{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,16]],"date-time":"2025-10-16T19:49:56Z","timestamp":1760644196008,"version":"3.37.3"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Glob Optim"],"published-print":{"date-parts":[[2024,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider global optimization problems, where the feasible region <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {X}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a compact subset of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$d \\ge 10$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>10<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For these problems, we demonstrate that the actual convergence of global random search algorithms is much slower than that given by the classical estimates, based on the asymptotic properties of random points, and that the usually recommended space exploration schemes are inefficient in the non-asymptotic regime. Moreover, we show that uniform sampling on entire\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {X}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is much less efficient than uniform sampling on a suitable subset of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {X}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>X<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and that the effect of replacement of random points by low-discrepancy sequences can be felt in small dimensions only.<\/jats:p>","DOI":"10.1007\/s10898-023-01308-6","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T03:10:47Z","timestamp":1690168247000},"page":"1-26","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Improving exploration strategies in large dimensions and rate of convergence of global random search algorithms"],"prefix":"10.1007","volume":"88","author":[{"given":"Jack","family":"Noonan","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0630-8279","authenticated-orcid":false,"given":"Anatoly","family":"Zhigljavsky","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,24]]},"reference":[{"key":"1308_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-84808-2","volume-title":"Discrete Energy on Rectifiable Sets","author":"S Borodachov","year":"2019","unstructured":"Borodachov, S., Hardin, D., Saff, E.: Discrete Energy on Rectifiable Sets. Springer, London (2019)"},{"issue":"4","key":"1308_CR2","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1137\/S0036144599352836","volume":"41","author":"Q Du","year":"1999","unstructured":"Du, Q., Faber, V., Gunzburger, M.: Centroidal voronoi tessellations: applications and algorithms. SIAM Rev. 41(4), 637\u2013676 (1999)","journal-title":"SIAM Rev."},{"key":"1308_CR3","volume-title":"Foundations of Quantization for Probability Distributions","author":"S Graf","year":"2007","unstructured":"Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Springer, London (2007)"},{"key":"1308_CR4","unstructured":"Grimmett, G., Stirzaker, D.: Probability and random processes. Oxford University Press (2020)"},{"issue":"1","key":"1308_CR5","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/BF02399201","volume":"156","author":"S Janson","year":"1986","unstructured":"Janson, S.: Random coverings in several dimensions. Acta Math. 156(1), 83\u2013118 (1986)","journal-title":"Acta Math."},{"key":"1308_CR6","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970081","volume-title":"Random Number Generation and Quasi-Monte Carlo Methods","author":"H Niederreiter","year":"1992","unstructured":"Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia, PA (1992)"},{"issue":"3","key":"1308_CR7","first-page":"1","volume":"1","author":"J Noonan","year":"2020","unstructured":"Noonan, J., Zhigljavsky, A.: Covering of high-dimensional cubes and quantization. SN Operat. Res. Forum 1(3), 1\u201332 (2020)","journal-title":"SN Operat. Res. Forum"},{"key":"1308_CR8","doi-asserted-by":"crossref","unstructured":"Noonan, J., Zhigljavsky, A.: Non-lattice covering and quantization of high dimensional sets. In Black Box Optimization, Machine Learning, and No-Free Lunch Theorems, pp. 273\u2013318. Springer: London (2021)","DOI":"10.1007\/978-3-030-66515-9_10"},{"key":"1308_CR9","doi-asserted-by":"crossref","unstructured":"Noonan, J., Zhigljavsky, A.: Efficient quantisation and weak covering of high dimensional cubes. Discrete & Computational Geometry, pp. 1\u201326, (2022)","DOI":"10.1007\/s00454-022-00396-7"},{"issue":"1","key":"1308_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0377-0427(97)00190-8","volume":"89","author":"G Pag\u00e8s","year":"1998","unstructured":"Pag\u00e8s, G.: A space quantization method for numerical integration. J. Computat. Appl. Math. 89(1), 1\u201338 (1998)","journal-title":"J. Computat. Appl. Math."},{"issue":"3\u20134","key":"1308_CR11","doi-asserted-by":"publisher","first-page":"747","DOI":"10.1007\/s00440-022-01182-5","volume":"185","author":"M Penrose","year":"2023","unstructured":"Penrose, M.: Random Euclidean coverage from within. Probabil. Theory Relat. Fields 185(3\u20134), 747\u2013814 (2023)","journal-title":"Probabil. Theory Relat. Fields"},{"issue":"1","key":"1308_CR12","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s10898-017-0535-8","volume":"71","author":"A Pepelyshev","year":"2018","unstructured":"Pepelyshev, A., Zhigljavsky, A., \u017dilinskas, A.: Performance of global random search algorithms for large dimensions. J. Global Optimiz. 71(1), 57\u201371 (2018)","journal-title":"J. Global Optimiz."},{"key":"1308_CR13","doi-asserted-by":"crossref","unstructured":"Petrov, V.: Sums of independent random variables. Springer-Verlag (1975)","DOI":"10.1515\/9783112573006"},{"key":"1308_CR14","doi-asserted-by":"crossref","unstructured":"Petrov, V.: Limit theorems of probability theory: sequences of independent random variables. Oxford Science Publications (1995)","DOI":"10.1093\/oso\/9780198534990.003.0002"},{"issue":"3","key":"1308_CR15","first-page":"405","volume":"15","author":"J Pint\u00e9r","year":"1984","unstructured":"Pint\u00e9r, J.: Convergence properties of stochastic optimization procedures. Optimization 15(3), 405\u2013427 (1984)","journal-title":"Optimization"},{"issue":"3","key":"1308_CR16","doi-asserted-by":"publisher","first-page":"959","DOI":"10.1137\/18M1210332","volume":"8","author":"L Pronzato","year":"2020","unstructured":"Pronzato, L., Zhigljavsky, A.: Bayesian quadrature, energy minimization, and space-filling design. SIAM\/ASA J. Uncert. Quantif. 8(3), 959\u20131011 (2020)","journal-title":"SIAM\/ASA J. Uncert. Quantif."},{"key":"1308_CR17","doi-asserted-by":"crossref","unstructured":"Santner, T., Williams, B., Notz, W., Williams, B.: The design and analysis of computer experiments, vol. 1. Springer (2003)","DOI":"10.1007\/978-1-4757-3799-8_1"},{"key":"1308_CR18","doi-asserted-by":"publisher","first-page":"543","DOI":"10.1017\/S0962492906270016","volume":"15","author":"R Schaback","year":"2006","unstructured":"Schaback, R., Wendland, H.: Kernel techniques: from machine learning to meshless methods. Acta Num. 15, 543\u2013639 (2006)","journal-title":"Acta Num."},{"issue":"1","key":"1308_CR19","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1287\/moor.6.1.19","volume":"6","author":"F Solis","year":"1981","unstructured":"Solis, F., Wets, R.: Minimization by random search techniques. Math. Operat. Res. 6(1), 19\u201330 (1981)","journal-title":"Math. Operat. Res."},{"issue":"4","key":"1308_CR20","first-page":"910","volume":"11","author":"A Sukharev","year":"1971","unstructured":"Sukharev, A.: Optimal strategies of search for an extremum. USSR Computat. Math. Math. Phys. 11(4), 910\u2013924 (1971)","journal-title":"USSR Computat. Math. Math. Phys."},{"key":"1308_CR21","doi-asserted-by":"crossref","unstructured":"Sukharev, A.: Minimax models in the theory of numerical methods. Springer (1992)","DOI":"10.1007\/978-94-011-2759-2"},{"issue":"2","key":"1308_CR22","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s10957-022-02097-w","volume":"195","author":"E Tsvetkov","year":"2022","unstructured":"Tsvetkov, E., Krymov, R.: Pure random search with virtual extension of feasible region. J. Optimizat. Theory Appl. 195(2), 575\u2013595 (2022)","journal-title":"J. Optimizat. Theory Appl."},{"key":"1308_CR23","doi-asserted-by":"crossref","unstructured":"Wendland, H.: Scattered data approximation, vol. 17. Cambridge University Press, UK (2004)","DOI":"10.1017\/CBO9780511617539"},{"issue":"2","key":"1308_CR24","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1109\/TIT.1982.1056490","volume":"28","author":"P Zador","year":"1982","unstructured":"Zador, P.: Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28(2), 139\u2013149 (1982)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"1308_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-3436-1","volume-title":"Theory of Global Random Search","author":"A Zhigljavsky","year":"1991","unstructured":"Zhigljavsky, A.: Theory of Global Random Search. Kluwer, Dordrecht (1991)"},{"key":"1308_CR26","doi-asserted-by":"publisher","unstructured":"Zhigljavsky, A., Noonan, J.: Covering of High-Dimensional Sets, pp 1\u20136. Springer: Cham, (2023). https:\/\/doi.org\/10.1007\/978-3-030-54621-2_770-1","DOI":"10.1007\/978-3-030-54621-2_770-1"},{"key":"1308_CR27","unstructured":"Zhigljavsky, A., \u017dilinskas, A.: Stochastic Global Optimization. Springer (2008)"},{"key":"1308_CR28","doi-asserted-by":"crossref","unstructured":"Zhigljavsky, A., \u017dilinskas, A.: Bayesian and High-Dimensional Global Optimization. Springer (2021)","DOI":"10.1007\/978-3-030-64712-4"},{"issue":"8","key":"1308_CR29","doi-asserted-by":"publisher","first-page":"1921","DOI":"10.1007\/s11590-012-0547-8","volume":"7","author":"A \u017dilinskas","year":"2013","unstructured":"\u017dilinskas, A.: On the worst-case optimal multi-objective global optimization. Optimizat. Lett. 7(8), 1921\u20131928 (2013)","journal-title":"Optimizat. Lett."}],"container-title":["Journal of Global Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-023-01308-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10898-023-01308-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10898-023-01308-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,10]],"date-time":"2024-01-10T10:04:20Z","timestamp":1704881060000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10898-023-01308-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,24]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2024,1]]}},"alternative-id":["1308"],"URL":"https:\/\/doi.org\/10.1007\/s10898-023-01308-6","relation":{},"ISSN":["0925-5001","1573-2916"],"issn-type":[{"type":"print","value":"0925-5001"},{"type":"electronic","value":"1573-2916"}],"subject":[],"published":{"date-parts":[[2023,7,24]]},"assertion":[{"value":"26 August 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 June 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 July 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}