{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T10:37:06Z","timestamp":1776335826187,"version":"3.51.2"},"reference-count":30,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,16]],"date-time":"2022-05-16T00:00:00Z","timestamp":1652659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The local optima network model has proved useful in the past in connection with combinatorial optimization problems. Here we examine its extension to the real continuous function domain. Through a sampling process, the model builds a weighted directed graph which captures the function\u2019s minima basin structure and its interconnection and which can be easily manipulated with the help of complex networks metrics. We show that the model provides a complementary view of function spaces that is easier to analyze and visualize, especially at higher dimensions. In particular, we show that function hardness as represented by algorithm performance is strongly related to several graph properties of the corresponding local optima network, opening the way for a classification of problem difficulty according to the corresponding graph structure and with possible extensions in the design of better metaheuristic approaches.<\/jats:p>","DOI":"10.3390\/e24050703","type":"journal-article","created":{"date-parts":[[2022,5,16]],"date-time":"2022-05-16T13:06:23Z","timestamp":1652706383000},"page":"703","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Local Optima Network View of Real Function Fitness Landscapes"],"prefix":"10.3390","volume":"24","author":[{"given":"Marco","family":"Tomassini","sequence":"first","affiliation":[{"name":"Department of Information Systems, University of Lausanne, 1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,16]]},"reference":[{"key":"ref_1","unstructured":"Liberti, L. (2008). Introduction to Global Optimization, LIX, Ecole Polytechnique."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Talbi, E.G. (2009). Metaheuristics: From Design to Implementation, John Wiley & Sons.","DOI":"10.1002\/9780470496916"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Richter, H., and Engelbrecht, A. (2014). Recent Advances in the Theory and Application of Fitness Landscapes, Springer.","DOI":"10.1007\/978-3-642-41888-4"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"066114","DOI":"10.1103\/PhysRevE.78.066114","article-title":"Complex-network analysis of combinatorial spaces: The NK landscape case","volume":"78","author":"Tomassini","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1109\/TEVC.2010.2046175","article-title":"Local optima networks of NK landscapes with neutrality","volume":"15","author":"Verel","year":"2011","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1684","DOI":"10.1016\/j.physa.2011.01.005","article-title":"Communities of minima in local optima networks of combinatorial spaces","volume":"390","author":"Daolio","year":"2011","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Hernando, L., Daolio, F., Veerapen, N., and Ochoa, G. (2017, January 5\u20138). Local optima networks of the permutation flowshop scheduling problem: Makespan vs. total flow time. Proceedings of the 2017 IEEE Congress on Evolutionary Computation (CEC), Donostia, Spain.","DOI":"10.1109\/CEC.2017.7969541"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"126089","DOI":"10.1016\/j.physa.2021.126089","article-title":"Complex networks analysis of the energy landscape of the low autocorrelation binary sequences problem","volume":"577","author":"Tomassini","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/j.ejor.2012.12.019","article-title":"On the structure of multiobjective combinatorial search space: Mnk-landscapes with correlated objectives","volume":"227","author":"Verel","year":"2013","journal-title":"Eur. J. Oper. Res."},{"key":"ref_10","unstructured":"Wales, D. (2003). Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2009","DOI":"10.1093\/bioinformatics\/btu156","article-title":"Basin hopping graph: A computational framework to characterize RNA folding landscapes","volume":"30","author":"Kucharik","year":"2014","journal-title":"Bioinformatics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1007\/s10100-017-0480-0","article-title":"Basin hopping networks of continuous global optimization problems","volume":"25","author":"Gelle","year":"2017","journal-title":"Cent. Eur. J. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Contreras-Cruz, M.A., Ochoa, G., and Ramirez-Paredes, J.P. (2020). Synthetic vs. real-world continuous landscapes: A local optima networks view. Lecture Notes in Computer Science, Proceedings of the International Conference on Bioinspired Methods and Their Applications, Brussels, Belgium, 19\u201320 November 2016, Springer.","DOI":"10.1007\/978-3-030-63710-1_1"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/S0036144501395952","article-title":"Combinatorial landscapes","volume":"44","author":"Reidys","year":"2002","journal-title":"SIAM Rev."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Thomson, S.L., Ochoa, G., and Verel, S. (2019). Clarifying the difference in local optima network sampling algorithms. Lecture Notes in Computer Science, Proceedings of the European Conference on Evolutionary Computation in Combinatorial Optimization (Part of EvoStar), Leipzig, Germany, 24\u201326 April 2019, Springer.","DOI":"10.1007\/978-3-030-16711-0_11"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10589-005-4561-y","article-title":"On the multilevel structure of global optimization problems","volume":"30","author":"Locatelli","year":"2005","journal-title":"Comput. Optim. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1368","DOI":"10.1126\/science.285.5432.1368","article-title":"Global optimization of clusters, crystals, and biomolecules","volume":"285","author":"Wales","year":"1999","journal-title":"Science"},{"key":"ref_18","unstructured":"Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P. (2007). Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press."},{"key":"ref_19","unstructured":"Bartz-Beielstein, T., Doerr, C., Berg, D., Bossek, J., Chandrasekaran, S., Eftimov, T., Fischbach, A., Kerschke, P., Cava, W.L., and Lopez-Ibanez, M. (2020). Benchmarking in optimization: Best practice and open issues. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential evolution\u2013a simple and efficient heuristic for global optimization over continuous spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1090\/S0025-5718-1967-0224273-2","article-title":"Quasi-newton methods and their application to function minimisation","volume":"21","author":"Broyden","year":"1967","journal-title":"Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Newman, M. (2018). Networks, Oxford University Press.","DOI":"10.1093\/oso\/9780198805090.001.0001"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"378","DOI":"10.1038\/35019019","article-title":"Error and attack tolerance of complex networks","volume":"406","author":"Albert","year":"2000","journal-title":"Nature"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0169-7552(98)00110-X","article-title":"The anatomy of a large-scale hypertextual web search engine","volume":"30","author":"Brin","year":"1998","journal-title":"Comput. Netw. Isdn Syst."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"8721","DOI":"10.1073\/pnas.89.18.8721","article-title":"Protein folding funnels: A kinetic approach to the sequence-structure relationship","volume":"89","author":"Leopold","year":"1992","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s10732-017-9334-0","article-title":"Mapping the global structure of TSP fitness landscapes","volume":"24","author":"Ochoa","year":"2018","journal-title":"J. Heuristics"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1080\/10556788.2020.1808977","article-title":"Coco: A platform for comparing continuous optimizers in a black-box setting","volume":"36","author":"Hansen","year":"2021","journal-title":"Optim. Methods Softw."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Homolya, V., and Vink\u00f3, T. (2019). Leveraging local optima network properties for memetic differential evolution. Advances in Intelligent Systems and Computing, Proceedings of the World Congress on Global Optimization, Metz, France, 8\u201310 July 2019, Springer.","DOI":"10.1007\/978-3-030-21803-4_11"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"12585","DOI":"10.1039\/C7CP01108C","article-title":"Energy landscapes for machine learning","volume":"19","author":"Ballard","year":"2017","journal-title":"Phys. Chem. Chem. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/703\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:11:08Z","timestamp":1760137868000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/703"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,16]]},"references-count":30,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["e24050703"],"URL":"https:\/\/doi.org\/10.3390\/e24050703","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,16]]}}}