{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T04:59:06Z","timestamp":1770958746302,"version":"3.50.1"},"reference-count":66,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,2,7]],"date-time":"2023-02-07T00:00:00Z","timestamp":1675728000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The systematic design of exact optimal designs of experiments is typically challenging, as it results in nonconvex optimization problems. The literature on the computation of model-based exact optimal designs of experiments via mathematical programming, when the covariates are categorical variables, is still scarce. We propose mixed-integer semidefinite programming formulations, to find exact D-, A- and I-optimal designs for linear models, and locally optimal designs for nonlinear models when the design domain is a finite set of points. The strategy requires: (i) the generation of a set of candidate treatments; (ii) the formulation of the optimal design problem as a mixed-integer semidefinite program; and (iii) its solution, employing appropriate solvers. For comparison, we use semidefinite programming-based formulations to find equivalent approximate optimal designs. We demonstrate the application of the algorithm with various models, considering both unconstrained and constrained setups. Equivalent approximate optimal designs are used for comparison.<\/jats:p>","DOI":"10.3390\/math11040854","type":"journal-article","created":{"date-parts":[[2023,2,8]],"date-time":"2023-02-08T04:57:41Z","timestamp":1675832261000},"page":"854","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2550-4320","authenticated-orcid":false,"given":"Belmiro P. M.","family":"Duarte","sequence":"first","affiliation":[{"name":"Department of Chemical and Biological Engineering, Coimbra Engineering Academia, Polytechnic Institute of Coimbra, Rua Pedro Nunes, Quinta da Nora, 3030-199 Coimbra, Portugal"},{"name":"Centro de Investiga\u00e7\u00e3o em Engenharia dos Processos Qu\u00edmicos e dos Produtos da Floresta, Department of Chemical Engineering, University of Coimbra, Rua S\u00edlvio Lima, P\u00f3lo II, 3030-790 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Fedorov, V.V., and Leonov, S.L. (2014). Optimal Design for Nonlinear Response Models, Chapman and Hall\/CRC Press.","DOI":"10.1201\/b15054"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1111\/j.2517-6161.1959.tb00338.x","article-title":"Optimum experimental designs","volume":"21","author":"Kiefer","year":"1959","journal-title":"J. R. Stat. Soc. Ser. 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