{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T08:57:49Z","timestamp":1772182669934,"version":"3.50.1"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T00:00:00Z","timestamp":1751673600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,5]],"date-time":"2025-07-05T00:00:00Z","timestamp":1751673600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Instituto Polit\u00e9cnico de Coimbra"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Eng"],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>An optimal experimental design represents a structured approach to collecting data with the aim of maximizing the information gleaned. Achieving this requires defining an optimality criterion tailored to the specific model under consideration and the purpose of the investigation. However, it is often observed that a design optimized for one criterion may not perform optimally when applied to another. To mitigate this, one strategy involves employing compound designs. These designs balance multiple criteria to create robust experimental plans that are versatile across different applications. In our study, we systematically tackle the challenge of constructing compound approximate optimal experimental designs using Semidefinite Programming. We focus on discretized design spaces, with the objective function being the geometric or the arithmetic mean of design efficiencies relative to individual criteria. We address two combinations of two criteria: concave-concave (illustrated by DE\u2013optimality) and convex-concave (such as DA\u2013optimality). To handle the latter, we reformulate the problem as a bilevel problem. Here, the outer problem is solved using Surrogate Based Optimization, while the inner problem is addressed with a Semidefinite Programming solver. We demonstrate our formulations using both linear and nonlinear models (for the response) of the Beta class, previously linearized to facilitate analysis and comparison.<\/jats:p>","DOI":"10.1007\/s11081-025-10001-4","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T22:47:00Z","timestamp":1751669220000},"page":"75-97","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Compound optimal design of experiments \u2013 Semidefinite Programming formulations"],"prefix":"10.1007","volume":"27","author":[{"given":"Belmiro P. M.","family":"Duarte","sequence":"first","affiliation":[]},{"given":"Anthony C.","family":"Atkinson","sequence":"additional","affiliation":[]},{"given":"Nuno M. C.","family":"Oliveira","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,5]]},"reference":[{"issue":"7","key":"10001_CR1","doi-asserted-by":"publisher","first-page":"1643","DOI":"10.1007\/s11590-020-01561-8","volume":"14","author":"A Agrawal","year":"2020","unstructured":"Agrawal A, Boyd S (2020) Disciplined quasiconvex programming. Optimization Letters 14(7):1643\u20131657. https:\/\/doi.org\/10.1007\/s11590-020-01561-8","journal-title":"Optimization Letters"},{"key":"10001_CR2","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s00163-020-00336-7","volume":"31","author":"R Alizadeh","year":"2020","unstructured":"Alizadeh R, Allen JK, Mistree F (2020) Managing computational complexity using surrogate models: a critical review. Research in Engineering Design 31:275\u2013298","journal-title":"Research in Engineering Design"},{"issue":"1","key":"10001_CR3","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/j.jspi.2007.05.024","volume":"138","author":"AC Atkinson","year":"2008","unstructured":"Atkinson AC (2008) DT-optimum designs for model discrimination and parameter estimation. Journal of Statistical planning and Inference 138(1):56\u201364","journal-title":"Journal of Statistical planning and Inference"},{"key":"10001_CR4","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780199296590.001.0001","volume-title":"Optimum Experimental Designs, with SAS","author":"AC Atkinson","year":"2007","unstructured":"Atkinson AC, Donev AN, Tobias RD (2007) Optimum Experimental Designs, with SAS. Oxford University Press, Oxford"},{"key":"10001_CR5","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718829","volume-title":"Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications","author":"A Ben-Tal","year":"2001","unstructured":"Ben-Tal A, Nemirovski AS (2001) Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics, Philadelphia"},{"key":"10001_CR6","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1016\/j.compchemeng.2017.09.017","volume":"108","author":"A Bhosekar","year":"2018","unstructured":"Bhosekar A, Ierapetritou M (2018) Advances in surrogate based modeling, feasibility analysis, and optimization: A review. Computers & Chemical Engineering 108:250\u2013267","journal-title":"Computers & Chemical Engineering"},{"key":"10001_CR7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441","volume-title":"Convex Optimization","author":"S Boyd","year":"2004","unstructured":"Boyd S, Vandenberghe L (2004) Convex Optimization. University Press, Cambridge"},{"key":"10001_CR8","volume-title":"Radial Basis Functions - Theory and Implementations","author":"MD Buhmann","year":"2009","unstructured":"Buhmann MD (2009) Radial Basis Functions - Theory and Implementations, vol 12. Cambridge University Press"},{"key":"10001_CR9","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/j.jspi.2018.09.007","volume":"200","author":"Q Cheng","year":"2019","unstructured":"Cheng Q, Yang M (2019) On multiple-objective optimal designs. Journal of Statistical Planning and Inference 200:87\u2013101. https:\/\/doi.org\/10.1016\/j.jspi.2018.09.007","journal-title":"Journal of Statistical Planning and Inference"},{"key":"10001_CR10","doi-asserted-by":"publisher","first-page":"586","DOI":"10.1214\/aoms\/1177728915","volume":"24","author":"H Chernoff","year":"1953","unstructured":"Chernoff H (1953) Locally optimal designs for estimating parameters. The Annals of Mathematical Statistics 24:586\u2013602","journal-title":"The Annals of Mathematical Statistics"},{"issue":"435","key":"10001_CR11","doi-asserted-by":"publisher","first-page":"1236","DOI":"10.1080\/01621459.1996.10476993","volume":"91","author":"M Clyde","year":"1996","unstructured":"Clyde M, Chaloner K (1996) The equivalence of constrained and weighted designs in multiple objective design problems. Journal of the American Statistical Association 91(435):1236\u20131244","journal-title":"Journal of the American Statistical Association"},{"issue":"426","key":"10001_CR12","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1080\/01621459.1994.10476794","volume":"89","author":"RD Cook","year":"1994","unstructured":"Cook RD, Wong WK (1994) On the equivalence of constrained and compound optimal designs. Journal of the American Statistical Association 89(426):687\u2013692","journal-title":"Journal of the American Statistical Association"},{"issue":"3","key":"10001_CR13","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1145\/29380.29864","volume":"13","author":"A Corana","year":"1987","unstructured":"Corana A, Marchesi M, Martini C, Ridella S (1987) Minimizing multimodal functions of continuous variables with the \u201csimulated annealing\u2019\u2019 algorithm - corrigenda for this article is available here. ACM Trans. Math. Softw. 13(3):262\u2013280. https:\/\/doi.org\/10.1145\/29380.29864","journal-title":"ACM Trans. Math. Softw."},{"issue":"2","key":"10001_CR14","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/BF03398807","volume":"45","author":"JP Crouzeix","year":"2008","unstructured":"Crouzeix JP, Ferland JA, Van Nguyen H (2008) Revisiting Dinkelbach-type algorithms for generalized fractional programs. Opsearch 45(2):97\u2013110","journal-title":"Opsearch"},{"issue":"7","key":"10001_CR15","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1287\/mnsc.13.7.492","volume":"13","author":"W Dinkelbach","year":"1967","unstructured":"Dinkelbach W (1967) On nonlinear fractional programming. Management Science 13(7):492\u2013498","journal-title":"Management Science"},{"key":"10001_CR16","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1007\/s00362-022-01334-8","volume":"64","author":"BPM Duarte","year":"2023","unstructured":"Duarte BPM, Atkinson AC, Singh SP, Reis MS (2023) Optimal design of experiments for hypothesis testing on ordered treatments via Intersection-Union Tests. Statistical Papers 64:587\u2013615. https:\/\/doi.org\/10.1007\/s00362-022-01334-8","journal-title":"Statistical Papers"},{"issue":"2","key":"10001_CR17","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1111\/insr.12073","volume":"83","author":"BPM Duarte","year":"2015","unstructured":"Duarte BPM, Wong WK (2015) Finding Bayesian optimal designs for nonlinear models: A semidefinite programming-based approach. International Statistical Review 83(2):239\u2013262","journal-title":"International Statistical Review"},{"key":"10001_CR18","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1016\/j.jmva.2014.11.006","volume":"135","author":"BPM Duarte","year":"2015","unstructured":"Duarte BPM, Wong WK, Atkinson AC (2015) A semi-infinite programming based algorithm for determining T-optimum designs for model discrimination. Journal of Multivariate Analysis 135:11\u201324","journal-title":"Journal of Multivariate Analysis"},{"key":"10001_CR19","unstructured":"Erikson D, Bindel D, Shoemaker CA (2019) pySOT and POAP: An event-driven asynchronous framework for surrogate optimization"},{"key":"10001_CR20","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1023\/A:1011255519438","volume":"19","author":"HM Gutmann","year":"2001","unstructured":"Gutmann HM (2001) A radial basis function method for global optimization. Journal of Global Optimization 19:201\u2013227","journal-title":"Journal of Global Optimization"},{"issue":"2","key":"10001_CR21","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.csda.2008.06.023","volume":"53","author":"R Harman","year":"2008","unstructured":"Harman R, Jur\u00edk T (2008) Computing $$c-$$optimal experimental designs using the Simplex method of linear programming. Comput. Stat. Data Anal. 53(2):247\u2013254","journal-title":"Comput. Stat. Data Anal."},{"key":"10001_CR22","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1080\/00224065.2004.11980273","volume":"36","author":"A Heredia-Langner","year":"2004","unstructured":"Heredia-Langner A, Montgomery DC, Carlyle WM, Borror CM (2004) Model-robust optimal designs: A Genetic Algorithm approach. Journal of Quality Technology 36:263\u2013279","journal-title":"Journal of Quality Technology"},{"issue":"6","key":"10001_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v035.i06","volume":"35","author":"J Hu","year":"2010","unstructured":"Hu J, Zhu W, Su Y, Wong WK (2010) Controlled optimal design program for the logit dose response model. Journal of Statistical Software 35(6):1\u201317. https:\/\/doi.org\/10.18637\/jss.v035.i06","journal-title":"Journal of Statistical Software"},{"key":"10001_CR24","volume-title":"Multiple objective decision making-methods and applications: a state-of-the-art survey","author":"CL Hwang","year":"2012","unstructured":"Hwang CL, Masud ASM (2012) Multiple objective decision making-methods and applications: a state-of-the-art survey, vol 164. Springer Science & Business Media"},{"issue":"5","key":"10001_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.18637\/jss.v083.i05","volume":"83","author":"SW Hyun","year":"2018","unstructured":"Hyun SW, Wong WK, Yang Y (2018) VNM: An R package for finding multiple-objective optimal designs for the 4-parameter logistic model. Journal of Statistical Software 83(5):1\u201319. https:\/\/doi.org\/10.18637\/jss.v083.i05","journal-title":"Journal of Statistical Software"},{"key":"10001_CR26","doi-asserted-by":"publisher","unstructured":"Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of ICNN\u201995 - International Conference on Neural Networks, vol.\u00a04, pp. 1942\u20131948 vol.4. https:\/\/doi.org\/10.1109\/ICNN.1995.488968","DOI":"10.1109\/ICNN.1995.488968"},{"key":"10001_CR27","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1214\/aos\/1176342810","volume":"2","author":"J Kiefer","year":"1974","unstructured":"Kiefer J (1974) General equivalence theory for optimum design (approximate theory). Annals of Statistics 2:849\u2013879","journal-title":"Annals of Statistics"},{"key":"10001_CR28","doi-asserted-by":"publisher","first-page":"106,847","DOI":"10.1016\/j.compchemeng.2020.106847","volume":"140","author":"SH Kim","year":"2020","unstructured":"Kim SH, Boukouvala F (2020) Surrogate-based optimization for mixed-integer nonlinear problems. Computers & Chemical Engineering 140:106,847","journal-title":"Computers & Chemical Engineering"},{"issue":"4\u20135","key":"10001_CR29","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1080\/02331887408801175","volume":"5","author":"E L\u00e4uter","year":"1974","unstructured":"L\u00e4uter E (1974) Experimental design in a class of models. Mathematische Operationsforschung und Statistik 5(4\u20135):379\u2013398. https:\/\/doi.org\/10.1080\/02331887408801175","journal-title":"Mathematische Operationsforschung und Statistik"},{"issue":"4","key":"10001_CR30","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1916461.1916468","volume":"37","author":"S Le Digabel","year":"2011","unstructured":"Le Digabel S (2011) Algorithm 909: NOMAD: Nonlinear optimization with the MADS algorithm. ACM Trans. Math. Softw. 37(4):1\u201344","journal-title":"ACM Trans. Math. Softw."},{"issue":"4","key":"10001_CR31","doi-asserted-by":"publisher","first-page":"853","DOI":"10.2514\/1.8650","volume":"43","author":"JD Martin","year":"2005","unstructured":"Martin JD, Simpson TW (2005) Use of kriging models to approximate deterministic computer models. AIAA Journal 43(4):853\u2013863","journal-title":"AIAA Journal"},{"issue":"4","key":"10001_CR32","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1080\/10618600.2019.1601097","volume":"28","author":"E Masoudi","year":"2019","unstructured":"Masoudi E, Holling H, Duarte BPM, Wong WK (2019) A metaheuristic adaptive cubature based algorithm to find bayesian optimal designs for nonlinear models. Journal of Computational and Graphical Statistics 28(4):861\u2013876","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"4","key":"10001_CR33","doi-asserted-by":"publisher","first-page":"646","DOI":"10.1080\/10543400802071352","volume":"18","author":"JM McGree","year":"2008","unstructured":"McGree JM, Eccleston JA, Duffull SB (2008) Compound optimal design criteria for nonlinear models. Journal of Biopharmaceutical Statistics 18(4):646\u2013661","journal-title":"Journal of Biopharmaceutical Statistics"},{"key":"10001_CR34","volume-title":"Nonlinear Multiobjective Optimization","author":"K Miettinen","year":"1999","unstructured":"Miettinen K (1999) Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston"},{"key":"10001_CR35","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1023\/A:1014878317736","volume":"12","author":"I Molchanov","year":"2002","unstructured":"Molchanov I, Zuyev S (2002) Steepest descent algorithm in a space of measures. Statistics and Computing 12:115\u2013123","journal-title":"Statistics and Computing"},{"key":"10001_CR36","unstructured":"MOSEK, A.: MOSEK Modeling Cookbook \u2013 Release 3.3.0. https:\/\/docs.mosek.com\/MOSEKModelingCookbook-a4paper.pdf 7, 25 (2024)"},{"key":"10001_CR37","unstructured":"M\u00fcller J (2014) MATSuMoTo: The Matlab surrogate model toolbox for computationally expensive black-box global optimization problems"},{"key":"10001_CR38","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s11081-015-9281-2","volume":"17","author":"J M\u00fcller","year":"2016","unstructured":"M\u00fcller J (2016) MISO: mixed-integer surrogate optimization framework. Optimization and Engineering 17:177\u2013203","journal-title":"Optimization and Engineering"},{"issue":"4","key":"10001_CR39","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1287\/ijoc.2018.0864","volume":"31","author":"J M\u00fcller","year":"2019","unstructured":"M\u00fcller J, Day M (2019) Surrogate optimization of computationally expensive black-box problems with hidden constraints. INFORMS Journal on Computing 31(4):689\u2013702","journal-title":"INFORMS Journal on Computing"},{"issue":"5","key":"10001_CR40","doi-asserted-by":"publisher","first-page":"1383","DOI":"10.1016\/j.cor.2012.08.022","volume":"40","author":"J M\u00fcller","year":"2013","unstructured":"M\u00fcller J, Shoemaker CA, Pich\u00e9 R (2013) SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems. Computers & Operations Research 40(5):1383\u20131400","journal-title":"Computers & Operations Research"},{"key":"10001_CR41","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/s10898-017-0496-y","volume":"69","author":"J M\u00fcller","year":"2017","unstructured":"M\u00fcller J, Woodbury JD (2017) GOSAC: global optimization with surrogate approximation of constraints. Journal of Global Optimization 69:117\u2013136","journal-title":"Journal of Global Optimization"},{"key":"10001_CR42","doi-asserted-by":"crossref","unstructured":"Nesterov YE, Nemirovski AS (1994) Interior-Point Polynomial Algorithms in Convex Programming (Studies in Applied and Numerical Mathematics). Society for Industrial Mathematics","DOI":"10.1137\/1.9781611970791"},{"key":"10001_CR43","first-page":"105","volume-title":"Advances in Numerical Analysis II: Wavelets, Subdivision, and Radial Functions","author":"MJD Powell","year":"1992","unstructured":"Powell MJD (1992) The theory of radial basis function approximation in 1990. In: Light WA (ed) Advances in Numerical Analysis II: Wavelets, Subdivision, and Radial Functions. Oxford University Press, Oxford, pp 105\u2013210"},{"key":"10001_CR44","volume-title":"Optimal Design of Experiments","author":"F Pukelsheim","year":"1993","unstructured":"Pukelsheim F (1993) Optimal Design of Experiments. SIAM, Philadelphia"},{"key":"10001_CR45","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.swevo.2014.06.003","volume":"18","author":"J Qiu","year":"2014","unstructured":"Qiu J, Chen RB, Wang W, Wong WK (2014) Using animal instincts to design efficient biomedical studies via particle swarm optimization. Swarm and evolutionary computation 18:1\u201310","journal-title":"Swarm and evolutionary computation"},{"key":"10001_CR46","doi-asserted-by":"publisher","unstructured":"Qiu J, Wong WK (2023) Nature-inspired metaheuristics for finding optimal designs for the continuation-ratio models. The New England Journal of Statistics in Data Science pp. 1\u201315. https:\/\/doi.org\/10.51387\/23-NEJSDS44","DOI":"10.51387\/23-NEJSDS44"},{"issue":"4","key":"10001_CR47","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1287\/ijoc.1060.0182","volume":"19","author":"RG Regis","year":"2007","unstructured":"Regis RG, Shoemaker CA (2007) A stochastic radial basis function method for the global optimization of expensive functions. INFORMS Journal on Computing 19(4):497\u2013509","journal-title":"INFORMS Journal on Computing"},{"issue":"5","key":"10001_CR48","doi-asserted-by":"publisher","first-page":"1684","DOI":"10.1016\/j.jspi.2010.11.031","volume":"141","author":"G Sagnol","year":"2011","unstructured":"Sagnol G (2011) Computing optimal designs of multiresponse experiments reduces to second-order cone programming. Journal of Statistical Planning and Inference 141(5):1684\u20131708","journal-title":"Journal of Statistical Planning and Inference"},{"issue":"10","key":"10001_CR49","doi-asserted-by":"publisher","first-page":"2829","DOI":"10.1016\/j.laa.2013.08.021","volume":"439","author":"G Sagnol","year":"2013","unstructured":"Sagnol G (2013) On the semidefinite representation of real functions applied to symmetric matrices. Linear Algebra and its Applications 439(10):2829\u20132843","journal-title":"Linear Algebra and its Applications"},{"key":"10001_CR50","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1023\/A:1008202821328","volume":"11","author":"R Storn","year":"1997","unstructured":"Storn R, Price K (1997) Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11:341\u2013359","journal-title":"Journal of Global Optimization"},{"key":"10001_CR51","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1137\/1038003","volume":"8","author":"L Vandenberghe","year":"1996","unstructured":"Vandenberghe L, Boyd S (1996) Semidefinite programming. SIAM Review 8:49\u201395","journal-title":"SIAM Review"},{"key":"10001_CR52","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1016\/S0168-9274(98)00098-1","volume":"29","author":"L Vandenberghe","year":"1999","unstructured":"Vandenberghe L, Boyd S (1999) Applications of semidefinite programming. Applied Numerical Mathematics 29:283\u2013299","journal-title":"Applied Numerical Mathematics"},{"issue":"2","key":"10001_CR53","doi-asserted-by":"publisher","first-page":"744","DOI":"10.1080\/10618600.2022.2104858","volume":"32","author":"WK Wong","year":"2023","unstructured":"Wong WK, Zhou J (2023) Using CVX to construct optimal designs for biomedical studies with multiple objectives. Journal of Computational and Graphical Statistics 32(2):744\u2013753. https:\/\/doi.org\/10.1080\/10618600.2022.2104858","journal-title":"Journal of Computational and Graphical Statistics"},{"issue":"504","key":"10001_CR54","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1080\/01621459.2013.806268","volume":"108","author":"M Yang","year":"2013","unstructured":"Yang M, Biedermann S, Tang E (2013) On optimal designs for nonlinear models: A general and efficient algorithm. Journal of the American Statistical Association 108(504):1411\u20131420","journal-title":"Journal of the American Statistical Association"},{"issue":"1","key":"10001_CR55","doi-asserted-by":"publisher","first-page":"666","DOI":"10.1137\/110850268","volume":"23","author":"JJ Ye","year":"2013","unstructured":"Ye JJ, Zhou J (2013) Minimizing the condition number to construct design points for polynomial regression models. SIAM Journal on Optimization 23(1):666\u2013686","journal-title":"SIAM Journal on Optimization"},{"key":"10001_CR56","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032701","volume-title":"Interior Point Algorithms: Theory and Analysis","author":"Y Ye","year":"1997","unstructured":"Ye Y (1997) Interior Point Algorithms: Theory and Analysis. John Wiley & Sons, New York"},{"issue":"4","key":"10001_CR57","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1007\/s11222-010-9183-2","volume":"21","author":"Y Yu","year":"2010","unstructured":"Yu Y (2010) D-optimal designs via a cocktail algorithm. Statistics and Computing 21(4):475\u2013481","journal-title":"Statistics and Computing"}],"container-title":["Optimization and Engineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11081-025-10001-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11081-025-10001-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11081-025-10001-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T08:03:12Z","timestamp":1772179392000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11081-025-10001-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,5]]},"references-count":57,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["10001"],"URL":"https:\/\/doi.org\/10.1007\/s11081-025-10001-4","relation":{},"ISSN":["1389-4420","1573-2924"],"issn-type":[{"value":"1389-4420","type":"print"},{"value":"1573-2924","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,5]]},"assertion":[{"value":"20 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval and consent to participate"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}]}}