{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T20:16:24Z","timestamp":1777407384896,"version":"3.51.4"},"reference-count":40,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2020,2,29]],"date-time":"2020-02-29T00:00:00Z","timestamp":1582934400000},"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>Surfing in rough waters is not always as fun as wave riding the \u201cbig one\u201d. Similarly, in optimization problems, fitness landscapes with a huge number of local optima make the search for the global optimum a hard and generally annoying game. Computational Intelligence optimization metaheuristics use a set of individuals that \u201csurf\u201d across the fitness landscape, sharing and exploiting pieces of information about local fitness values in a joint effort to find out the global optimum. In this context, we designed surF, a novel surrogate modeling technique that leverages the discrete Fourier transform to generate a smoother, and possibly easier to explore, fitness landscape. The rationale behind this idea is that filtering out the high frequencies of the fitness function and keeping only its partial information (i.e., the low frequencies) can actually be beneficial in the optimization process. We prove our theory by combining surF with a settings free variant of Particle Swarm Optimization (PSO) based on Fuzzy Logic, called Fuzzy Self-Tuning PSO. Specifically, we introduce a new algorithm, named F3ST-PSO, which performs a preliminary exploration on the surrogate model followed by a second optimization using the actual fitness function. We show that F3ST-PSO can lead to improved performances, notably using the same budget of fitness evaluations.<\/jats:p>","DOI":"10.3390\/e22030285","type":"journal-article","created":{"date-parts":[[2020,3,2]],"date-time":"2020-03-02T04:16:16Z","timestamp":1583122576000},"page":"285","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Surfing on Fitness Landscapes: A Boost on Optimization by Fourier Surrogate Modeling"],"prefix":"10.3390","volume":"22","author":[{"given":"Luca","family":"Manzoni","sequence":"first","affiliation":[{"name":"Department of Mathematics and Geosciences, University of Trieste, 34127 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniele M.","family":"Papetti","sequence":"additional","affiliation":[{"name":"Department of Informatics, Systems and Communication, University of Milano-Bicocca, 20126 Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7780-0434","authenticated-orcid":false,"given":"Paolo","family":"Cazzaniga","sequence":"additional","affiliation":[{"name":"Department of Human and Social Sciences, University of Bergamo, 24129 Bergamo, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3383-367X","authenticated-orcid":false,"given":"Simone","family":"Spolaor","sequence":"additional","affiliation":[{"name":"Department of Informatics, Systems and Communication, University of Milano-Bicocca, 20126 Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3520-4022","authenticated-orcid":false,"given":"Giancarlo","family":"Mauri","sequence":"additional","affiliation":[{"name":"Department of Informatics, Systems and Communication, University of Milano-Bicocca, 20126 Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5532-3059","authenticated-orcid":false,"given":"Daniela","family":"Besozzi","sequence":"additional","affiliation":[{"name":"Department of Informatics, Systems and Communication, University of Milano-Bicocca, 20126 Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7692-7203","authenticated-orcid":false,"given":"Marco S.","family":"Nobile","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering &amp; Innovation Sciences, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1016\/j.compchemeng.2017.09.017","article-title":"Advances in surrogate based modeling, feasibility analysis, and optimization: A review","volume":"108","author":"Bhosekar","year":"2018","journal-title":"Comput. Chem. Eng."},{"key":"ref_2","unstructured":"Box, G.E., and Draper, N.R. (1987). Empirical Model-Building and Response Surfaces, John Wiley & Sons."},{"key":"ref_3","first-page":"409","article-title":"Design and analysis of computer experiments","volume":"4","author":"Sacks","year":"1989","journal-title":"Stat. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1023\/B:STCO.0000035301.49549.88","article-title":"A tutorial on support vector regression","volume":"14","author":"Smola","year":"2004","journal-title":"Stat. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1002\/aic.15362","article-title":"A novel feasibility analysis method for black-box processes using a radial basis function adaptive sampling approach","volume":"63","author":"Wang","year":"2017","journal-title":"AIChE J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.compchemeng.2014.05.021","article-title":"Adaptive sequential sampling for surrogate model generation with artificial neural networks","volume":"68","author":"Eason","year":"2014","journal-title":"Comput. Chem. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1016\/j.ymssp.2005.12.003","article-title":"Identification of response surface models using genetic programming","volume":"20","author":"Lew","year":"2006","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"301","DOI":"10.2514\/1.28999","article-title":"Multiple surrogate modeling for axial compressor blade shape optimization","volume":"24","author":"Samad","year":"2008","journal-title":"J. Propuls. Power"},{"key":"ref_9","first-page":"3251","article-title":"Multi-fidelity optimization via surrogate modelling","volume":"463","author":"Forrester","year":"2007","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1007\/s10898-012-9892-5","article-title":"Efficient global optimization algorithm assisted by multiple surrogate techniques","volume":"56","author":"Viana","year":"2013","journal-title":"J. Glob. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1109\/TSMCC.2005.855506","article-title":"Combining global and local surrogate models to accelerate evolutionary optimization","volume":"37","author":"Zhou","year":"2006","journal-title":"IEEE Trans. Syst. Man Cybern. Part C"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.paerosci.2008.11.001","article-title":"Recent advances in surrogate-based optimization","volume":"45","author":"Forrester","year":"2009","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.paerosci.2005.02.001","article-title":"Surrogate-based analysis and optimization","volume":"41","author":"Queipo","year":"2005","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1109\/TEVC.2013.2248012","article-title":"A Gaussian process surrogate model assisted evolutionary algorithm for medium scale expensive optimization problems","volume":"18","author":"Liu","year":"2013","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1177\/0954409715617213","article-title":"Optimization of the suspension parameters of a rail vehicle based on a virtual prototype Kriging surrogate model","volume":"230","author":"Yang","year":"2016","journal-title":"Proc. Inst. Mech. Eng. Part J. Rail Rapid Transit"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.swevo.2011.05.001","article-title":"Surrogate-assisted evolutionary computation: Recent advances and future challenges","volume":"1","author":"Jin","year":"2011","journal-title":"Swarm Evol. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1109\/TEVC.2017.2675628","article-title":"Surrogate-assisted cooperative swarm optimization of high-dimensional expensive problems","volume":"21","author":"Sun","year":"2017","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1080\/0305215X.2012.690759","article-title":"A surrogate-based particle swarm optimization algorithm for solving optimization problems with expensive black box functions","volume":"45","author":"Tang","year":"2013","journal-title":"Eng. Optim."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Branke, J. (1998). Creating robust solutions by means of evolutionary algorithms. International Conference on Parallel Problem Solving from Nature, Springer.","DOI":"10.1007\/BFb0056855"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Yu, X., Jin, Y., Tang, K., and Yao, X. (2010, January 18\u201323). Robust optimization over time\u2014A new perspective on dynamic optimization problems. Proceedings of the IEEE Congress on evolutionary computation, Barcelona, Spain.","DOI":"10.1109\/CEC.2010.5586024"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Bhattacharya, M. (2008, January 21\u201324). Reduced computation for evolutionary optimization in noisy environment. Proceedings of the 10th Annual Conference Companion on Genetic and Evolutionary Computation, New York, NY, USA.","DOI":"10.1145\/1388969.1389033"},{"key":"ref_22","unstructured":"Yang, D., and Flockton, S.J. (December, January 29). Evolutionary algorithms with a coarse-to-fine function smoothing. Proceedings of the 1995 IEEE International Conference on Evolutionary Computation, Perth, WA, Australia."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.swevo.2017.09.001","article-title":"Fuzzy Self-Tuning PSO: A settings-free algorithm for global optimization","volume":"39","author":"Nobile","year":"2018","journal-title":"Swarm Evol. Comp."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s11721-007-0002-0","article-title":"Particle swarm optimization","volume":"1","author":"Poli","year":"2007","journal-title":"Swarm Intell."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"105494","DOI":"10.1016\/j.asoc.2019.105494","article-title":"Biochemical parameter estimation vs. benchmark functions: A comparative study of optimization performance and representation design","volume":"81","author":"Tangherloni","year":"2019","journal-title":"Appl. Soft Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.pepi.2019.05.009","article-title":"An improved 1-D crustal velocity model for the Central Alborz (Iran) using Particle Swarm Optimization algorithm","volume":"292","author":"SoltaniMoghadam","year":"2019","journal-title":"Phys. Earth Planet. Inter."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Fuchs, C., Spolaor, S., Nobile, M.S., and Kaymak, U. (2019, January 23\u201326). A Swarm Intelligence approach to avoid local optima in fuzzy C-Means clustering. Proceedings of the 2019 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), New Orleans, LA, USA.","DOI":"10.1109\/FUZZ-IEEE.2019.8858940"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1090\/S0025-5718-1965-0178586-1","article-title":"An algorithm for the machine calculation of complex Fourier series","volume":"19","author":"Cooley","year":"1965","journal-title":"Math. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Nobile, M.S., Tangherloni, A., Besozzi, D., and Cazzaniga, P. (2016, January 24\u201329). GPU-powered and settings-free parameter estimation of biochemical systems. Proceedings of the 2016 IEEE Congress on Evolutionary Computation (CEC), Vancouver, BC, Canada.","DOI":"10.1109\/CEC.2016.7743775"},{"key":"ref_30","unstructured":"Oliphant, T.E. (2006). A Guide to NumPy, Trelgol Publishing."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., and Bright, J. (2019). SciPy 1.0\u2013Fundamental Algorithms for Scientific Computing in Python. arXiv.","DOI":"10.1038\/s41592-020-0772-5"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1145\/272991.272995","article-title":"Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator","volume":"8","author":"Matsumoto","year":"1998","journal-title":"ACM Trans. Model. Comput. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"606","DOI":"10.1038\/059606a0","article-title":"Fourier\u2019s series","volume":"59","author":"Gibbs","year":"1899","journal-title":"Nature"},{"key":"ref_34","unstructured":"Schwefel, H.P. (1981). Numerical Optimization of Computer Models, John Wiley & Sons."},{"key":"ref_35","first-page":"2005","article-title":"Problem definitions and evaluation criteria for the CEC 2005 special session on real-parameter optimization","volume":"2005005","author":"Suganthan","year":"2005","journal-title":"KanGAL Rep."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Nobile, M.S., Cazzaniga, P., and Ashlock, D.A. (2019, January 10\u201313). Dilation functions in global optimization. Proceedings of the 2019 IEEE Congress on Evolutionary Computation (CEC), Wellington, New Zealand.","DOI":"10.1109\/CEC.2019.8790247"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Nobile, M.S., Besozzi, D., Cazzaniga, P., Mauri, G., and Pescini, D. (2012). A GPU-based multi-swarm PSO method for parameter estimation in stochastic biological systems exploiting discrete-time target series. European Conference on Evolutionary Computation, Machine Learning and Data Mining in Bioinformatics, Springer.","DOI":"10.1007\/978-3-642-29066-4_7"},{"key":"ref_38","first-page":"784","article-title":"On the distribution of points in a cube and the approximate evaluation of integrals","volume":"7","year":"1967","journal-title":"Zhurnal Vychislitel\u2019Noi Mat. Mat. Fiz."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Manzoni, L., and Mariot, L. (2018). Cellular Automata pseudo-random number generators and their resistance to asynchrony. International Conference on Cellular Automata, Springer.","DOI":"10.1007\/978-3-319-99813-8_39"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1430","DOI":"10.1080\/01621459.1998.10473803","article-title":"Orthogonal column Latin hypercubes and their application in computer experiments","volume":"93","author":"Ye","year":"1998","journal-title":"J. Am. Stat. Assoc."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/285\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:02:45Z","timestamp":1760173365000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/3\/285"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,29]]},"references-count":40,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,3]]}},"alternative-id":["e22030285"],"URL":"https:\/\/doi.org\/10.3390\/e22030285","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,29]]}}}