{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T01:54:51Z","timestamp":1778896491048,"version":"3.51.4"},"reference-count":23,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2018,10,1]],"date-time":"2018-10-01T00:00:00Z","timestamp":1538352000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100005989","name":"Ministrstvo za Izobra\u017eevanje, Znanost in \u0160port","doi-asserted-by":"publisher","award":["P2-0246"],"award-info":[{"award-number":["P2-0246"]}],"id":[{"id":"10.13039\/501100005989","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The problem of the creation of numerical constants has haunted the Genetic Programming (GP) community for a long time and is still considered one of the principal open research issues. Many problems tackled by GP include finding mathematical formulas, which often contain numerical constants. It is, however, a great challenge for GP to create highly accurate constants as their values are normally continuous, while GP is intrinsically suited for combinatorial optimization. The prevailing attempts to resolve this issue either employ separate real-valued local optimizers or special numeric mutations. While the former yield better accuracy than the latter, they add to implementation complexity and significantly increase computational cost. In this paper, we propose a special numeric crossover operator for use with Robust Gene Expression Programming (RGEP). RGEP is a type of genotype\/phenotype evolutionary algorithm closely related to GP, but employing linear chromosomes. Using normalized least squares error as a fitness measure, we show that the proposed operator is significantly better in finding highly accurate solutions than the existing numeric mutation operators on several symbolic regression problems. Another two important advantages of the proposed operator are that it is extremely simple to implement, and it comes at no additional computational cost. The latter is true because the operator is integrated into an existing crossover operator and does not call for an additional cost function evaluation.<\/jats:p>","DOI":"10.3390\/e20100756","type":"journal-article","created":{"date-parts":[[2018,10,2]],"date-time":"2018-10-02T11:30:02Z","timestamp":1538479802000},"page":"756","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Creation of Numerical Constants in Robust Gene Expression Programming"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4416-5432","authenticated-orcid":false,"given":"Iztok","family":"Fajfar","sequence":"first","affiliation":[{"name":"University of Ljubljana, Faculty of Electrical Engineering, Tr\u017ea\u0161ka 25, 1000 Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6030-0820","authenticated-orcid":false,"given":"Tadej","family":"Tuma","sequence":"additional","affiliation":[{"name":"University of Ljubljana, Faculty of Electrical Engineering, Tr\u017ea\u0161ka 25, 1000 Ljubljana, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,10,1]]},"reference":[{"key":"ref_1","first-page":"87","article-title":"Gene expression programming: A new adaptive algorithm for solving Problems","volume":"13","author":"Ferreira","year":"2001","journal-title":"Complex Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1109\/MCI.2017.2708618","article-title":"Gene expression programming: A survey","volume":"12","author":"Zhong","year":"2017","journal-title":"IEEE Comp. Int. Mag."},{"key":"ref_3","unstructured":"Koza, J.R. (1992). Genetic Programming: On the Programming of Computers by means of Natural Selection, MIT Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ben\u00edtez, J.M., Cord\u00f3n, O., Hoffmann, F., and Roy, R. (2003). Function finding and the creation of numerical constants in gene expression programming. Advances in Soft Computing, Springer.","DOI":"10.1007\/978-1-4471-3744-3"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/j.neucom.2013.05.062","article-title":"An improved gene expression programming approach for symbolic regression problems","volume":"137","author":"Peng","year":"2014","journal-title":"Neurocomputing"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1007\/3-540-36599-0_38","article-title":"An analysis of diversity of constants of genetic programming","volume":"Volume 2610","author":"Ryan","year":"2003","journal-title":"Proceedings of the 6th European Conference on Genetic Programming"},{"key":"ref_7","unstructured":"Evett, M., and Fernandez, T. (1998, January 22\u201325). Numeric mutation improves the discovery of numeric constants in genetic programming. Proceedings of the Third Annual Conference on Genetic Programming 1998, San Francisco, CA, USA."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/TEVC.2015.2424410","article-title":"Self-Learning gene expression programming","volume":"20","author":"Zhong","year":"2016","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_9","unstructured":"Li, X., Zhou, C., Nelson, P.C., and Tirpak, T.M. (2004, January 26\u201330). Investigation of constant creation techniques in the context of gene expression programming. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2004), Seattle, WA, USA."},{"key":"ref_10","unstructured":"Zhang, Q., Zhou, C., Xiao, W., Nelson, P.C., and Li, X. (2006, January 8\u201312). Using differential evolution for GEP constant creation. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2006), Seattle, WA, USA."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Zhang, Q., Zhou, C., Xiao, W., and Nelson, P.C. (2007, January 13\u201315). Improving gene expression programming performance by using differential evolution. Proceedings of the Sixth International Conference on Machine Learning and Applications, Cincinnati, OH, USA.","DOI":"10.1109\/ICMLA.2007.62"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Cerny, B.M., Nelson, P.C., and Zhou, C. (2008, January 12\u201316). Using differential evolution for symbolic regression and numerical constant creation. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2008), Seattle, WA, USA.","DOI":"10.1145\/1389095.1389331"},{"key":"ref_13","unstructured":"Li, T., Dong, T., Wu, J., and He, T. (2009, January 8\u201311). Function mining based on gene Expression Programming and Particle Swarm Optimization. Proceedings of the 2009 2nd IEEE International Conference on Computer Science and Information Technology, Beijing, China."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1016\/j.ijrmms.2016.07.028","article-title":"Genetic programming and gene expression programming for flyrock assessment due to mine blasting","volume":"88","author":"Faradonbeh","year":"2016","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s00366-017-0526-x","article-title":"Performance Prediction of Tunnel Boring Machine Through Developing a Gene Expression Programming Equation","volume":"34","author":"Faradonbeh","year":"2018","journal-title":"Eng. Comput."},{"key":"ref_16","first-page":"23","article-title":"Constant Creation in Grammatical Evolution","volume":"1","author":"Dempsey","year":"2007","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.procs.2011.08.032","article-title":"Robust Gene Expression Programming","volume":"6","author":"Ryan","year":"2011","journal-title":"Procedia Comput. Sci."},{"key":"ref_18","unstructured":"Li, X., Zhou, C., Xiao, W., and Nelson, P.C. (2005, January 25\u201329). Prefix gene expression programming. Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2005), Washington, DC, USA."},{"key":"ref_19","first-page":"375","article-title":"EGIPSYS: An Enhanced Gene Expression Programming Approach for Symbolic Regression Problems","volume":"14","author":"Lopes","year":"2004","journal-title":"Int. J. Rock Mech. Min. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Rothlauf, F., and Oetzel, M. (2006, January 10\u201312). On the Locality of Grammatical Evolution. Proceedings of the 9th European Conference on Genetic Programming, Budapest, Hungary.","DOI":"10.1007\/11729976_29"},{"key":"ref_21","first-page":"182","article-title":"Positional Effect of Crossover and Mutation in grammatical evolution","volume":"Volume 6021","author":"Castle","year":"2003","journal-title":"Proceedings of the 13rd European Conference on Genetic Programming (EuroGP 2010)"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Riolo, R., Vladislavleva, E., and Moore, J.H. (2011). Accuracy in Symbolic Regression. Genetic Programming Theory and Practice IX, Springer.","DOI":"10.1007\/978-1-4614-1770-5"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Keijzer, M. (2003, January 14\u201316). Improving Symbolic Regression with Interval Arithmetic and Linear Scaling. Proceedings of the 6th European Conference on Genetic Programming, Essex, UK.","DOI":"10.1007\/3-540-36599-0_7"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/10\/756\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:23:39Z","timestamp":1760196219000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/10\/756"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,1]]},"references-count":23,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2018,10]]}},"alternative-id":["e20100756"],"URL":"https:\/\/doi.org\/10.3390\/e20100756","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,1]]}}}