/10.3390/math10244790 - 16 Dec 2022

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Optimization algorithms are an irreplaceable core component of many contemporary tools used in science and engineering [...] Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

Research

Jump to: Editorial

17 pages, 330 KiB

Open AccessArticle

An Efficient Methodology for Locating and Sizing PV Generators in Radial Distribution Networks Using a Mixed-Integer Conic Relaxation

by Oscar Danilo Montoya, Carlos Andrés Ramos-Paja and Luis Fernando Grisales-Noreña

Mathematics 2022, 10(15), 2626; https://doi.org/10.3390/math10152626 - 27 Jul 2022

Cited by 4 | Viewed by 984

Abstract

This paper proposes a new solution methodology based on a mixed-integer conic formulation to locate and size photovoltaic (PV) generation units in AC distribution networks with a radial structure. The objective function comprises the annual expected energy costs of the conventional substation in [...] Read more.

This paper proposes a new solution methodology based on a mixed-integer conic formulation to locate and size photovoltaic (PV) generation units in AC distribution networks with a radial structure. The objective function comprises the annual expected energy costs of the conventional substation in addition to the investment and operating costs of PV sources. The original optimization model that represents this problem belongs to the family of mixed-integer nonlinear programming (MINLP); however, the complexity of the power balance constraints make it difficult to find the global optimum. In order to improve the quality of the optimization model, a mixed-integer conic (MIC) formulation is proposed in this research in order to represent the studied problem. Numerical results in two test feeders composed of 33 and 69 nodes demonstrate the effectiveness of the proposed MIC model when compared to multiple metaheuristic optimizers such as the Chu and Beasley Genetic Algorithm, the Newton Metaheuristic Algorithm, the Vortex Search Algorithm, the Gradient-Based Metaheuristic Optimization Algorithm, and the Arithmetic Optimization Algorithm, among others. The final results obtained with the MIC model show improvements greater than USD 100,000 per year of operation. All simulations were run in the MATLAB programming environment, using its own scripts for all the metaheuristic algorithms and the disciplined convex tool known as CVX with the Gurobi solver in order to solve the proposed MIC model. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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16 pages, 740 KiB

Open AccessArticle

Meta-Optimization of Dimension Adaptive Parameter Schema for Nelder–Mead Algorithm in High-Dimensional Problems

by Žiga Rojec, Tadej Tuma, Jernej Olenšek, Árpád Bűrmen and Janez Puhan

Mathematics 2022, 10(13), 2288; https://doi.org/10.3390/math10132288 - 30 Jun 2022

Cited by 2 | Viewed by 1196

Abstract

Although proposed more than half a century ago, the Nelder–Mead simplex search algorithm is still widely used. Four numeric constants define the operations and behavior of the algorithm. The algorithm with the original constant values performs fine on most low-dimensional, but poorly on [...] Read more.

Although proposed more than half a century ago, the Nelder–Mead simplex search algorithm is still widely used. Four numeric constants define the operations and behavior of the algorithm. The algorithm with the original constant values performs fine on most low-dimensional, but poorly on high-dimensional, problems. Therefore, to improve its behavior in high dimensions, several adaptive schemas setting the constants according to the problem dimension were proposed in the past. In this work, we present a novel adaptive schema obtained by a meta-optimization procedure. We describe a schema candidate with eight parameters subject to meta-optimization and define an objective function evaluating the candidate’s performance. The schema is optimized on up to 100-dimensional problems using the Parallel Simulated Annealing with Differential Evolution global method. The obtained global minimum represents the proposed schema. We compare the performance of the optimized schema with the existing adaptive schemas. The data profiles on the Gao–Han modified quadratic, Moré–Garbow–Hilstrom, and CUTEr (Constrained and Unconstrained Testing Environment, revisited) benchmark problem sets show that the obtained schema outperforms the existing adaptive schemas in terms of accuracy and convergence speed. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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17 pages, 3720 KiB

Open AccessArticle

A Comprehensive Approach for an Approximative Integration of Nonlinear-Bivariate Functions in Mixed-Integer Linear Programming Models

by Maximilian Roth, Georg Franke and Stephan Rinderknecht

Mathematics 2022, 10(13), 2226; https://doi.org/10.3390/math10132226 - 25 Jun 2022

Cited by 3 | Viewed by 1801

Abstract

As decentralized energy supply units, microgrids can make a decisive contribution to achieving climate targets. In this context, it is particularly important to determine the optimal size of the energy components contained in the microgrids and their optimal operating schedule. Hence, mathematical optimization [...] Read more.

As decentralized energy supply units, microgrids can make a decisive contribution to achieving climate targets. In this context, it is particularly important to determine the optimal size of the energy components contained in the microgrids and their optimal operating schedule. Hence, mathematical optimization methods are often used in association with such tasks. In particular, mixed-integer linear programming (MILP) has proven to be a useful tool. Due to the versatility of the different energetic components (e.g., storages, solar modules) and their special technical characteristics, linear relationships can often only inadequately describe the real processes. In order to take advantage of linear solution techniques but at the same time better represent these real-world processes, accurate and efficient approximation techniques need to be applied in system modeling. In particular, nonlinear-bivariate functions represent a major challenge, which is why this paper derives and implements a method that addresses this issue. The advantage of this method is that any bivariate mixed-integer nonlinear programming (MINLP) formulation can be transformed into a MILP formulation using this comprehensive method. For a performance comparison, a mixed-integer quadratic constrained programming (MIQCP) model—as an MINLP special case—is applied and transformed into a MILP, and the solution of the transformed problem is compared with the one of the MIQCP. Since there are good off-the-shelf solvers for MIQCP problems available, the comparison is conservative. The results for an exemplary microgrid sizing task show that the method delivers a strong performance, both in terms of approximation error (0.08%) and computation time. The method and its implementation can serve as a general user-tool but also as a basis for further methodological developments and research. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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40 pages, 570 KiB

Open AccessArticle

Algorithmic Aspect and Convergence Analysis for System of Generalized Multivalued Variational-like Inequalities

by Javad Balooee, Shih-Sen Chang, Lin Wang and Zhaoli Ma

Mathematics 2022, 10(12), 2016; https://doi.org/10.3390/math10122016 - 11 Jun 2022

Cited by 1 | Viewed by 902

Abstract

The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P- [...] Read more.

The main aim of this paper is twofold. Our first objective is to study a new system of generalized multivalued variational-like inequalities in Banach spaces and to establish its equivalence with a system of fixed point problems utilizing the concept of P-

η

-proximal mapping. The obtained alternative equivalent formulation is used and a new iterative algorithm for finding its approximate solution is suggested. Under some appropriate assumptions imposed on the mappings and parameters involved in the system of generalized multivalued variational-like inequalities, the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences generated by our proposed iterative algorithm is discussed. The second objective of this work is to investigate and analyze the notion M-

η

-proximal mapping defined in the literature. Taking into account of the assumptions considered for such a mapping, we prove that every M-

η

-proximal mapping is actually P-

η

-proximal and is not a new one. At the same time, some comments relating to some existing results are pointed out. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

14 pages, 317 KiB

Open AccessArticle

Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation

by Oscar Danilo Montoya, Francisco David Moya and Arul Rajagopalan

Mathematics 2022, 10(9), 1600; https://doi.org/10.3390/math10091600 - 08 May 2022

Cited by 4 | Viewed by 1484

Abstract

The minimization of annual operating costs in radial distribution networks with the optimal selection and siting of fixed-step capacitor banks is addressed in this research by means of a two-stage optimization approach. The first stage proposes an approximated mixed-integer quadratic model to select [...] Read more.

The minimization of annual operating costs in radial distribution networks with the optimal selection and siting of fixed-step capacitor banks is addressed in this research by means of a two-stage optimization approach. The first stage proposes an approximated mixed-integer quadratic model to select the nodes where the capacitor banks must be installed. In the second stage, a recursive power flow method is employed to make an exhaustive evaluation of the solution space. The main contribution of this research is the use of the expected load curve to estimate the equivalent annual grid operating costs. Numerical simulations in the IEEE 33- and IEEE 69-bus systems demonstrate the effectiveness of the proposed methodology in comparison with the solution of the exact optimization model in the General Algebraic Modeling System software. Reductions of

33.04 %

and

34.29 %

with respect to the benchmark case are obtained with the proposed two-stage approach, with minimum investments in capacitor banks. All numerical implementations are performed in the MATLAB software using the convex tool known as CVX and the Gurobi solver. The main advantage of the proposed hybrid optimization method lies in the possibility of dealing with radial and meshed distribution system topologies without any modification on the MIQC model and the recursive power flow approach. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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13 pages, 1616 KiB

Open AccessArticle

A Quick Search Dynamic Vector-Evaluated Particle Swarm Optimization Algorithm Based on Fitness Distance

by Suyu Wang, Dengcheng Ma and Miao Wu

Mathematics 2022, 10(9), 1587; https://doi.org/10.3390/math10091587 - 07 May 2022

Cited by 1 | Viewed by 1238

Abstract

A quick search dynamic vector-evaluated particle swarm optimization algorithm based on fitness distance (DVEPSO/FD) is proposed according to the fact that some dynamic multi-objective optimization methods, such as the DVEPSO, cannot achieve a very accurate Pareto optimal front (POF) tracked after each objective [...] Read more.

A quick search dynamic vector-evaluated particle swarm optimization algorithm based on fitness distance (DVEPSO/FD) is proposed according to the fact that some dynamic multi-objective optimization methods, such as the DVEPSO, cannot achieve a very accurate Pareto optimal front (POF) tracked after each objective changes, although they exhibit advantages in multi-objective optimization. Featuring a repository update mechanism using the fitness distance together with a quick search mechanism, the DVEPSO/FD is capable of obtaining the optimal values that are closer to the real POF. The fitness distance is used to streamline the repository to improve the distribution of nondominant solutions, and the flight parameters of the particles are adjusted dynamically to improve the search speed. Groups of the standard benchmark experiments are conducted and the results show that, compared with the DVEPSO method, from the figures generated by the test functions, DVEPSO/FD achieves a higher accuracy and clearness with the POF dynamically changing; from the values of performance indexes, the DVEPSO/FD effectively improves the accuracy of the tracked POF without destroying the stability. The proposed DVEPSO/FD method shows a good dynamic change adaptability and solving set ability of the dynamic multi-objective optimization problem. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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20 pages, 1024 KiB

Open AccessArticle

Out of the Niche: Using Direct Search Methods to Find Multiple Global Optima

by Javier Cano, Cesar Alfaro, Javier Gomez and Abraham Duarte

Mathematics 2022, 10(9), 1494; https://doi.org/10.3390/math10091494 - 30 Apr 2022

Cited by 2 | Viewed by 1356

Abstract

Multimodal optimization deals with problems where multiple feasible global solutions coexist. Despite sharing a common objective function value, some global optima may be preferred to others for various reasons. In such cases, it is paramount to devise methods that are able to find [...] Read more.

Multimodal optimization deals with problems where multiple feasible global solutions coexist. Despite sharing a common objective function value, some global optima may be preferred to others for various reasons. In such cases, it is paramount to devise methods that are able to find as many global optima as possible within an affordable computational budget. Niching strategies have received an overwhelming attention in recent years as the most suitable technique to tackle these kinds of problems. In this paper we explore a different approach, based on a systematic yet versatile use of traditional direct search methods. When tested over reference benchmark functions, our proposal, despite its apparent simplicity, noticeably resists the comparison with state-of-the-art niching methods in most cases, both in the number of global optima found and in the number of function evaluations required. However, rather than trying to outperform niching methods—far more elaborated—our aim is to enrich them with the knowledge gained from exploiting the distinctive features of direct search methods. To that end, we propose two new performance measures that can be used to evaluate, compare and monitor the progress of optimization algorithms of (possibly) very different nature in their effort to find as many global optima of a given multimodal objective function as possible. We believe that adopting these metrics as reference criteria could lead to more sophisticated and computationally-efficient algorithms, which could benefit from the brute force of derivative-free local search methods. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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22 pages, 409 KiB

Open AccessArticle

Automatic Grammatical Evolution-Based Optimization of Matrix Factorization Algorithm

by Matevž Kunaver, Árpád Bűrmen and Iztok Fajfar

Mathematics 2022, 10(7), 1139; https://doi.org/10.3390/math10071139 - 01 Apr 2022

Cited by 1 | Viewed by 1358

Abstract

Nowadays, recommender systems are vital in lessening the information overload by filtering out unnecessary information, thus increasing comfort and quality of life. Matrix factorization (MF) is a well-known recommender system algorithm that offers good results but requires a certain level of system knowledge [...] Read more.

Nowadays, recommender systems are vital in lessening the information overload by filtering out unnecessary information, thus increasing comfort and quality of life. Matrix factorization (MF) is a well-known recommender system algorithm that offers good results but requires a certain level of system knowledge and some effort on part of the user before use. In this article, we proposed an improvement using grammatical evolution (GE) to automatically initialize and optimize the algorithm and some of its settings. This enables the algorithm to produce optimal results without requiring any prior or in-depth knowledge, thus making it possible for an average user to use the system without going through a lengthy initialization phase. We tested the approach on several well-known datasets. We found our results to be comparable to those of others while requiring a lot less set-up. Finally, we also found out that our approach can detect the occurrence of over-saturation in large datasets. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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20 pages, 1740 KiB

Open AccessArticle

Evolutionary Synthesis of Failure-Resilient Analog Circuits

by Žiga Rojec, Iztok Fajfar and Árpád Burmen

Mathematics 2022, 10(1), 156; https://doi.org/10.3390/math10010156 - 05 Jan 2022

Cited by 6 | Viewed by 1336

Abstract

Analog circuit design requires large amounts of human knowledge. A special case of circuit design is the synthesis of robust and failure-resilient electronics. Evolutionary algorithms can aid designers in exploring topologies with new properties. Here, we show how to encode a circuit topology [...] Read more.

Analog circuit design requires large amounts of human knowledge. A special case of circuit design is the synthesis of robust and failure-resilient electronics. Evolutionary algorithms can aid designers in exploring topologies with new properties. Here, we show how to encode a circuit topology with an upper-triangular incident matrix and use the NSGA-II algorithm to find computational circuits that are robust to component failure. Techniques for robustness evaluation and evolutionary algorithm guidances are described. As a result, we evolve square root and natural logarithm computational circuits that are robust to high-impedance or short-circuit malfunction of an arbitrary rectifying diode. We confirm the simulation results by hardware circuit implementation and measurements. We think that our research will inspire further searches for failure-resilient topologies. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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22 pages, 2111 KiB

Open AccessArticle

Optimal Control Applied to Vaccination and Testing Policies for COVID-19

by Alberto Olivares and Ernesto Staffetti

Mathematics 2021, 9(23), 3100; https://doi.org/10.3390/math9233100 - 01 Dec 2021

Cited by 5 | Viewed by 1518

Abstract

In this paper, several policies for controlling the spread of SARS-CoV-2 are determined under the assumption that a limited number of effective COVID-19 vaccines and tests are available. These policies are calculated for different vaccination scenarios representing vaccine supply and administration restrictions, plus [...] Read more.

In this paper, several policies for controlling the spread of SARS-CoV-2 are determined under the assumption that a limited number of effective COVID-19 vaccines and tests are available. These policies are calculated for different vaccination scenarios representing vaccine supply and administration restrictions, plus their impacts on the disease transmission are analyzed. The policies are determined by solving optimal control problems of a compartmental epidemic model, in which the control variables are the vaccination rate and the testing rate for the detection of asymptomatic infected people. A combination of the proportion of threatened and deceased people together with the cost of vaccination of susceptible people, and detection of asymptomatic infected people, is taken as the objective functional to be minimized, whereas different types of algebraic constraints are considered to represent several vaccination scenarios. A direct transcription method is employed to solve these optimal control problems. More specifically, the Hermite–Simpson collocation technique is used. The results of the numerical experiments show that the optimal control approach offers healthcare system managers a helpful resource for designing vaccination programs and testing plans to prevent COVID-19 transmission. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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10 pages, 268 KiB

Open AccessArticle

Efficient Covering of Thin Convex Domains Using Congruent Discs

by Shai Gul and Reuven Cohen

Mathematics 2021, 9(23), 3056; https://doi.org/10.3390/math9233056 - 28 Nov 2021

Cited by 2 | Viewed by 1566

Abstract

We present efficient strategies for covering classes of thin domains in the plane using unit discs. We start with efficient covering of narrow domains using a single row of covering discs. We then move to efficient covering of general rectangles by discs centered [...] Read more.

We present efficient strategies for covering classes of thin domains in the plane using unit discs. We start with efficient covering of narrow domains using a single row of covering discs. We then move to efficient covering of general rectangles by discs centered at the lattice points of an irregular hexagonal lattice. This optimization uses a lattice that leads to a covering using a small number of discs. We compare the bounds on the covering using the presented strategies to the bounds obtained from the standard honeycomb covering, which is asymptotically optimal for fat domains, and show the improvement for thin domains. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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26 pages, 26423 KiB

Open AccessArticle

Snow Leopard Optimization Algorithm: A New Nature-Based Optimization Algorithm for Solving Optimization Problems

by Petr Coufal, Štěpán Hubálovský, Marie Hubálovská and Zoltan Balogh

Mathematics 2021, 9(21), 2832; https://doi.org/10.3390/math9212832 - 08 Nov 2021

Cited by 16 | Viewed by 2348

Abstract

Numerous optimization problems have been defined in different disciplines of science that must be optimized using effective techniques. Optimization algorithms are an effective and widely used method of solving optimization problems that are able to provide suitable solutions for optimization problems. In this [...] Read more.

Numerous optimization problems have been defined in different disciplines of science that must be optimized using effective techniques. Optimization algorithms are an effective and widely used method of solving optimization problems that are able to provide suitable solutions for optimization problems. In this paper, a new nature-based optimization algorithm called Snow Leopard Optimization Algorithm (SLOA) is designed that mimics the natural behaviors of snow leopards. SLOA is simulated in four phases including travel routes, hunting, reproduction, and mortality. The different phases of the proposed algorithm are described and then the mathematical modeling of the SLOA is presented in order to implement it on different optimization problems. A standard set of objective functions, including twenty-three functions, is used to evaluate the ability of the proposed algorithm to optimize and provide appropriate solutions for optimization problems. Also, the optimization results obtained from the proposed SLOA are compared with eight other well-known optimization algorithms. The optimization results show that the proposed SLOA has a high ability to solve various optimization problems. Also, the analysis and comparison of the optimization results obtained from the SLOA with the other eight algorithms shows that the SLOA is able to provide more appropriate quasi-optimal solutions and closer to the global optimal, and with better performance, it is much more competitive than similar algorithms. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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11 pages, 323 KiB

Open AccessArticle

Outer Approximation Method for the Unit Commitment Problem with Wind Curtailment and Pollutant Emission

by Xiali Pang, Haiyan Zheng, Liying Huang and Yumei Liang

Mathematics 2021, 9(21), 2686; https://doi.org/10.3390/math9212686 - 22 Oct 2021

Cited by 2 | Viewed by 1120

Abstract

This paper considers the fast and effective solving method for the unit commitment (UC) problem with wind curtailment and pollutant emission in power systems. Firstly, a suitable mixed-integer quadratic programming (MIQP) model of the corresponding UC problem is presented by some linearization techniques, [...] Read more.

This paper considers the fast and effective solving method for the unit commitment (UC) problem with wind curtailment and pollutant emission in power systems. Firstly, a suitable mixed-integer quadratic programming (MIQP) model of the corresponding UC problem is presented by some linearization techniques, which is difficult to solve directly. Then, the MIQP model is solved by the outer approximation method (OAM), which decomposes the MIQP into a mixed-integer linear programming (MILP) master problem and a nonlinear programming (NLP) subproblem for alternate iterative solving. Finally, simulation results for six systems with up to 100 thermal units and one wind unit in 24 periods are presented, which show the practicality of MIQP model and the effectiveness of OAM. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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9 pages, 253 KiB

Open AccessArticle

Arcwise Connectedness of the Solution Sets for Generalized Vector Equilibrium Problems

by Qiuying Li and Sanhua Wang

Mathematics 2021, 9(20), 2532; https://doi.org/10.3390/math9202532 - 09 Oct 2021

Cited by 1 | Viewed by 960

Abstract

In this research, by means of the scalarization method, arcwise connectedness results were established for the sets of globally efficient solutions, weakly efficient solutions, Henig efficient solutions and superefficient solutions for the generalized vector equilibrium problem under suitable assumptions of natural quasi cone-convexity [...] Read more.

In this research, by means of the scalarization method, arcwise connectedness results were established for the sets of globally efficient solutions, weakly efficient solutions, Henig efficient solutions and superefficient solutions for the generalized vector equilibrium problem under suitable assumptions of natural quasi cone-convexity and natural quasi cone-concavity. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

28 pages, 1465 KiB

Open AccessArticle

Dynamic Model of Contingency Flight Crew Planning Extending to Crew Formation

by Vojtech Graf, Dusan Teichmann, Michal Dorda and Lenka Kontrikova

Mathematics 2021, 9(17), 2138; https://doi.org/10.3390/math9172138 - 02 Sep 2021

Cited by 4 | Viewed by 2471

Abstract

The creation of a flight schedule and the associated crew planning are clearly among the most complicated tasks in terms of traffic preparation. Even with a relatively small number of pilots and aircraft, numerous specific constraints arising from real operations must be included [...] Read more.

The creation of a flight schedule and the associated crew planning are clearly among the most complicated tasks in terms of traffic preparation. Even with a relatively small number of pilots and aircraft, numerous specific constraints arising from real operations must be included in the calculation, thus increasing the complexity of the planning process. However, even in a precision-planned operation, non-standard situations often occur, which must be addressed flexibly. It is at this point that an operational solution must be applied, the aims of which are to stabilize the flight schedule as soon as possible and minimize the financial impacts resulting from the non-standard situation. These problems are resolved by the airline’s Operational Control Center, which also uses various software approaches to solve the problem. The problem is approached differently by large air carriers, which use software products to address it, and small and medium-sized air carriers, which resolve the issue of operational rescheduling intuitively, based on the experience of dispatchers. However, this intuitive approach can lead to inaccuracies that can lead to unnecessary financial losses. In this paper, we present an optimization model that can serve as a tool to support the decision-making of employees of the operations centers of smaller and medium-sized air carriers. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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15 pages, 2917 KiB

Open AccessArticle

Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators

by Martín Alejandro Valencia-Ponce , Esteban Tlelo-Cuautle and Luis Gerardo de la Fraga

Mathematics 2021, 9(16), 1938; https://doi.org/10.3390/math9161938 - 13 Aug 2021

Cited by 16 | Viewed by 2223

Abstract

The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step h. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest h in either one-step or multi-step methods. [...] Read more.

The execution time that takes to perform numerical simulation of a chaotic oscillator mainly depends on the time-step h. This paper shows that the optimization of chaotic oscillators can be enhanced by estimating the highest h in either one-step or multi-step methods. Four chaotic oscillators are used as a case study, and the optimization of their Kaplan-Yorke dimension (

D_{K Y}

) is performed by applying three metaheuristics, namely: particle swarm optimization (PSO), many optimizing liaison (MOL), and differential evolution (DE) algorithms. Three representative one-step and three multi-step methods are used to solve the four chaotic oscillators, for which the estimation of the highest h is obtained from their stability analysis. The optimization results show the effectiveness of using a high h value for the six numerical methods in reducing execution time while maximizing the positive Lyapunov exponent (

L E +

) and

D_{K Y}

of the chaotic oscillators by applying PSO, MOL, and DE algorithms. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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18 pages, 371 KiB

Open AccessFeature PaperArticle

Randomized Simplicial Hessian Update

by Árpád Bűrmen, Tadej Tuma and Jernej Olenšek

Mathematics 2021, 9(15), 1775; https://doi.org/10.3390/math9151775 - 27 Jul 2021

Cited by 1 | Viewed by 1254

Abstract

Recently, a derivative-free optimization algorithm was proposed that utilizes a minimum Frobenius norm (MFN) Hessian update for estimating the second derivative information, which in turn is used for accelerating the search. The proposed update formula relies only on computed function values and is [...] Read more.

Recently, a derivative-free optimization algorithm was proposed that utilizes a minimum Frobenius norm (MFN) Hessian update for estimating the second derivative information, which in turn is used for accelerating the search. The proposed update formula relies only on computed function values and is a closed-form expression for a special case of a more general approach first published by Powell. This paper analyzes the convergence of the proposed update formula under the assumption that the points from

R^{n}

where the function value is known are random. The analysis assumes that the

N + 2

points used by the update formula are obtained by adding

N + 1

vectors to a central point. The vectors are obtained by transforming a prototype set of

N + 1

vectors with a random orthogonal matrix from the Haar measure. The prototype set must positively span a

N \leq n

dimensional subspace. Because the update is random by nature we can estimate a lower bound on the expected improvement of the approximate Hessian. This lower bound was derived for a special case of the proposed update by Leventhal and Lewis. We generalize their result and show that the amount of improvement greatly depends on N as well as the choice of the vectors in the prototype set. The obtained result is then used for analyzing the performance of the update based on various commonly used prototype sets. One of the results obtained by this analysis states that a regular n-simplex is a bad choice for a prototype set because it does not guarantee any improvement of the approximate Hessian. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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22 pages, 3520 KiB

Open AccessEditor’s ChoiceArticle

Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms

by Jose-Cruz Nuñez-Perez, Vincent-Ademola Adeyemi, Yuma Sandoval-Ibarra, Francisco-Javier Perez-Pinal and Esteban Tlelo-Cuautle

Mathematics 2021, 9(11), 1194; https://doi.org/10.3390/math9111194 - 25 May 2021

Cited by 12 | Viewed by 2825

Abstract

This paper presents the application of three optimization algorithms to increase the chaotic behavior of the fractional order chaotic Chen system. This is achieved by optimizing the maximum Lyapunov exponent (MLE). The applied optimization techniques are evolutionary algorithms (EAs), namely: differential evolution (DE), [...] Read more.

This paper presents the application of three optimization algorithms to increase the chaotic behavior of the fractional order chaotic Chen system. This is achieved by optimizing the maximum Lyapunov exponent (MLE). The applied optimization techniques are evolutionary algorithms (EAs), namely: differential evolution (DE), particle swarm optimization (PSO), and invasive weed optimization (IWO). In each algorithm, the optimization process is performed using 100 individuals and generations from 50 to 500, with a step of 50, which makes a total of ten independent runs. The results show that the optimized fractional order chaotic Chen systems have higher maximum Lyapunov exponents than the non-optimized system, with the DE giving the highest MLE. Additionally, the results indicate that the chaotic behavior of the fractional order Chen system is multifaceted with respect to the parameter and fractional order values. The dynamical behavior and complexity of the optimized systems are verified using properties, such as bifurcation, LE spectrum, equilibrium point, eigenvalue, and sample entropy. Moreover, the optimized systems are compared with a hyper-chaotic Chen system on the basis of their prediction times. The results show that the optimized systems have a shorter prediction time than the hyper-chaotic system. The optimized results are suitable for developing a secure communication system and a random number generator. Finally, the Halstead parameters measure the complexity of the three optimization algorithms that were implemented in MATLAB. The results reveal that the invasive weed optimization has the simplest implementation. Full article

(This article belongs to the Special Issue Optimization Theory and Applications)

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