Special Issue "Optimization Theory and Applications"

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

Deadline for manuscript submissions: 31 May 2022 | Viewed by 7114

Special Issue Editors

Prof. Dr. Árpád Bűrmen
E-Mail Website
Guest Editor
Faculty of Electrical Engineering, University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia
Interests: optimization algorithms; derivative-free optimization; electronic circuit design automation; electronic circuit simulation
Prof. Dr. Tadej Tuma
E-Mail Website
Guest Editor
Faculty of Electrical Engineering, University of Ljubljana, Tržaška cesta 25, 1000 Ljubljana, Slovenia
Interests: optimization in EDA; embedded systems

Special Issue Information

Dear Colleagues,

Optimization algorithms lie at the core of many contemporary tools used in science and engineering. They represent the engine behind design automation in electrical and mechanical engineering, protein-folding simulations, drug design, machine learning, scheduling and timetable design, traffic management, resource allocation, decision making, model predictive control, geophysical-parameter estimation, portfolio management, asset-price modelling, etc.

Nature can be a great source of inspiration for designing optimization algorithms. In the past, algorithms were devised so that they mimicked the annealing of metals, the motion of objects in gravitational fields, evolution, animal behavior, and many more inventions of mother nature.

The theoretical analysis of optimization algorithms is important not only because it confirms the appropriateness of an algorithm. It also provides insight into the algorithm's limitations and hints for future research. In this sense, not only are positive results significant but counterexamples that point out the algorithm's weaknesses are also important.

You are cordially invited to submit papers related to all aspects of optimization, both theoretical and applicational. This involves (but is not limited to) linear, quadratic, convex, nonconvex, nonlinear, and integer programming; combinatorial optimization; robust optimization; stochastic programming; quasi-Newton methods; interior point methods; successive quadratic programming; derivative-free methods; approximation algorithms; and evolutionary algorithms.

Prof. Dr. Árpád Bűrmen
Prof. Dr. Tadej Tuma
Guest Editors

Manuscript Submission Information

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Keywords

  • Optimization
  • Optimization applications
  • Mathematical programming
  • Stochastic programming
  • Integer programming
  • Approximation algorithms
  • Derivative-free optimization
  • Robust optimization
  • Combinatorial optimization
  • Optimality conditions
  • Optimal control

Published Papers (14 papers)

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Research

Article
Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation
Mathematics 2022, 10(9), 1600; https://doi.org/10.3390/math10091600 - 08 May 2022
Viewed by 308
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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Article
A Quick Search Dynamic Vector-Evaluated Particle Swarm Optimization Algorithm Based on Fitness Distance
Mathematics 2022, 10(9), 1587; https://doi.org/10.3390/math10091587 - 07 May 2022
Viewed by 251
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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Article
Out of the Niche: Using Direct Search Methods to Find Multiple Global Optima
Mathematics 2022, 10(9), 1494; https://doi.org/10.3390/math10091494 - 30 Apr 2022
Viewed by 262
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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Article
Automatic Grammatical Evolution-Based Optimization of Matrix Factorization Algorithm
Mathematics 2022, 10(7), 1139; https://doi.org/10.3390/math10071139 - 01 Apr 2022
Viewed by 356
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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Article
Evolutionary Synthesis of Failure-Resilient Analog Circuits
Mathematics 2022, 10(1), 156; https://doi.org/10.3390/math10010156 - 05 Jan 2022
Viewed by 261
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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Article
Optimal Control Applied to Vaccination and Testing Policies for COVID-19
Mathematics 2021, 9(23), 3100; https://doi.org/10.3390/math9233100 - 01 Dec 2021
Viewed by 440
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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Article
Efficient Covering of Thin Convex Domains Using Congruent Discs
Mathematics 2021, 9(23), 3056; https://doi.org/10.3390/math9233056 - 28 Nov 2021
Viewed by 315
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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Article
Snow Leopard Optimization Algorithm: A New Nature-Based Optimization Algorithm for Solving Optimization Problems
Mathematics 2021, 9(21), 2832; https://doi.org/10.3390/math9212832 - 08 Nov 2021
Viewed by 563
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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Article
Outer Approximation Method for the Unit Commitment Problem with Wind Curtailment and Pollutant Emission
Mathematics 2021, 9(21), 2686; https://doi.org/10.3390/math9212686 - 22 Oct 2021
Viewed by 373
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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Article
Arcwise Connectedness of the Solution Sets for Generalized Vector Equilibrium Problems
Mathematics 2021, 9(20), 2532; https://doi.org/10.3390/math9202532 - 09 Oct 2021
Viewed by 313
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)
Article
Dynamic Model of Contingency Flight Crew Planning Extending to Crew Formation
Mathematics 2021, 9(17), 2138; https://doi.org/10.3390/math9172138 - 02 Sep 2021
Viewed by 528
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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Article
Estimating the Highest Time-Step in Numerical Methods to Enhance the Optimization of Chaotic Oscillators
Mathematics 2021, 9(16), 1938; https://doi.org/10.3390/math9161938 - 13 Aug 2021
Cited by 5 | Viewed by 471
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 (DKY) 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 (LE+) and DKY 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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Article
Randomized Simplicial Hessian Update
Mathematics 2021, 9(15), 1775; https://doi.org/10.3390/math9151775 - 27 Jul 2021
Viewed by 462
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 Rn 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 Nn 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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Article
Maximizing the Chaotic Behavior of Fractional Order Chen System by Evolutionary Algorithms
Mathematics 2021, 9(11), 1194; https://doi.org/10.3390/math9111194 - 25 May 2021
Cited by 5 | Viewed by 731
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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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: Distributed and Parallel NSGA-II using the Divide-and Conquer Method and Migration for Compensation on Many-core Environments
Authors: Yuji Sato and Mikiko Sato
Affiliation: Graduate School of Engineering, The University of Tokyo: Bunkyo, Tokyo, JP
Abstract: In this paper proposes a novel way to distribute and parallelsthe computation of NSGA-II on many-core environments. A recent trend in multi-objective evolutionary algorithms is to increase the population size to approximate the Pareto front with high accuracy. On the other hand, the NSGA-II algorithm widely used in multi-objective optimization performs nondominated sorting in solution ranking, which means an increase in computational complexity proportional to the square of the population. This execution time becomes a problem in engineering applications. In this paper, we propose distributed, high-speed NSGA-II using a many-core environment to obtain a Pareto-optimal solution set excelling in convergence and diversity. This method improves performance while maintaining the accuracy of the Pareto-optimal solution set by repeating NSGA-II distributed processing in a manycore environment inspired by the divide-and-conquer method together with migration processing for compensation of the nondominated solution set obtained by distributed processing. Using test functions attached to the NSGA-II source code and a two-objective and three-objective constrained knapsack problem for evaluation, and on comparing with NSGA-II executing on a single CPU and parallel, high-speed NSGA-II using a standard island model, it was found that the proposed method greatly shortened the execution time for obtaining a Pareto-optimal solution set with equivalent hypervolume while increasing the accuracy of solution searching.It also shows that the proposed methodis especially useful for complex discrete-shaped PFsuch as CTP2 and CTP5, and for problems such as the knapsack problem where it is difficult to find solutions at both ends of the PF.

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