{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:50:09Z","timestamp":1760143809018,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T00:00:00Z","timestamp":1709251200000},"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>We consider the Golden Section and Parabola Methods for solving univariate optimization problems. For multivariate problems, we use these methods as line search procedures in combination with well-known zero-order methods such as the coordinate descent method, the Hooke and Jeeves method, and the Rosenbrock method. A comprehensive numerical comparison of the obtained versions of zero-order methods is given in the present work. The set of test problems includes nonconvex functions with a large number of local and global optimum points. Zero-order methods combined with the Parabola method demonstrate high performance and quite frequently find the global optimum even for large problems (up to 100 variables).<\/jats:p>","DOI":"10.3390\/a17030107","type":"journal-article","created":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T03:31:23Z","timestamp":1709263883000},"page":"107","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Application of the Parabola Method in Nonconvex Optimization"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3964-1940","authenticated-orcid":false,"given":"Anton","family":"Kolosnitsyn","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Melentiev Energy Systems Institute, Lermontov St. 130, 664033 Irkutsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8021-0205","authenticated-orcid":false,"given":"Oleg","family":"Khamisov","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Melentiev Energy Systems Institute, Lermontov St. 130, 664033 Irkutsk, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3776-5707","authenticated-orcid":false,"given":"Eugene","family":"Semenkin","sequence":"additional","affiliation":[{"name":"Scientific and Educational Center \u201cArtificial Intellegence Technologies\u201d, Bauman Moscow State Technical University, 2nd Baumanskaya, Str. 5, 105005 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4263-2367","authenticated-orcid":false,"given":"Vladimir","family":"Nelyub","sequence":"additional","affiliation":[{"name":"Scientific and Educational Center \u201cArtificial Intellegence Technologies\u201d, Bauman Moscow State Technical University, 2nd Baumanskaya, Str. 5, 105005 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,1]]},"reference":[{"unstructured":"Pinter, J. (1995). Global Optimization in Action, Kluwer Academic Publishers.","key":"ref_1"},{"doi-asserted-by":"crossref","unstructured":"Strongin, R.G., and Sergeev, Y.D. (2000). Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms, Springer.","key":"ref_2","DOI":"10.1007\/978-1-4615-4677-1"},{"key":"ref_3","first-page":"91","article-title":"Estimation of the Lipschitz constant of a function","volume":"8","author":"Wood","year":"1998","journal-title":"J. Glob. Optim."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1023\/A:1008791528195","article-title":"Evaluating Lipschitz Constants for Functions Given by Algorithms","volume":"16","author":"Oliveira","year":"2000","journal-title":"Comput. Optim. Appl."},{"doi-asserted-by":"crossref","unstructured":"Tuy, H. (1998). Convex Analysis and Global Optimization, Kluwer Academic Publishers.","key":"ref_5","DOI":"10.1007\/978-1-4757-2809-5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1023\/A:1008343420713","article-title":"A Method for Converting a Class of Univariate Functions into d.c. Functions","volume":"15","author":"Lamar","year":"1999","journal-title":"J. Glob. Optim."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1007\/s10898-012-9964-6","article-title":"Characterization and recognition of d.c. functions","volume":"57","author":"Ginchev","year":"2013","journal-title":"J. Glob. Optim."},{"doi-asserted-by":"crossref","unstructured":"Horst, R., and Tuy, H. (1996). Global Optimization: Deterministic Approaches, Springer.","key":"ref_8","DOI":"10.1007\/978-3-662-03199-5"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1023\/A:1011287523150","article-title":"Convexity of Nonlinear Image of a Small Ball with Application to Optimization","volume":"9","author":"Polyak","year":"2001","journal-title":"Set-Valued Anal."},{"doi-asserted-by":"crossref","unstructured":"Conn, A., Scheinberg, K., and Vicente, L. (2009). Introduction to Derivative-Free Optimization, Society for Industrial and Applied Mathematics.","key":"ref_10","DOI":"10.1137\/1.9780898718768"},{"doi-asserted-by":"crossref","unstructured":"Bazaraa, M.S., Sherali, H.D., and Shetty, C.M. (2006). Nonlinear Programming: Theory and Algorithms, John Wiley & Sons. [3rd ed.].","key":"ref_11","DOI":"10.1002\/0471787779"},{"doi-asserted-by":"crossref","unstructured":"Eiselt, H.A., and Sandblom, C.-L. (2010). Operations Research: A Model-Based Approach, Springer.","key":"ref_12","DOI":"10.1007\/978-3-642-10326-1"},{"doi-asserted-by":"crossref","unstructured":"Grouzeix, J.-P., and Martinez-Legaz, J.-E. (1998). Generalized Convexity, Generalized Monotonicity: Recent Results, Kluwer Academic Publishers.","key":"ref_13","DOI":"10.1007\/978-1-4613-3341-8"},{"doi-asserted-by":"crossref","unstructured":"Horst, R., and Pardalos, P.M. (1994). Handbook on Global Optimization, Kluwer Academic Publishers.","key":"ref_14","DOI":"10.1007\/978-1-4615-2025-2"},{"doi-asserted-by":"crossref","unstructured":"Walter, \u00c9. (2014). Numerical Methods and Optimization, Springer.","key":"ref_15","DOI":"10.1007\/978-3-319-07671-3"},{"doi-asserted-by":"crossref","unstructured":"Snyman, J.A., and Wilke, D.N. (2018). Practical Mathematical Optimization, Springer. [2nd ed.].","key":"ref_16","DOI":"10.1007\/978-3-319-77586-9"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1247","DOI":"10.1007\/s10898-012-9951-y","article-title":"Derivative-free optimization: A review of algorithms and comparison of software implementations","volume":"56","author":"Rios","year":"2012","journal-title":"J. Glob. Optim."},{"key":"ref_18","first-page":"61","article-title":"Derivative-Free Optimization","volume":"Volume 356","author":"Koziel","year":"2011","journal-title":"Computational Optimization, Methods and Algorithms. Studies in Computational Intelligence"},{"unstructured":"Minoux, M. (1986). Mathematical Programming: Theory and Algorithms, John Wiley & Sons Ltd.","key":"ref_19"}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/3\/107\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:07:46Z","timestamp":1760105266000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/17\/3\/107"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,1]]},"references-count":19,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["a17030107"],"URL":"https:\/\/doi.org\/10.3390\/a17030107","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2024,3,1]]}}}