Multi-objective Benchmark Functions Research Articles

Distributed Multi-objective Optimization in Cyber-Physical Energy Systems

Managing complex Cyber-Physical Energy Systems (CPES) requires solving various optimization problems with multiple objectives and constraints. As distributed control architectures are becoming more popular in CPES for certain tasks due to their flexibility, robustness, and privacy protection, multi-objective optimization must also be distributed. For this purpose, we present MO-COHDA, a fully distributed, agent-based algorithm, for solving multi-objective optimization problems of CPES. MO-COHDA allows an easy and flexible adaptation to different use cases and integration of custom functionality. To evaluate the effectiveness of MO-COHDA, we compare it to a central NSGA-2 algorithm using multi-objective benchmark functions from the ZDT problem suite. The results show that MO-COHDA can approximate the reference front of the benchmark problems well and is suitable for solving multi-objective optimization problems. In addition, an example use case of scheduling a group of generation units while optimizing three different objectives was evaluated to show how MO-COHDA can be easily applied to real-world optimization problems in CPES.

An enhanced multi-objective artificial bee colony algorithm with non-dominated sorting strategy

<p>This paper presents an improved metaheuristic technique inspired by the foundational concepts of the artificial bee colony (ABC) algorithm adapted to deal with multi-objective optimization challenges. Our approach combines the main ideas of ABC with a non-dominated sorting strategy including aspects of Pareto dominance, crowding distance, and greedy selection method. Furthermore, the chosen non-dominated solutions are archived in a repository with a static size. The presented approach, multi-objective artificial bee colony (MOABC), is compared to other state-of-the-art algorithms including the non-dominated sorting genetic algorithm II (NSGA II) and the multi-objective particle swarm optimization (MOPSO). MOABC and selected algorithms from the literature are applied to five zitzler-deb-thiele (ZDT) Multi-objective benchmark functions. Then three key metrics are employed for performance evaluations: generational distance (GD), spread (SP), and hypervolume (HV). The simulation results suggest that the proposed method is competitive and presents an effective choice for tackling multi-objective optimization problems.</p>

Optimizing brushless direct current motor design: An application of the multi-objective generalized normal distribution optimization

In this study, we tackle the challenge of optimizing the design of a Brushless Direct Current (BLDC) motor. Utilizing an established analytical model, we introduced the Multi-Objective Generalized Normal Distribution Optimization (MOGNDO) method, a biomimetic approach based on Pareto optimality, dominance, and external archiving. We initially tested MOGNDO on standard multi-objective benchmark functions, where it showed strong performance. When applied to the BLDC motor design with the objectives of either maximizing operational efficiency or minimizing motor mass, the MOGNDO algorithm consistently outperformed other techniques like Ant Lion Optimizer (ALO), Ion Motion Optimization (IMO), and Sine Cosine Algorithm (SCA). Specifically, MOGNDO yielded the most optimal values across efficiency and mass metrics, providing practical solutions for real-world BLDC motor design. The MOGNDO source code is available at: https://github.com/kanak02/MOGNDO.

Double-space environmental change detection and response strategy for dynamic multi-objective optimize problem

Dynamic multi-objective optimization problems (DMOPs) which contain various Pareto-optimal front (PF) and Pareto-optimal set (PS) have gained much attention. Accurate environmental change detection reveals the change degree of DMOPs and contributes the algorithm to quickly respond to the environment changes. In order to fully detect environmental changes and efficiently track front, a double-space environmental change detection and response strategy (DSDRS) is proposed. It could detect whether the environment has changed while explore the change intensity of PF and PS, respectively. Moreover, different response strategies are implemented for PF and PS. For PF environmental changes, a multiple knee points-guided evolutionary strategy (MKGES) is proposed, which is driven by front shape information and adaptively responds to different PF change intensities. For PS environmental changes, a knowledge guided memory strategy (KGMS) is proposed, which guides population evolution based on environmental information. The effectiveness of DSDRS is confirmed by comparison with five evolutionary algorithms on 20 dynamic multiobjective benchmark functions. Simulation results demonstrate that the performance of proposed algorithm is outstanding on test functions with complex changing PF and PS.

A multi-objective cooperation search algorithm for cascade reservoirs operation optimization considering power generation and ecological flows

With growing attention focused on energy generation and environmental conservation, the operation of cascade reservoirs that considers power generation benefits and ecological flow requirements plays an increasingly important role in water resource and power systems. However, the traditional optimization methods may fail to solve complex cascade reservoirs operation models with strong spatial-temporal coupling physical constraints. To effectively address this problem, this study proposes an efficient multi-objective cooperation search algorithm (MOCSA) that utilizes the modified team communication, reflective learning operator and internal competition operators to enhance global exploration and local exploitation. MOCSA is tested on a group of multi-objective benchmark functions and real-world constrained engineering problems. The results demonstrate that MOCSA can provide better results for most benchmark test functions in comparison with the competitive algorithms. Besides, MOCSA exhibits excellent search ability to find feasible solutions of constrained engineering problems. The proposed method is then applied to resolve a real-world reservoir system under different operation scenarios. Simulations indicate that MOCSA provides diversified decision options for the operation of cascade reservoir system and find a more diverse set of non-dominated solutions within the feasible solution space. Overall, this research presents MOCSA as an effective multi-objective optimization tool for complex cascade reservoir operation problems.

A dual-population multi-objective evolutionary algorithm driven by generative adversarial networks for benchmarking and protein-peptide docking

Multi-objective optimization problems (MOPs) are characterized as optimization problems in which multiple conflicting objective functions are optimized simultaneously. To solve MOPs, some algorithms used machine learning models to drive the evolutionary algorithms, leading to the design of a variety of model-based evolutionary algorithms. However, model collapse occurs during the generation of candidate solutions, which results in local optima and poor diversity in model-based evolutionary algorithms. To address this problem, we propose a dual-population multi-objective evolutionary algorithm driven by Wasserstein generative adversarial network with gradient penalty (DGMOEA), where the dual-populations coordinate and cooperate to generate high-quality solutions, thus improving the performance of the evolutionary algorithm. We compare the proposed algorithm with the 7 state-of-the-art algorithms on 20 multi-objective benchmark functions. Experimental results indicate that DGMOEA achieves significant results in solving MOPs, where the metrics IGD and HV outperform the other comparative algorithms on 15 and 18 out of 20 benchmarks, respectively. Our algorithm is evaluated on the LEADS-PEP dataset containing 53 protein-peptide complexes, and the experimental results on solving the protein-peptide docking problem indicated that DGMOEA can effectively reduce the RMSD between the generated and the original peptide's 3D poses and achieve more competitive results.

Parameter estimation of the hydraulic turbine regulator system using multi-objective lion algorithm

Accurate modeling of electric power generating unit and its hydraulic turbine regulation systems provides support for the speed controller synthesis and stability analysis. It is however a difficult task due to the presence of many non-linear factors in this system. an approach to estimate the parameters of hydraulic turbine regulatory system models is to derive the physical representation of each component and, through simulation, to compare to compare their models, outputs with real data obtained from a hydroelectric plant located in Brazil. The objective of this paper is to find the best values that will represent the system under study as a whole. This problem can be seen as an optimization problem. To find its feasible and optimal solution, this work proposes a new metaheuristics multi-objective based on the Lion Algorithm (LA), called the Multi-Objective Lion Algorithm (MOLA), and its application in the estimation of parameters of the system under study. In addition, the new metaheuristic proposed is validated by using a set of benchmark cases. The results have demonstrated that MOLA outperforms or at least performs similarly to Multi-objective Grey Wolf Optimizer (MOGWO), Multiple Objective Particle Swarm Optimization (MOPSO), Multi-objective Salp Swarm Algorithm (MSSA), Multiobjective Evolutionary Algorithm Based on Decomposition (MOEA/D), and Non-dominated Sorting Genetic Algorithm III (NSGA-III) in the optimization of multi-objective benchmark functions. These results, suggest that the proposed MOLA algorithm works efficiently.

Enhanced multi-objective marine predator algorithm for dynamic economic-grid fluctuation dispatch with plug-in electric vehicles

The optimization of multi-objective dynamic economic-grid fluctuation dispatch (MODEGD) involves simultaneous optimization of two competing objectives, fuel cost and grid fluctuation, while fulfilling constraints. In this paper, an improved version of the multi-objective marine predator algorithm (IMOMPA), is proposed for solving MODEGD considering plug-in electric vehicles. The IMOMPA algorithm improves upon the existing multi-objective marine predator algorithm in three key ways. Firstly, the initial population's diversity is enhanced using opposition learning combined with chaotic mapping. Secondly, a nonlinear convergence factor is introduced to speed up the convergence. Finally, the spiral search mechanism from the whale optimization algorithm (WOA) is incorporated to improve the algorithm's global search ability. By testing the performance of IMOMPA on ZDT series and DTLZ series multi-objective benchmark functions, IMOMPA shows strong local search capability, better robustness, and better convergence speed and convergence accuracy. Moreover, connecting plug-in electric vehicles (PEVs) to the grid (V2G), can mitigate grid fluctuations and balance peak demand periods. To validate the feasibility of the proposed IMOMPA, numerical experiments are performed on multi-objective benchmark functions and generation units of varying scales. The results demonstrate the superiority of the IMOMPA algorithm compared to other alternatives.

Manifold Interpolation for Large-Scale Multiobjective Optimization via Generative Adversarial Networks.

Large-scale multiobjective optimization problems (LSMOPs) are characterized as optimization problems involving hundreds or even thousands of decision variables and multiple conflicting objectives. To solve LSMOPs, some algorithms designed a variety of strategies to track Pareto-optimal solutions (POSs) by assuming that the distribution of POSs follows a low-dimensional manifold. However, traditional genetic operators for solving LSMOPs have some deficiencies in dealing with the manifold, which often results in poor diversity, local optima, and inefficient searches. In this work, a generative adversarial network (GAN)-based manifold interpolation framework is proposed to learn the manifold and generate high-quality solutions on the manifold, thereby improving the optimization performance of evolutionary algorithms. We compare the proposed approach with several state-of-the-art algorithms on various large-scale multiobjective benchmark functions. The experimental results demonstrate that significant improvements have been achieved by the proposed framework in solving LSMOPs.

Multi-strategy competitive-cooperative co-evolutionary algorithm and its application

In order to effectively solve multi-objective optimization problems (MOPs) and fully balance uniformity and convergence, a multi-strategy competitive-cooperative co-evolutionary algorithm based on adaptive random competition and neighborhood crossover, namely MSCOEA is developed in this paper. In the MSCOEA, a new adaptive random competition strategy is designed to determine whether one sub-population loses diversity through the performance. A random competition process is executed to increase the sub-population diversity in order to compete for participation opportunities in the next iteration. And the extra population is employed to store the found non-dominated solutions. A new neighborhood crossover strategy is designed to enhance the local search ability. Finally, three different types of multi-objective benchmark functions are selected to verify the effectiveness of the MSCOEA. The experiment results show that the MSCOEA can effectively balance convergence and uniformity, and obtains better optimization performance and robustness by comparing with other algorithms. The convergence performance of the adaptive random competition and the neighborhood crossover strategies are also analyzed in detail.

Production Wastage Avoidance Using Modified Multi-Objective Teaching Learning Based Optimization Embedded With Refined Learning Scheme

Teaching learning-based optimization (TLBO) is a popular algorithm used to solve various optimization problems. Nevertheless, conventional TLBO and some improved variants tends to suffer with premature convergence due to rapid loss of population diversity, especially when handling the challenging optimization problems. Furthermore, it is not practical to tackle real-world multiobjective problems using prior approach given the frequent changes of customers’ requirements. Motivated by these challenges, an improved variant known as Modified Multi-objective Teaching Learning Based Optimization-Refined Learning Scheme (MMTLBO-RLS) was proposed as a posterior approach to solve challenging multiobjective optimization problems, including the prediction of optimum turning parameters to machine Polyether ether ketone material (PEEK). Substantial modifications were introduced for teacher and learner phases of MMTLBO-RLS to achieve better balancing of exploration and exploitation searches without incurring excessive computational cost. For modified teacher phase of MMTLBO-RLS, each learner was guided by a unique teacher solution and unique mean position to perform searching with better diversity. Meanwhile, two new learning strategies are incorporated into the modified learner phase of MMTLBO-RLS, enabling all learners to enhance their knowledge more efficiently based on their learning preferences. A systematic approach was followed to develop modelling equations required for optimization. The developed algorithm was then employed in single objective optimization as well as multiobjective optimization to cater its performances in any real-world environment. The prediction model reports that surface roughness of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.1042~\mu m$ </tex-math></inline-formula> and material removal rate of 22.8991 cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> /minute can be achieved. The predicted results differ from validation results by less than 2.69% in any case of optimization. A benchmarking on the performance of MMTLBO-RLS in solving CEC 2009 multiobjective benchmark functions was further carried out with other seven meta-heuristic algorithms. The superior performance of MMTLBO-RLS proves that it is not only suitable to be used in industries to produce the parts of PEEK with supportive quality and quantity, but it is also able to solve other multiobjective optimization problems with competitive performances.

Handling multiple objectives using k‐means clustering guided multiobjective evolutionary algorithm

AbstractMultiobjective optimization problems (MOPs) are very popular these days and have gained continuous research attention. These problems involve a minimum of two conflicting objectives that need to be optimized simultaneously. Multiobjective evolutionary algorithms (MOEAs) optimize these objectives by targeting solutions on the Pareto front. To judge the performance of different MOEAs, the convergence rate and population diversity are used. In order to create a good algorithm with a high convergence rate and diversity, the k‐means clustering algorithm is combined with a real parameter version of the improved environmental adaptation method (IEAM‐RP) in this paper. This is done as IEAM‐RP converges very fast and the k‐means clustering algorithm is used to provide diversity in decision search space. To check the performance of the proposed algorithm, it has been compared with nine state‐of‐the‐art multiobjective algorithms (MOPSOtridist, MOEA/D‐STM, MOEA/D, MOPSO, NSGA‐II, MOPSOses, MOGOA, MODA, MOALO) using standard multiobjective benchmark functions. Further, the performance of the proposed algorithm is compared against IBEA and SMS‐EMOA using walking fish group (WFG) test suit. The experimental results prove the effectiveness of the proposed algorithm.

An optimal nuclei segmentation method based on enhanced multi-objective GWO

In breast cancer image analysis, reliable segmentation of the nuclei is still an open-ended research problem. In this paper, a new clustering-based nuclei segmentation method is presented. First, the proposed method pre-processes the histopathology image through SLIC method. Then, a novel variant of multi-objective grey wolf optimizer is employed to group the obtained super-pixels into optimal clusters. Lastly, the optimal cluster with minimum value is segmented as the nuclei region. The experimental results demonstrates that the proposed variant of multi-objective grey wolf algorithm surpasses the existing multi-objective algorithms over ten standard multi-objective benchmark functions belonging to different categories. Particularly, the proposed variant has achieved best fitness value of more than 0.90 on 90% of the considered functions. Further, the nuclei segmentation accuracy of the proposed method is validated on H&E-stained estrogen receptor positive (ER+) breast cancer images. Experimental results illustrates that the proposed method has attained dice-coefficient value of more than 0.52 on 80% of the images. This illustrates that the proposed method is efficient in producing efficacious segmenting over histology images of Breast cancer.

MOMPA: Multi-objective marine predator algorithm

In this paper, a multi-objective version of the recently proposed marine predator algorithm (MPA) is presented, which is called the multi-objective marine predator algorithm (MOMPA). In this algorithm, an external archive component is introduced to store the non dominated Pareto optimal solutions found so far. Based on the elite selection method, a top predator selection mechanism is proposed, which selects the effective solutions from the archive as the top predators to simulate the predator’s foraging behavior. The CEC2019 multi-modal multi-objective benchmark functions are utilized to evaluate the performance of the proposed algorithm and compared with nine state-of-the-art multi-objective meta-heuristics algorithms. In addition, seven multi-objective engineering design problems (car side impact problem, gear train design problem, welded beam design problem, disk brake design problem, two bar truss design problem, spring design problem and cantilever beam design problem) are used to further verify the effectiveness of the proposed algorithm. The results demonstrate that the proposed MOMPA algorithm not only provides very competitive results but also outperforms other algorithms.

Golden eagle optimizer: A nature-inspired metaheuristic algorithm

This paper proposes a nature-inspired swarm-based metaheuristic for solving global optimization problems called Golden Eagle Optimizer (GEO). The core inspiration of GEO is the intelligence of golden eagles in tuning speed at different stages of their spiral trajectory for hunting. They show more propensity to cruise around and search for prey in the initial stages of hunting and more propensity to attack in the final stages. A golden eagle adjusts these two components to catch the best possible prey in feasible region the shortest possible time. This behavior is mathematically modeled to highlight exploration and exploitation for a global optimization method. The performance of the proposed algorithm is tested and confirmed using 33 benchmark test functions and a scalability test. Results were compared to that of six other well-known algorithms, which revealed GEO’s superiority, which indicates that it can find the global optimum and avoid local optima effectively. The Multi-Objective Golden Eagle Optimizer (MOGEO) is also proposed to solve multi-objective problems. The performance of MOGEO is also tested and verified on ten multi-objective benchmark functions. Results were compared to that of two other multi-objective algorithms, which showed that it can approximate true Pareto optimal solutions better than the other two algorithms. The software (toolbox) and source code for GEO and MOGEO are also provided, which are publicly available.

Pioneer Pareto artificial bee colony algorithm for three-dimensional objective space optimization of composite-based layered radar absorber

A three-dimensional objective space (3DOS) optimization strategy using an enhanced multi-objective artificial bee colony (ABC) algorithm for the design optimization of layered radar absorbing material (LRAM) is presented in this study. The multi-objective exploitation ability of ABC is improved with regard to the convergence and diversity by integrating a pioneer Pareto (PP) solution to the onlooker bee phase, which is selected from the Pareto optimal set. Initially, the performance of PP-ABC is successfully verified by a comparison with ABC and the well-known multi-objective counterparts like particle swarm optimization (PSO) and differential evolution (DE) algorithms. The comparison is carried out through five multi-objective benchmark functions with respect to three favorable and reliable multi-objective indicators such as hypervolume (HV), HV ratio and Pareto sets proximity (PSP). The employed three objective functions to be the dimensions of 3DOS are weighted bandwidth-based total reflection coefficient involving sub-reflection waves of a wide oblique incident angular range 0°–75°, the total thickness and the number of layers. By using PP-ABC, a 3D designed LRAM operating at a large frequency band of 2–18 GHz is then designed for synchronously minimizing the three objective vectors by finding out the design variables: thickness and material types. Meanwhile, the material types of the proposed LRAM are optimally picked up from a composite material database with 51 specimens from 9 previously reported studies (51/9#database). In order to point out the effectiveness of the proposed 3DOS optimization strategy, three LRAMs are also compared with respective reported designs whose material type is selected from a database with 6 specimens (6/1#database). The results show that the proposed LRAMs are hence the global optimal designs in terms of all objective functions thanks to the proposed 3DOS optimization strategy based on PP-ABC.

A Triple-Objective Optimization Scheme Using Butterfly-Integrated ABC Algorithm for Design of Multilayer RAM

A triple-objective scheme with a novel enhanced artificial bee colony (ABC) algorithm is presented for fully optimized design of multilayer radar-absorbing material (MRAM). The exploitation ability of ABC is further enhanced by integrating with butterfly optimization algorithm (BOA), denoted as butterfly-integrated ABC (BiABC). The performance of BiABC is well verified through three multiobjective benchmark functions. A triple-objective electromagnetic (EM) model is constituted, which involves the total reflection (TR), total thickness (TT), and number of layers (NL). Pareto optimality is employed to explore the global optimal designs within the three-dimensional objective space by simultaneously determining the design variables for synchronously minimizing the three objectives by assuring tradeoff among the objectives. The model is verified via a full-wave EM simulator. Five fully optimized broadband and broad-angle MRAMs (BBMs) operating at different frequency bands for 0° – 75° are then designed using an artificial material database, and thus, the proposed scheme is corroborated by comparing them with the state-of-the-art designs. Furthermore, a fully optimized BBM designed using a composite-based real material database is validated through a successful comparison. Therefore, the TT is reduced as 65% and the NL is decreased while the bandwidth is widened thanks to the proposed robust and versatile triple-objective scheme with BiABC.

Adaptive online data-driven closed-loop parameter control strategy for swarm intelligence algorithm

Parameter control is critical for the performance of any swarm intelligence algorithm. In this study, we propose an adaptive online data-driven closed-loop parameter control (CLPC) strategy for a swarm intelligence algorithm to solve both single-objective and multi-objective optimization problems with better performance. The proposed CLPC strategy involves three key parts: controller design, feedback selection, and reference determination. First, based on the control theory, we adopt a proportional integral derivative (PID) controller in the CLPC strategy, which can adaptively adjust the value of parameter according to the difference between reference and feedback. Second, to reflect and monitor the evolution state in real time, we use the mean shift clustering method and define the convergence entropy and the extension entropy to generate feedback. Finally, the reference should provide useful guidance for parameter control according to the features of optimization problems. Thus, in single-objective optimization, we propose a new lossless fitness landscape analysis method and design a decision tree to determine the reference; in multi-objective optimization, the range of the convergence entropy and the extension entropy are regarded as the reference. In addition, to illustrate the effectiveness of the CLPC strategy, two groups of experiments are performed based on the particle swarm optimization (PSO) algorithm in single-objective and multi-objective optimization. At the optimization algorithm level, we compare our proposed CLPC-PSO algorithm with three standard PSO algorithms, three decreasing inertia weight PSO algorithms, and two adaptive PSO algorithms. At the optimization problem level, we perform abundant experiments based on five single-objective benchmark functions, five multi-objective benchmark functions, and twelve scheduling instances. The statistical results show that the performance of the proposed CLPC-PSO algorithm is considerably better and more stable than those of the other eight PSO variants when faced with different problems having various features.

Grid-based dynamic robust multi-objective brain storm optimization algorithm

Rich works have been done on brain storm optimization algorithm solving static single- or multi-objective optimization problems, but less reports for dynamic multi-objective optimization problems. Based on this, a grid-based multi-objective brain storming algorithm with hybrid mutation operation is proposed to find the robust Pareto-optimal solution set over time. Grid-based clustering method partitions the objective space evenly along each objective and classifies the individuals located in the same grid into a cluster. Its computational complexity is less than k-means- and group-based clustering strategies. Traditional Gaussian-, Cauchy- and Chaotic-based mutation operators have different mutation steps and generate the new individuals with various diversity. In order to enhance the diversity and avoiding the premature convergence, a hybrid mutation strategy integrating above three mutation operators is presented. Experimental results for eight dynamic multi-objective benchmark functions show that the proposed algorithm can find robust Pareto-optimal solutions approximating the true Pareto front under more subsequent environments with the acceptable fitness threshold. The longer survival time also indicates that grid-based clustering method and hybrid mutation strategy are apt to better robustness.

P-Optimality-Based Multiobjective Root System Growth Algorithms for Multiobjective Applications

In this paper, a multiobjective root system growth algorithm-based p-optimality (p-MORSGA) is proposed. The proposed p-MORSGA extended original root system growth algorithm with multiobjective nondomination strategy. To enhance its effect of convergence of solution groups, the p-optimality criterion is employed to determine the solutions of last nondominated front into the next generation group. In the evolution process, global general (GG), concerning the margin information and population density, is selected as the suitable optimality criterion of evaluating the performance of solutions. Application of the new p-MORSGA on several multiobjective benchmark functions shows a marked improvement in performance over the modified classical MOEAs with such criterion. Finally, the proposed p-MORSGA is applied to solve two real-world problems, multiobjective portfolio optimization problems (MOPOPs) and multiobjective optimal power flow (OPF) problems. The experimental results demonstrate that p-MORSGA is extremely effective for real-world application problems.