{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:38:46Z","timestamp":1760060326194,"version":"build-2065373602"},"reference-count":98,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,17]],"date-time":"2025-08-17T00:00:00Z","timestamp":1755388800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>It is crucial to understand how fitness landscape characteristics (FLCs) are associated with the performance and behavior of the differential evolution (DE) algorithm to optimize its application across various optimization problems. Although previous studies have explored DE performance in relation to FLCs, these studies have limitations. Specifically, the narrow range of FLC metrics considered for problem characterization and the lack of research exploring the relationship between the search behavior of the DE algorithm and FLCs represent two major concerns. This study investigates the impact of five FLCs, namely ruggedness, gradients, funnels, deception, and searchability, on DE performance and behavior across various problems and dimensions. Two experiments were conducted: the first assesses DE performance using three performance metrics, i.e., solution quality, success rate, and success speed. The first experiment reveals that DE exhibits stronger associations with FLCs for higher-dimensional problems. Moreover, the presence of multiple funnels and high deception levels are linked to performance degradation, while high searchability is significantly associated with improved performance. The second experiment analyzes the DE search behavior using the diversity rate-of-change (DRoC) behavioral measure. The second experiment shows that the speed at which the DE algorithm transitions from exploration to exploitation varies with different FLCs and the problem dimensionality. The analysis reveals that DE reduces its diversity more slowly in landscapes with multiple funnels and resists deception, but faces excessively slow convergence for high-dimensional problems. Overall, the results elucidate that multiple funnels and high deception levels are the FLCs most strongly associated with the performance and search behavior of the DE algorithm. These findings contribute to a deeper understanding of how FLCs interact with both the performance and search behavior of the DE algorithm and suggest avenues to optimize DE for real-world applications.<\/jats:p>","DOI":"10.3390\/a18080520","type":"journal-article","created":{"date-parts":[[2025,8,18]],"date-time":"2025-08-18T15:34:53Z","timestamp":1755531293000},"page":"520","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Fitness Landscape Analysis for the Differential Evolution Algorithm"],"prefix":"10.3390","volume":"18","author":[{"given":"Amani","family":"Saad","sequence":"first","affiliation":[{"name":"Department of Computer Science, Stellenbosch University, Stellenbosch 7600, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0242-3539","authenticated-orcid":false,"given":"Andries P.","family":"Engelbrecht","sequence":"additional","affiliation":[{"name":"Industrial Engineering, and Computer Science Division, Stellenbosch University, Stellenbosch 7600, South Africa"},{"name":"GUST Engineering and Applied Innovation Research Center, Gulf University of Science and Technology, West Mishref 15453, Kuwait"}]},{"given":"Salman A.","family":"Khan","sequence":"additional","affiliation":[{"name":"College of Computing and Information Sciences, Karachi Institute of Economics and Technology, Karachi 75190, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,17]]},"reference":[{"key":"ref_1","unstructured":"Storn, R. (1996, January 8\u201311). On the Usage of Differential Evolution for Function Optimization. Proceedings of the IEEE International Conference on Fuzzy Systems, New Orleans, LA, USA."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential Evolution\u2014A Simple and Efficient Heuristic for Global Optimization Over Continuous Spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_3","unstructured":"Price, K., Storn, R.M., and Lampinen, J.A. (2006). Differential Evolution: A Practical Approach to Global Optimization, Springer Science & Business Media. [1st ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chakraborty, U.K. (2008). A Review of Major Application Areas of Differential Evolution. Advances in Differential Evolution, Springer.","DOI":"10.1007\/978-3-540-68830-3"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.swevo.2016.01.004","article-title":"Recent Advances in Differential Evolution\u2014An Updated Survey","volume":"27","author":"Das","year":"2016","journal-title":"Swarm Evol. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3831","DOI":"10.1016\/j.aej.2021.09.013","article-title":"Differential Evolution: A Recent Review Based on State-of-the-Art Works","volume":"61","author":"Ahmad","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1007\/s11831-022-09825-5","article-title":"Differential Evolution and Its Applications in Image Processing Problems: A Comprehensive Review","volume":"30","author":"Chakraborty","year":"2022","journal-title":"Arch. Comput. Methods Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1109\/TEVC.2006.886448","article-title":"Evolving Problems to Learn About Particle Swarm Optimizers and Other Search Algorithms","volume":"11","author":"Langdon","year":"2007","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Talbi, E. (2009). Metaheuristics: From Design to Implementation, John Wiley & Sons.","DOI":"10.1002\/9780470496916"},{"key":"ref_10","unstructured":"Yang, X. (2010). Nature-Inspired Metaheuristic Algorithms, Luniver Press."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Gendreau, M., and Potvin, J. (2010). Handbook of Metaheuristics, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4419-1665-5"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Michalewicz, Z., and Fogel, D.B. (2004). Tuning the Algorithm to the Problem. How to Solve It: Modern Heuristics, Springer. [2nd ed.].","DOI":"10.1007\/978-3-662-07807-5"},{"key":"ref_13","unstructured":"Wright, S. (1932, January 24\u201331). The Roles of Mutation, Inbreeding, Crossbreeding, and Selection in Evolution. Proceedings of the Sixth International Congress on Genetics, Ithaca, NY, USA."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1086\/284777","article-title":"Surfaces of Selective Value Revisited","volume":"131","author":"Wright","year":"1988","journal-title":"Am. Nat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.neucom.2022.06.084","article-title":"A Survey of Fitness Landscape Analysis for Optimization","volume":"503","author":"Zou","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Malan, K.M., and Engelbrecht, A.P. (2009, January 18\u201321). Quantifying Ruggedness of Continuous Landscapes Using Entropy. Proceedings of the IEEE Congress on Evolutionary Computation, Trondheim, Norway.","DOI":"10.1109\/CEC.2009.4983112"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Malan, K.M., and Engelbrecht, A.P. (2013, January 20\u201323). Ruggedness, Funnels and Gradients in Fitness Landscapes and the Effect on PSO Performance. Proceedings of the IEEE Congress on Evolutionary Computation, Cancun, Mexico.","DOI":"10.1109\/CEC.2013.6557671"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s11721-014-0099-x","article-title":"Characterising the Searchability of Continuous Optimisation Problems for PSO","volume":"8","author":"Malan","year":"2014","journal-title":"Swarm Intell."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/978-3-642-41888-4_4","article-title":"Fitness Landscape Analysis for Metaheuristic Performance Prediction","volume":"Volume 6","author":"Richter","year":"2014","journal-title":"Recent Advances in the Theory and Application of Fitness Landscapes"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Malan, K.M., and Engelbrecht, A.P. (2014, January 9\u201312). Particle Swarm Optimisation Failure Prediction Based on Fitness Landscape Characteristics. Proceedings of the Symposium on Swarm Intelligence, Orlando, FL, USA.","DOI":"10.1109\/SIS.2014.7011789"},{"key":"ref_21","unstructured":"Dennis, C., Ombuki-Berman, B.M., and Engelbrecht, A.P. (July, January 28). Predicting Particle Swarm Optimization Control Parameters from Fitness Landscape Characteristics. Proceedings of the IEEE Congress on Evolutionary Computation, Krakow, Poland."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Uluda\u011f, G., and Uyar, A.\u015e. (2009, January 9\u201311). Fitness Landscape Analysis of Differential Evolution Algorithms. Proceedings of the International Conference on Soft Computing, Computing with Words and Perceptions in System Analysis, Decision and Control, Antalya, Turkey.","DOI":"10.1109\/ICSCCW.2009.5379477"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Yang, S., Li, K., Li, W., Chen, W., and Chen, Y. (2016, January 28\u201330). Dynamic Fitness Landscape Analysis on Differential Evolution Algorithm. Proceedings of the Bio-Inspired Computing\u2013Theories and Applications, Xi\u2019an, China.","DOI":"10.1007\/978-981-10-3614-9_23"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zhang, Z., Duan, N., Zou, K., and Sun, Z. (2018, January 25\u201327). Predictive Models of Problem Difficulties for Differential Evolutionary Algorithm Based on Fitness Landscape Analysis. Proceedings of the 37th Chinese Control Conference, Wuhan, China.","DOI":"10.23919\/ChiCC.2018.8483931"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"7773","DOI":"10.1007\/s00500-018-3448-7","article-title":"A Self-Feedback Strategy Differential Evolution with Fitness Landscape Analysis","volume":"22","author":"Huang","year":"2018","journal-title":"Soft Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1151","DOI":"10.1007\/s00500-017-2833-y","article-title":"Self-Feedback Differential Evolution Adapting to Fitness Landscape Characteristics","volume":"23","author":"Li","year":"2019","journal-title":"Soft Comput."},{"key":"ref_27","first-page":"36","article-title":"Performance Analyses of Differential Evolution Algorithm Based on Dynamic Fitness Landscape","volume":"13","author":"Li","year":"2019","journal-title":"Int. J. Cogn. Inform. Nat. Intell."},{"key":"ref_28","first-page":"284","article-title":"Mutation Strategy Selection Based on Fitness Landscape Analysis: A Preliminary Study","volume":"Volume 1159","author":"Pan","year":"2020","journal-title":"Bio-Inspired Computing: Theories and Applications"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"106693","DOI":"10.1016\/j.asoc.2020.106693","article-title":"A Fitness Landscape Ruggedness Multiobjective Differential Evolution Algorithm with a Reinforcement Learning Strategy","volume":"96","author":"Huang","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5726","DOI":"10.1007\/s11227-020-03482-w","article-title":"A Novel Mutation Strategy Selection Mechanism for Differential Evolution Based on Local Fitness Landscape","volume":"77","author":"Tan","year":"2021","journal-title":"J. Supercomput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/j.ins.2020.11.023","article-title":"Differential Evolution with Adaptive Mutation Strategy Based on Fitness Landscape Analysis","volume":"549","author":"Tan","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"5251","DOI":"10.1007\/s40747-023-01005-7","article-title":"Keenness for Characterizing Continuous Optimization Problems and Predicting Differential Evolution Algorithm Performance","volume":"9","author":"Li","year":"2023","journal-title":"Complex Intell. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.ins.2022.11.071","article-title":"A New Evolving Operator Selector by Using Fitness Landscape in Differential Evolution Algorithm","volume":"624","author":"Li","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1587\/transinf.2022DLP0010","article-title":"A Novel Differential Evolution Algorithm Based on Local Fitness Landscape Information for Optimization Problems","volume":"106","author":"Liang","year":"2023","journal-title":"Trans. Inf. Syst."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Zheng, L., and Luo, S. (2022). Adaptive Differential Evolution Algorithm Based on Fitness Landscape Characteristic. Mathematics, 10.","DOI":"10.3390\/math10091511"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s11047-020-09835-x","article-title":"The Influence of Fitness Landscape Characteristics on Particle Swarm Optimisers","volume":"21","author":"Engelbrecht","year":"2022","journal-title":"Nat. Comput."},{"key":"ref_37","unstructured":"Ghosh, A., and Tsutsui, S. (2003). Smoothness, Ruggedness and Neutrality of Fitness Landscapes: From Theory to Application. Advances in Evolutionary Computing: Theory and Applications, Springer."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Lunacek, M., and Whitley, D. (2006, January 8\u201312). The Dispersion Metric and the CMA Evolution Strategy. Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, Seattle, WA, USA.","DOI":"10.1145\/1143997.1144085"},{"key":"ref_39","unstructured":"Jones, T., and Forrest, S. (1995, January 15\u201319). Fitness Distance Correlation as a Measure of Problem Difficulty for Genetic Algorithms. Proceedings of the Sixth International Conference on Genetic Algorithms, Pittsburgh, PA, USA."},{"key":"ref_40","unstructured":"Verel, S., Collard, P., and Clergue, M. (2003, January 8\u201312). Where Are Bottlenecks in NK Fitness Landscapes?. Proceedings of the IEEE Congress on Evolutionary Computation, Canberra, Australia."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Bosman, P., and Engelbrecht, A.P. (2014, January 10\u201312). Diversity Rate of Change Measurement for Particle Swarm Optimisers. Proceedings of the 9th International Conference on Swarm Intelligence, Brussels, Belgium.","DOI":"10.1007\/978-3-319-09952-1_8"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1109\/TEVC.2023.3346672","article-title":"Determining Metaheuristic Similarity Using Behavioral Analysis","volume":"29","author":"Hayward","year":"2025","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_43","first-page":"161","article-title":"A Comprehensive Survey on Fitness Landscape Analysis","volume":"Volume 378","author":"Fodor","year":"2012","journal-title":"Recent Advances in Intelligent Engineering Systems"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Ochoa, G., and Malan, K. (2019, January 13\u201317). Recent Advances in Fitness Landscape Analysis. Proceedings of the Genetic and Evolutionary Computation Conference Companion, Prague, Czech Republic.","DOI":"10.1145\/3319619.3323383"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Malan, K.M. (2021). A Survey of Advances in Landscape Analysis for Optimisation. Algorithms, 14.","DOI":"10.3390\/a14020040"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Malan, K., and Ochoa, G. (2023, January 15\u201319). Landscape Analysis of Optimization Problems and Algorithms. Proceedings of the Companion Conference on Genetic and Evolutionary Computation, Lisbon, Portugal.","DOI":"10.1145\/3583133.3595051"},{"key":"ref_47","unstructured":"Jones, T. (1995). Evolutionary Algorithms, Fitness Landscapes and Search. [Ph.D. Thesis, University of New Mexico]."},{"key":"ref_48","unstructured":"Barnett, L. (2003). Evolutionary Search on Fitness Landscapes with Neutral Networks. [Ph.D. Thesis, University of Sussex]."},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Derbel, B., and Verel, S. (2020, January 8\u201312). Fitness Landscape Analysis to Understand and Predict Algorithm Performance for Single- and Multi-Objective Optimization. Proceedings of the Genetic and Evolutionary Computation Conference Companion, Cancun, Mexico.","DOI":"10.1145\/3377929.3389893"},{"key":"ref_50","unstructured":"Bolshakov, V., Pitzer, E., and Affenzeller, M. (April, January 30). Fitness Landscape Analysis of a Simulation Optimisation Problem with HeuristicLab. Proceedings of the UKSim 5th European Symposium on Computer Modeling and Simulation, Cambridge, UK."},{"key":"ref_51","first-page":"3","article-title":"Fitness Landscape Analysis and Metaheuristics Efficiency","volume":"12","author":"Marmion","year":"2013","journal-title":"J. Math. Model. Algorithms Oper. Res."},{"key":"ref_52","unstructured":"B\u00e4ck, T., Preuss, M., Deutz, A., Wang, H., Kallel, S.A., Juez, J.L., and Sim, K. (2020, January 5\u20139). On Stochastic Fitness Landscapes: Local Optimality and Fitness Landscape Analysis for Stochastic Search Operators. Proceedings of the 16th International Conference on Parallel Problem Solving from Nature, Leiden, The Netherlands. Lecture Notes in Computer Science."},{"key":"ref_53","unstructured":"Russell, S., and Norvig, P. (2020). Artificial Intelligence: A Modern Approach, Pearson. [4th ed.]."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1016\/j.asoc.2016.11.041","article-title":"Measures in the Time and Frequency Domains for Fitness Landscape Analysis of Dynamic Optimization Problems","volume":"51","author":"Lu","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1016\/j.ejor.2018.01.051","article-title":"Analysis of the Similarities and Differences of Job-Based Scheduling Problems","volume":"270","author":"Lu","year":"2018","journal-title":"Eur. J. Oper. Res."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Tanabe, R., and Fukunaga, A. (2013, January 20\u201323). Success-history Based Parameter Adaptation for Differential Evolution. Proceedings of the IEEE Congress on Evolutionary Computation, Cancun, Mexico.","DOI":"10.1109\/CEC.2013.6557555"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Tanabe, R., and Fukunaga, A. (2014, January 6\u201311). Improving the Search Performance of SHADE Using Linear Population Size Reduction. Proceedings of the IEEE Congress on Evolutionary Computation), Beijing, China.","DOI":"10.1109\/CEC.2014.6900380"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1162\/106365602317301754","article-title":"Fitness Landscapes and Evolvability","volume":"10","author":"Smith","year":"2002","journal-title":"Evol. Comput."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1","DOI":"10.4018\/IJSIR.352060","article-title":"A Fitness Distance Correlation-Based Adaptive Differential Evolution for Nonlinear Equations Systems","volume":"15","author":"Xiaowang","year":"2024","journal-title":"Int. J. Swarm Intell. Res."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"121842","DOI":"10.1016\/j.ins.2024.121842","article-title":"Adaptive niching differential evolution algorithm with landscape analysis for multimodal optimization","volume":"700","author":"Xinyu","year":"2025","journal-title":"Inf. Sci."},{"key":"ref_61","doi-asserted-by":"crossref","unstructured":"Sun, Y., Halgamuge, S.K., Kirley, M., and Munoz, M.A. (2014, January 22\u201324). On the Selection of Fitness Landscape Analysis Metrics for Continuous Optimization Problems. Proceedings of the International Conference on Information and Automation for Sustainability, Colombo, Sri Lanka.","DOI":"10.1109\/ICIAFS.2014.7069635"},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Saad, A.D., Engelbrecht, A.P., and Khan, S.A. (2024). An Analysis of Differential Evolution Population Size. Appl. Sci., 14.","DOI":"10.3390\/app14219976"},{"key":"ref_63","first-page":"293","article-title":"A Parameter Study for Differential Evolution","volume":"10","author":"Koumoutsakos","year":"2002","journal-title":"Adv. Intell. Syst. Fuzzy Syst. Evol. Comput."},{"key":"ref_64","unstructured":"Ronkkonen, J., Kukkonen, S., and Price, K. (2005, January 2\u20135). Real-Parameter Optimization with Differential Evolution. Proceedings of the IEEE Congress on Evolutionary Computation, Edinburgh, Scotland, UK."},{"key":"ref_65","doi-asserted-by":"crossref","unstructured":"Montgomery, J., and Chen, S. (2010, January 18\u201323). An Analysis of the Operation of Differential Evolution at High and Low Crossover Rates. Proceedings of the IEEE Congress on Evolutionary Computation, Barcelona, Spain.","DOI":"10.1109\/CEC.2010.5586128"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1016\/S0305-0548(03)00116-3","article-title":"Population Set-Based Global Optimization Algorithms: Some Modifications and Numerical Studies","volume":"31","author":"Ali","year":"2004","journal-title":"Comput. Oper. Res."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"991","DOI":"10.1007\/s00500-010-0655-2","article-title":"Improving the Performance of Differential Evolution Algorithm Using Cauchy Mutation","volume":"15","author":"Ali","year":"2011","journal-title":"Soft Comput."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1016\/j.ins.2014.11.026","article-title":"Cluster-Based Population Initialization for Differential Evolution Frameworks","volume":"297","author":"Poikolainen","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.swevo.2017.12.007","article-title":"Comparison of Mutation Strategies in Differential Evolution\u2014A Probabilistic Perspective","volume":"39","author":"Opara","year":"2018","journal-title":"Swarm Evol. Comput."},{"key":"ref_70","first-page":"31","article-title":"Enhancing Differential Evolution Utilizing Eigenvector-Based Crossover Operator","volume":"19","author":"Guo","year":"2014","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"20035","DOI":"10.1109\/ACCESS.2021.3051264","article-title":"A Self-Adaptive Differential Evolution Algorithm Using Oppositional Solutions and Elitist Sharing","volume":"9","author":"Song","year":"2021","journal-title":"IEEE Access"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1109\/TCYB.2016.2536167","article-title":"Multiple Exponential Recombination for Differential Evolution","volume":"47","author":"Qiu","year":"2016","journal-title":"IEEE Trans. Cybern."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"2742","DOI":"10.1109\/TCYB.2017.2676882","article-title":"Adaptive Differential Evolution with Sorting Crossover Rate for Continuous Optimization Problems","volume":"47","author":"Zhou","year":"2017","journal-title":"IEEE Trans. Cybern."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.ins.2017.08.028","article-title":"Landscape-Based Adaptive Operator Selection Mechanism for Differential Evolution","volume":"418","author":"Sallam","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"286","DOI":"10.1016\/j.asoc.2017.03.010","article-title":"Differential Evolution with Improved Individual-Based Parameter Setting and Selection Strategy","volume":"56","author":"Tian","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.ejor.2015.10.043","article-title":"A Differential Evolution Algorithm with Self-Adaptive Strategy and Control Parameters Based on Symmetric Latin Hypercube Design for Unconstrained Optimization Problems","volume":"250","author":"Zhao","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_77","doi-asserted-by":"crossref","unstructured":"Takahama, T., and Sakai, S. (2014, January 5\u20138). An Adaptive Differential Evolution Considering Correlation of Two Algorithm Parameters. Proceedings of the 7th International Conference on Soft Computing and Intelligent Systems and 15th International Symposium on Advanced Intelligent Systems, Kitakyushu, Japan.","DOI":"10.1109\/SCIS-ISIS.2014.7044698"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/4235.771163","article-title":"Evolutionary Programming Made Faster","volume":"3","author":"Yao","year":"1999","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"1605","DOI":"10.1016\/j.camwa.2006.07.013","article-title":"A Novel Population Initialization Method for Accelerating Evolutionary Algorithms","volume":"53","author":"Rahnamayan","year":"2007","journal-title":"Comput. Math. Appl."},{"key":"ref_80","doi-asserted-by":"crossref","unstructured":"Mishra, S.K. (2006). Performance of Repulsive Particle Swarm Method in Global Optimization of Some Important Test Functions: A Fortran Program. SSRN Electron. J.","DOI":"10.2139\/ssrn.924339"},{"key":"ref_81","doi-asserted-by":"crossref","unstructured":"Hansen, N., and Kern, S. (2004, January 18\u201322). Evaluating the CMA Evolution Strategy on Multimodal Test Functions. Proceedings of the International Conference on Parallel Problem Solving from Nature, Birmingham, UK.","DOI":"10.1007\/978-3-540-30217-9_29"},{"key":"ref_82","doi-asserted-by":"crossref","unstructured":"Mishra, S.K. (2006). Some New Test Functions for Global Optimization and Performance of Repulsive Particle Swarm Method, North-Eastern Hill University. MPRA Paper 2718.","DOI":"10.2139\/ssrn.926132"},{"key":"ref_83","unstructured":"Price, K.V., Storn, R.M., and Lampinen, J.A. (2005). Appendix A.1: Unconstrained Uni-modal Test Functions. Differential Evolution: A Practical Approach to Global Optimization, Springer."},{"key":"ref_84","unstructured":"Jong, K.A.D. (1975). An Analysis of the Behavior of a Class of Genetic Adaptive Systems. [Ph.D. Thesis, University of Michigan]."},{"key":"ref_85","unstructured":"(2023, March 26). CIlib Fitness Landscape Analysis. Available online: https:\/\/github.com\/ciren\/fla."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"72","DOI":"10.2307\/1412159","article-title":"The Proof and Measurement of Association Between Two Things","volume":"15","author":"Spearman","year":"1904","journal-title":"Am. J. Psychol."},{"key":"ref_87","doi-asserted-by":"crossref","unstructured":"Malan, K.M. (2014). Characterising Continuous Optimisation Problems for Particle Swarm Optimisation Performance Prediction. [Ph.D. Thesis, University of Pretoria].","DOI":"10.1007\/s11721-014-0099-x"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1023\/A:1021956306041","article-title":"A Note on the Griewank Test Function","volume":"25","author":"Locatelli","year":"2003","journal-title":"J. Glob. Optim."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1080\/01621459.1972.10481251","article-title":"Significance Testing of the Spearman Rank Correlation Coefficient","volume":"67","author":"Zar","year":"1972","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_90","unstructured":"Lampinen, J., and Zelinka, I. (2000, January 7\u20139). On Stagnation of the Differential Evolution Algorithm. Proceedings of the 6th International Mendel Conference on Soft Computing, Brno, Czech Republic."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/j.ins.2020.12.076","article-title":"Average convergence rate of evolutionary algorithms in continuous optimization","volume":"562","author":"Yu","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_92","doi-asserted-by":"crossref","unstructured":"Morales-Casta\u00f1eda, B., Maciel-Castillo, O., Navarro, M.A., Aranguren, I., Valdivia, A., Ramos-Michel, A., Oliva, D., and Hinojosa, S. (2022, January 18\u201323). Handling stagnation through diversity analysis: A new set of operators for evolutionary algorithms. Proceedings of the IEEE Congress on Evolutionary Computation, Padua, Italy.","DOI":"10.1109\/CEC55065.2022.9870284"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"105496","DOI":"10.1016\/j.asoc.2019.105496","article-title":"Self-Adaptive Mutation Differential Evolution Algorithm Based on Particle Swarm Optimization","volume":"81","author":"Wang","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Xiao, P., Zou, D., Xia, Z., and Shen, X. (2019, January 19\u201320). Multi-strategy different dimensional mutation differential evolution algorithm. Proceedings of the 3rd International Conference on Advances in Materials, Machinery, and Electronics, Wuhan, China.","DOI":"10.1063\/1.5090756"},{"key":"ref_95","doi-asserted-by":"crossref","unstructured":"Mersmann, O., Bischl, B., Trautmann, H., Preuss, M., Weihs, C., and Rudolph, G. (2011, January 12\u201316). Exploratory Landscape Analysis. Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation, Dublin, Ireland.","DOI":"10.1145\/2001576.2001690"},{"key":"ref_96","doi-asserted-by":"crossref","unstructured":"Kerschke, P., Preuss, M., Hern\u00e1ndez, C., Sch\u00fctze, O., Sun, J., Grimme, C., Rudolph, G., Bischl, B., and Trautmann, H. (2014). Cell Mapping Techniques for Exploratory Landscape Analysis. Advances in Intelligent Systems and Computing, Springer International Publishing.","DOI":"10.1007\/978-3-319-07494-8_9"},{"key":"ref_97","doi-asserted-by":"crossref","unstructured":"Lang, R.D., and Engelbrecht, A.P. (2019, January 6\u20139). On the Robustness of Random Walks for Fitness Landscape Analysis. Proceedings of the IEEE Symposium Series on Computational Intelligence, Xiamen, China.","DOI":"10.1109\/SSCI44817.2019.9002761"},{"key":"ref_98","doi-asserted-by":"crossref","unstructured":"Lang, R.D., and Engelbrecht, A.P. (2020, January 19\u201324). Decision Space Coverage of Random Walks. Proceedings of the IEEE Congress on Evolutionary Computation, Glasgow, UK.","DOI":"10.1109\/CEC48606.2020.9185623"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/8\/520\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:29:29Z","timestamp":1760034569000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/8\/520"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,17]]},"references-count":98,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["a18080520"],"URL":"https:\/\/doi.org\/10.3390\/a18080520","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2025,8,17]]}}}