{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:23:30Z","timestamp":1740108210389,"version":"3.37.3"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T00:00:00Z","timestamp":1687219200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T00:00:00Z","timestamp":1687219200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Fraunhofer-Institut f\u00fcr Techno- und Wirtschaftsmathematik ITWM"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Stat"],"published-print":{"date-parts":[[2024,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Modelling real processes often results in several suitable models. In order to be able to distinguish, or discriminate, which model best represents a phenomenon, one is interested, e.g., in so-called T-optimal designs. These consist of the (design) points from a generally continuous design space at which the models deviate most from each other under the condition that they are best fitted to those points. Thus, the T-criterion represents a bi-level optimization problem, which can be transferred into a semi-infinite one but whose solution is very unstable or time consuming for non-linear models and non-convex lower- and upper-level problems. If one considers only a finite number of possible design points, a numerically well tractable linear semi-infinite optimization problem arises. Since this is only an approximation of the original model discrimination problem, we propose an algorithm which alternately and adaptively refines discretizations of the parameter as well as of the design space and, thus, solves a sequence of linear semi-infinite programs. We prove convergence of our method and its subroutine and show on the basis of discrimination tasks from process engineering that our approach is stable and can outperform the known methods.<\/jats:p>","DOI":"10.1007\/s00180-023-01370-4","type":"journal-article","created":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T16:01:47Z","timestamp":1687276907000},"page":"2451-2478","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computing T-optimal designs via nested semi-infinite programming and twofold adaptive discretization"],"prefix":"10.1007","volume":"39","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8962-4666","authenticated-orcid":false,"given":"David","family":"Mogalle","sequence":"first","affiliation":[]},{"given":"Philipp","family":"Seufert","sequence":"additional","affiliation":[]},{"given":"Jan","family":"Schwientek","sequence":"additional","affiliation":[]},{"given":"Michael","family":"Bortz","sequence":"additional","affiliation":[]},{"given":"Karl-Heinz","family":"K\u00fcfer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,20]]},"reference":[{"issue":"1","key":"1370_CR1","doi-asserted-by":"publisher","first-page":"57","DOI":"10.2307\/2334487","volume":"62","author":"AC Atkinson","year":"1975","unstructured":"Atkinson AC, Fedorov VV (1975a) The design of experiments for discriminating between two rival models. Biometrika 62(1):57\u201370. https:\/\/doi.org\/10.2307\/2334487","journal-title":"Biometrika"},{"issue":"2","key":"1370_CR2","doi-asserted-by":"publisher","first-page":"289","DOI":"10.2307\/2335364","volume":"62","author":"AC Atkinson","year":"1975","unstructured":"Atkinson AC, Fedorov VV (1975b) Optimal design: experiments for discriminating between several models. Biometrika 62(2):289\u2013303. https:\/\/doi.org\/10.2307\/2335364","journal-title":"Biometrika"},{"key":"1370_CR3","unstructured":"Bishop CM (2016) Pattern recognition and machine learning, reprint of the original, 1st edn (2006). Information Science and Statistics. Springer, New York"},{"issue":"2","key":"1370_CR4","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/BF00934096","volume":"19","author":"JW Blankenship","year":"1976","unstructured":"Blankenship JW, Falk JE (1976) Infinitely constrained optimization problems. J Optim Theory Appl 19(2):261\u2013281. https:\/\/doi.org\/10.1007\/BF00934096","journal-title":"J Optim Theory Appl"},{"issue":"1","key":"1370_CR5","doi-asserted-by":"publisher","first-page":"57","DOI":"10.2307\/1266318","volume":"9","author":"GEP Box","year":"1967","unstructured":"Box GEP, Hill WJ (1967) Discrimination among mechanistic models. Technometrics 9(1):57\u201371. https:\/\/doi.org\/10.2307\/1266318","journal-title":"Technometrics"},{"key":"1370_CR6","doi-asserted-by":"publisher","DOI":"10.1214\/13-AOS1103","author":"D Braess","year":"2013","unstructured":"Braess D, Dette H (2013) Optimal discriminating designs for several competing regression models. Ann Stat. https:\/\/doi.org\/10.1214\/13-AOS1103","journal-title":"Ann Stat"},{"key":"1370_CR7","doi-asserted-by":"crossref","unstructured":"Byrd RH, Nocedal J, Waltz RA (2006) Knitro: an integrated package for nonlinear optimization. In: Large-scale nonlinear optimization. Springer, p 35\u201359","DOI":"10.1007\/0-387-30065-1_4"},{"key":"1370_CR8","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0239864","author":"RB Chen","year":"2020","unstructured":"Chen RB, Chen PY, Hsu CL et al (2020) Hybrid algorithms for generating optimal designs for discriminating multiple nonlinear models under various error distributional assumptions. PLoS ONE. https:\/\/doi.org\/10.1371\/journal.pone.0239864","journal-title":"PLoS ONE"},{"key":"1370_CR9","doi-asserted-by":"publisher","first-page":"4","DOI":"10.1214\/08-AOS635","volume":"37","author":"H Dette","year":"2009","unstructured":"Dette H, Titoff S (2009) Optimal discrimination designs. Ann Stat 37:4. https:\/\/doi.org\/10.1214\/08-AOS635","journal-title":"Ann Stat"},{"key":"1370_CR10","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOS956","author":"H Dette","year":"2012","unstructured":"Dette H, Melas VB, Shpilev P (2012) T-optimal designs for discrimination between two polynomial models. Ann Stat. https:\/\/doi.org\/10.1214\/11-AOS956","journal-title":"Ann Stat"},{"issue":"5","key":"1370_CR11","doi-asserted-by":"publisher","first-page":"1959","DOI":"10.1214\/15-AOS1333","volume":"43","author":"H Dette","year":"2015","unstructured":"Dette H, Melas VB, Guchenko R (2015) Bayesian T-optimal discriminating designs. Ann Stat 43(5):1959\u20131985. https:\/\/doi.org\/10.1214\/15-AOS1333","journal-title":"Ann Stat"},{"issue":"1","key":"1370_CR12","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1093\/biomet\/asx058","volume":"105","author":"H Dette","year":"2018","unstructured":"Dette H, Guchenko R, Melas VB et al (2018) Optimal discrimination designs for semiparametric models. Biometrika 105(1):185\u2013197. https:\/\/doi.org\/10.1093\/biomet\/asx058","journal-title":"Biometrika"},{"key":"1370_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejco.2021.100006","volume":"9","author":"H Djelassi","year":"2021","unstructured":"Djelassi H, Mitsos A, Stein O (2021) Recent advances in nonconvex semi-infinite programming: applications and algorithms. EURO J Comput Optim 9:100006. https:\/\/doi.org\/10.1016\/j.ejco.2021.100006","journal-title":"EURO J Comput Optim"},{"key":"1370_CR14","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.jmva.2014.11.006","volume":"135","author":"BP Duarte","year":"2015","unstructured":"Duarte BP, Wong WK, Atkinson AC (2015) A semi-infinite programming based algorithm for determining T-optimum designs for model discrimination. J Multivar Anal 135:11\u201324. https:\/\/doi.org\/10.1016\/j.jmva.2014.11.006","journal-title":"J Multivar Anal"},{"key":"1370_CR15","volume-title":"Optimal design for nonlinear response models","author":"VV Fedorov","year":"2014","unstructured":"Fedorov VV, Leonov SL (2014) Optimal design for nonlinear response models. CRC Press, Chapman & Hall\/CRC biostatistics series, Boca Raton"},{"issue":"2","key":"1370_CR16","doi-asserted-by":"publisher","first-page":"122","DOI":"10.3103\/S1063454117020054","volume":"50","author":"RA Guchenko","year":"2017","unstructured":"Guchenko RA, Melas VB (2017) T-optimal designs for discrimination between rational and polynomial models. Vestnik St Petersburg Univ Math 50(2):122\u2013131. https:\/\/doi.org\/10.3103\/S1063454117020054","journal-title":"Vestnik St Petersburg Univ Math"},{"issue":"3","key":"1370_CR17","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1080\/00401706.1965.10490265","volume":"7","author":"WG Hunter","year":"1965","unstructured":"Hunter WG, Reiner AM (1965) Designs for discriminating between two rival models. Technometrics 7(3):307\u2013323","journal-title":"Technometrics"},{"key":"1370_CR18","unstructured":"Kuczewski B (2006) Computational aspects of discrimination between models of dynamic systems. Ph.D. thesis, University of Zielona Gora. https:\/\/zbc.uz.zgora.pl\/dlibra\/publication\/11162\/edition\/10408?language=en#description"},{"issue":"2","key":"1370_CR19","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1016\/j.ejor.2006.08.045","volume":"180","author":"M L\u00f3pez","year":"2007","unstructured":"L\u00f3pez M, Still G (2007) Semi-infinite programming. Eur J Oper Res 180(2):491\u2013518. https:\/\/doi.org\/10.1016\/j.ejor.2006.08.045","journal-title":"Eur J Oper Res"},{"issue":"2","key":"1370_CR20","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1111\/j.1467-9868.2007.00586.x","volume":"69","author":"J L\u00f3pez-Fidalgo","year":"2007","unstructured":"L\u00f3pez-Fidalgo J, Tommasi C, Trandafir PC (2007) An optimal experimental design criterion for discriminating between non-normal models. J Roy Stat Soc Ser B (Stat Methodol) 69(2):231\u2013242. https:\/\/doi.org\/10.1111\/j.1467-9868.2007.00586.x","journal-title":"J Roy Stat Soc Ser B (Stat Methodol)"},{"key":"1370_CR21","unstructured":"MOSEK ApS (2021) MOSEK Optimizer API for Python 9.2.49. https:\/\/docs.mosek.com\/latest\/pythonapi\/index.html"},{"key":"1370_CR22","volume-title":"Probability measures on metric spaces","author":"KR Parthasarathy","year":"2005","unstructured":"Parthasarathy KR (2005) Probability measures on metric spaces. AMS Chelsea Publishing, Providence"},{"key":"1370_CR23","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/BF00138693","volume":"8","author":"NV Sahinidis","year":"1996","unstructured":"Sahinidis NV (1996) BARON: a general purpose global optimization software package. J Glob Optim 8:201\u2013205","journal-title":"J Glob Optim"},{"issue":"2","key":"1370_CR24","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1080\/02331930902730070","volume":"58","author":"A Shapiro","year":"2009","unstructured":"Shapiro A (2009) Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58(2):133\u2013161. https:\/\/doi.org\/10.1080\/02331930902730070","journal-title":"Optimization"},{"issue":"4","key":"1370_CR25","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/0041-5553(67)90144-9","volume":"7","author":"IM Sobol","year":"1967","unstructured":"Sobol IM (1967) On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys 7(4):86\u2013112. https:\/\/doi.org\/10.1016\/0041-5553(67)90144-9","journal-title":"USSR Comput Math Math Phys"},{"issue":"20","key":"1370_CR26","doi-asserted-by":"publisher","first-page":"6086","DOI":"10.1111\/febs.16124","volume":"289","author":"B Srinivasan","year":"2022","unstructured":"Srinivasan B (2022) A guide to the Michaelis\u2013Menten equation: steady state and beyond. FEBS J 289(20):6086\u20136098. https:\/\/doi.org\/10.1111\/febs.16124","journal-title":"FEBS J"},{"issue":"1","key":"1370_CR27","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1111\/j.1467-9868.2005.00485.x","volume":"67","author":"D Ucinski","year":"2005","unstructured":"Ucinski D, Bogacka B (2005) T-optimum designs for discrimination between two multiresponse dynamic models. J Roy Stat Soc Ser B (Stat Methodol) 67(1):3\u201318. https:\/\/doi.org\/10.1111\/j.1467-9868.2005.00485.x","journal-title":"J Roy Stat Soc Ser B (Stat Methodol)"},{"key":"1370_CR28","volume-title":"Python 3 reference manual","author":"G Van Rossum","year":"2009","unstructured":"Van Rossum G, Drake FL (2009) Python 3 reference manual. CreateSpace, Scotts Valley"},{"key":"1370_CR29","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1038\/s41592-019-0686-2","volume":"17","author":"P Virtanen","year":"2020","unstructured":"Virtanen P, Gommers R, Oliphant TE et al (2020) SciPy 1.0: fundamental algorithms for scientific computing in python. Nat Methods 17:261\u2013272. https:\/\/doi.org\/10.1038\/s41592-019-0686-2","journal-title":"Nat Methods"},{"issue":"5","key":"1370_CR30","doi-asserted-by":"publisher","first-page":"1655","DOI":"10.1214\/aoms\/1177696809","volume":"41","author":"HP Wynn","year":"1970","unstructured":"Wynn HP (1970) The sequential generation of D-optimum experimental designs. Ann Math Stat 41(5):1655\u20131664. https:\/\/doi.org\/10.1214\/aoms\/1177696809","journal-title":"Ann Math Stat"},{"issue":"4","key":"1370_CR31","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1007\/s11222-018-9834-2","volume":"29","author":"Y Yue","year":"2019","unstructured":"Yue Y, Vandenberghe L, Wong WK (2019) T-optimal designs for multi-factor polynomial regression models via a semidefinite relaxation method. Stat Comput 29(4):725\u2013738. https:\/\/doi.org\/10.1007\/s11222-018-9834-2","journal-title":"Stat Comput"}],"container-title":["Computational Statistics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-023-01370-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00180-023-01370-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00180-023-01370-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,14]],"date-time":"2024-06-14T15:02:49Z","timestamp":1718377369000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00180-023-01370-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,20]]},"references-count":31,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,7]]}},"alternative-id":["1370"],"URL":"https:\/\/doi.org\/10.1007\/s00180-023-01370-4","relation":{},"ISSN":["0943-4062","1613-9658"],"issn-type":[{"type":"print","value":"0943-4062"},{"type":"electronic","value":"1613-9658"}],"subject":[],"published":{"date-parts":[[2023,6,20]]},"assertion":[{"value":"27 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 May 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 June 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}