{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T21:01:07Z","timestamp":1770411667535,"version":"3.49.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"1","content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["BMC Bioinformatics"],"published-print":{"date-parts":[[2012,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:sec>\n            <jats:title>Background<\/jats:title>\n            <jats:p>The estimation of parameter values for mathematical models of biological systems is an optimization problem that is particularly challenging due to the nonlinearities involved. One major difficulty is the existence of multiple minima in which standard optimization methods may fall during the search. Deterministic global optimization methods overcome this limitation, ensuring convergence to the global optimum within a desired tolerance. Global optimization techniques are usually classified into stochastic and deterministic. The former typically lead to lower CPU times but offer no guarantee of convergence to the global minimum in a finite number of iterations. In contrast, deterministic methods provide solutions of a given quality (i.e., optimality gap), but tend to lead to large computational burdens.<\/jats:p>\n          <\/jats:sec>\n          <jats:sec>\n            <jats:title>Results<\/jats:title>\n            <jats:p>This work presents a deterministic outer approximation-based algorithm for the global optimization of dynamic problems arising in the parameter estimation of models of biological systems. Our approach, which offers a theoretical guarantee of convergence to global minimum, is based on reformulating the set of ordinary differential equations into an equivalent set of algebraic equations through the use of orthogonal collocation methods, giving rise to a nonconvex nonlinear programming (NLP) problem. This nonconvex NLP is decomposed into two hierarchical levels: a master mixed-integer linear programming problem (MILP) that provides a rigorous lower bound on the optimal solution, and a reduced-space slave NLP that yields an upper bound. The algorithm iterates between these two levels until a termination criterion is satisfied.<\/jats:p>\n          <\/jats:sec>\n          <jats:sec>\n            <jats:title>Conclusion<\/jats:title>\n            <jats:p>The capabilities of our approach were tested in two benchmark problems, in which the performance of our algorithm was compared with that of the commercial global optimization package BARON. The proposed strategy produced near optimal solutions (i.e., within a desired tolerance) in a fraction of the CPU time required by BARON.<\/jats:p>\n          <\/jats:sec>","DOI":"10.1186\/1471-2105-13-90","type":"journal-article","created":{"date-parts":[[2012,5,10]],"date-time":"2012-05-10T12:18:28Z","timestamp":1336652308000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":38,"title":["Deterministic global optimization algorithm based on outer approximation for the parameter estimation of nonlinear dynamic biological systems"],"prefix":"10.1186","volume":"13","author":[{"given":"Anton","family":"Mir\u00f3","sequence":"first","affiliation":[]},{"given":"Carlos","family":"Pozo","sequence":"additional","affiliation":[]},{"given":"Gonzalo","family":"Guill\u00e9n-Gos\u00e1lbez","sequence":"additional","affiliation":[]},{"given":"Jose A","family":"Egea","sequence":"additional","affiliation":[]},{"given":"Laureano","family":"Jim\u00e9nez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2012,5,10]]},"reference":[{"issue":"10-12","key":"5552_CR1","doi-asserted-by":"publisher","first-page":"1560","DOI":"10.1016\/j.compchemeng.2006.05.034","volume":"30","author":"S Kameswaran","year":"2006","unstructured":"Kameswaran S, Biegler L: Simultaneous dynamic optimization strategies: recent advances and challenges. Comput & Chem Eng 2006, 30(10\u201312):1560\u20131575. 10.1016\/j.compchemeng.2006.05.034","journal-title":"Comput & Chem Eng"},{"issue":"5","key":"5552_CR2","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1021\/ie990486w","volume":"39","author":"W Esposito","year":"2000","unstructured":"Esposito W, Floudas C: Global optimization for the parameter estimation of differential-algebraic systems. Ind & Eng Chem Res 2000, 39(5):1291\u20131310. 10.1021\/ie990486w","journal-title":"Ind & Eng Chem Res"},{"key":"5552_CR3","unstructured":"Cizniar M, Salhi D, Fikar M, Latifi M: A MATLAB package for orthogonal collocations on finite elements in dynamic optimisation. Proc 15 Int Conference Process Control, Volume 5 058f-058f."},{"key":"5552_CR4","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1186\/1471-2105-7-483","volume":"7","author":"M Rodriguez-Fernandez","year":"2006","unstructured":"Rodriguez-Fernandez M, Egea J, Banga J: Novel metaheuristic for parameter estimation in nonlinear dynamic biological systems. BMC Bioinf 2006, 7: 483. 10.1186\/1471-2105-7-483","journal-title":"BMC Bioinf"},{"key":"5552_CR5","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1023\/A:1026578104213","volume":"17","author":"W Esposito","year":"2000","unstructured":"Esposito W, Floudas C: Deterministic global optimization in nonlinear optimal control problems. J Global Optimization 2000, 17: 97\u2013126. 10.1023\/A:1026578104213","journal-title":"J Global Optimization"},{"key":"5552_CR6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1023\/A:1016259507911","volume":"24","author":"I Papamichail","year":"2002","unstructured":"Papamichail I, Adjiman C: A rigorous global optimization algorithm for problems with ordinary differential equations. J Global Optimization 2002, 24: 1\u201333. 10.1023\/A:1016259507911","journal-title":"J Global Optimization"},{"issue":"3","key":"5552_CR7","doi-asserted-by":"publisher","first-page":"613","DOI":"10.1023\/B:JOTA.0000037606.79050.a7","volume":"121","author":"A Singer","year":"2004","unstructured":"Singer A, Barton P: Global solution of optimization problems with parameter-embedded linear dynamic systems. J Optimization Theory and Appl 2004, 121(3):613\u2013646.","journal-title":"J Optimization Theory and Appl"},{"issue":"3","key":"5552_CR8","doi-asserted-by":"publisher","first-page":"517","DOI":"10.1007\/s10107-004-0503-1","volume":"100","author":"P Kesavan","year":"2004","unstructured":"Kesavan P, Allgor R, Gatzke E, Barton P: Outer approximation algorithms for separable nonconvex mixed-integer nonlinear programs. Math Programming 2004, 100(3):517\u2013535.","journal-title":"Math Programming"},{"issue":"8","key":"5552_CR9","doi-asserted-by":"publisher","first-page":"1169","DOI":"10.1016\/j.compchemeng.2003.11.003","volume":"28","author":"L Biegler","year":"2004","unstructured":"Biegler L, Grossmann I: Retrospective on optimization. Comput & Chem Eng 2004, 28(8):1169\u20131192. 10.1016\/j.compchemeng.2003.11.003","journal-title":"Comput & Chem Eng"},{"key":"5552_CR10","volume-title":"The method of weighted residuals and variational principles: with application in fluid mechanics, heat and mass transfer, Volume 87","author":"B Finlayson","year":"1972","unstructured":"Finlayson B: The method of weighted residuals and variational principles: with application in fluid mechanics, heat and mass transfer, Volume 87. Academic Pr; 1972."},{"issue":"8","key":"5552_CR11","doi-asserted-by":"publisher","first-page":"1257","DOI":"10.1002\/aic.690330804","volume":"33","author":"J Cuthrell","year":"1987","unstructured":"Cuthrell J, Biegler L: On the optimization of differential-algebraic process systems. AIChE J 1987, 33(8):1257\u20131270. 10.1002\/aic.690330804","journal-title":"AIChE J"},{"issue":"4","key":"5552_CR12","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1016\/0098-1354(94)00064-U","volume":"19","author":"D Tieu","year":"1995","unstructured":"Tieu D, Cluett W, Penlidis A: A comparison of collocation methods for solving dynamic optimization problems. Comput & Chem Eng 1995, 19(4):375\u2013381. 10.1016\/0098-1354(94)00064-U","journal-title":"Comput & Chem Eng"},{"issue":"10","key":"5552_CR13","doi-asserted-by":"publisher","first-page":"1719","DOI":"10.1016\/j.compchemeng.2010.03.001","volume":"34","author":"C Pozo","year":"2010","unstructured":"Pozo C, Guill\u00e9n-Gos\u00e1lbez G, Sorribas A, Jim\u00e9nez L: Outer approximation-based algorithm for biotechnology studies in systems biology. Comp & Chem Eng 2010, 34(10):1719\u20131730. 10.1016\/j.compchemeng.2010.03.001","journal-title":"Comp & Chem Eng"},{"key":"5552_CR14","unstructured":"Carlos P, Alberto M, Rui A, Gonzalo G, Laureano J, Albert S: Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models. BMC Syst Biol 5: 137."},{"key":"5552_CR15","volume-title":"Ind & Eng Chem Res","author":"C Pozo","year":"2010","unstructured":"Pozo C, Guill\u00e9n-Gos\u00e1lbez G, Sorribas A, Jim\u00e9nez L: A spatial branch-and-bound framework for the global optimization of kinetic models of metabolic networks. Ind & Eng Chem Res 2010."},{"issue":"3","key":"5552_CR16","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1016\/j.jbiotec.2010.01.026","volume":"149","author":"A Sorribas","year":"2010","unstructured":"Sorribas A, Pozo C, Vilaprinyo E, Guill\u00e9n-Gos\u00e1lbez G, Jim\u00e9nez L, Alves R: Optimization and evolution in metabolic pathways: Global optimization techniques in Generalized Mass Action models. J Biotechnol 2010, 149(3):141\u2013153. 10.1016\/j.jbiotec.2010.01.026","journal-title":"J Biotechnol"},{"key":"5552_CR17","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1186\/1471-2105-10-386","volume":"10","author":"G Guill\u00e9n-Gos\u00e1lbez","year":"2009","unstructured":"Guill\u00e9n-Gos\u00e1lbez G, Sorribas A: Identifying quantitative operation principles in metabolic pathways: a systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses. BMC Bioinf 2009, 10: 386. 10.1186\/1471-2105-10-386","journal-title":"BMC Bioinf"},{"issue":"4","key":"5552_CR18","doi-asserted-by":"publisher","first-page":"991","DOI":"10.1002\/aic.11425","volume":"54","author":"D Wicaksono","year":"2008","unstructured":"Wicaksono D, Karimi I: Piecewise MILP under-and overestimators for global optimization of bilinear programs. AIChE J 2008, 54(4):991\u20131008. 10.1002\/aic.11425","journal-title":"AIChE J"},{"issue":"4","key":"5552_CR19","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/BF01099647","volume":"7","author":"I Androulakis","year":"1995","unstructured":"Androulakis I, Maranas C, Floudas C: \u03b1BB: A global optimization method for general constrained nonconvex problems. J Global Optimization 1995, 7(4):337\u2013363. 10.1007\/BF01099647","journal-title":"J Global Optimization"},{"issue":"4-5","key":"5552_CR20","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1016\/S0098-1354(98)00286-5","volume":"23","author":"E Smith","year":"1999","unstructured":"Smith E, Pantelides C: A symbolic reformulation\/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs. Comput & Chem Eng 1999, 23(4\u20135):457\u2013478. 10.1016\/S0098-1354(98)00286-5","journal-title":"Comput & Chem Eng"},{"issue":"3","key":"5552_CR21","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1021\/ie00015a012","volume":"32","author":"I Quesada","year":"1993","unstructured":"Quesada I, Grossmann I: Global optimization algorithm for heat exchanger networks. Ind & Eng Chem Res 1993, 32(3):487\u2013499. 10.1021\/ie00015a012","journal-title":"Ind & Eng Chem Res"},{"key":"5552_CR22","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/978-1-4757-5331-8_12","volume":"9","author":"E Smith","year":"1996","unstructured":"Smith E, Pantelides C: Global optimisation of general process models. NONCONVEX OPTIMIZATION APPL 1996, 9: 355\u2013384.","journal-title":"NONCONVEX OPTIMIZATION APPL"},{"key":"5552_CR23","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/BF01580665","volume":"10","author":"G McCormick","year":"1976","unstructured":"McCormick G: Computability of global solutions to factorable nonconvex programs: Part I Convex underestimating problems. Math Programming 1976, 10: 147\u2013175. 10.1007\/BF01580665","journal-title":"Math Programming"},{"key":"5552_CR24","volume-title":"Nonlinear programming: Theory, algorithms, and applications","author":"G McCormick","year":"1983","unstructured":"McCormick G: Nonlinear programming: Theory, algorithms, and applications. 1983."},{"key":"5552_CR25","doi-asserted-by":"publisher","first-page":"876","DOI":"10.1016\/j.compchemeng.2011.01.026","volume":"35","author":"R Misener","year":"2011","unstructured":"Misener R, Thompson J, Floudas C: APOGEE: Global optimization of standard, generalized, and extended pooling problems via linear and logarithmic partitioning schemes. Comput & Chem Eng 2011, 35: 876\u2013892. 10.1016\/j.compchemeng.2011.01.026","journal-title":"Comput & Chem Eng"},{"issue":"6","key":"5552_CR26","doi-asserted-by":"publisher","first-page":"2167","DOI":"10.1137\/040604388","volume":"27","author":"A Singer","year":"2006","unstructured":"Singer A, Barton P: Bounding the solutions of parameter dependent nonlinear ordinary differential equations. SIAM J Sci Comput 2006, 27(6):2167\u20132184. 10.1137\/040604388","journal-title":"SIAM J Sci Comput"},{"issue":"2","key":"5552_CR27","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/s10898-005-7074-4","volume":"34","author":"A Singer","year":"2006","unstructured":"Singer A, Barton P: Global optimization with nonlinear ordinary differential equations. J Global Optimization 2006, 34(2):159\u2013190. 10.1007\/s10898-005-7074-4","journal-title":"J Global Optimization"},{"issue":"4","key":"5552_CR28","doi-asserted-by":"publisher","first-page":"650","DOI":"10.1016\/j.compchemeng.2005.11.005","volume":"30","author":"R Karuppiah","year":"2006","unstructured":"Karuppiah R, Grossmann I: Global optimization for the synthesis of integrated water systems in chemical processes. Comput & Chem Eng 2006, 30(4):650\u2013673. 10.1016\/j.compchemeng.2005.11.005","journal-title":"Comput & Chem Eng"},{"issue":"2","key":"5552_CR29","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1021\/ja01194a047","volume":"69","author":"R Fuguitt","year":"1947","unstructured":"Fuguitt R, Hawkins J: Rate of the thermal isomerization of \u03b1-Pinene in the liquid phase1. J Am Chem Soc 1947, 69(2):319\u2013322. 10.1021\/ja01194a047","journal-title":"J Am Chem Soc"},{"key":"5552_CR30","volume-title":"Unpublished Report","author":"W Hunter","year":"1967","unstructured":"Hunter W, McGregor J: The estimation of common parameters from several responses: Some actual examples. Unpublished Report. 1967."},{"issue":"8","key":"5552_CR31","doi-asserted-by":"publisher","first-page":"1193","DOI":"10.1016\/j.compchemeng.2003.11.006","volume":"28","author":"I Grossmann","year":"2004","unstructured":"Grossmann I, Biegler L: Part II. Future perspective on optimization. Comput & Chem Eng 2004, 28(8):1193\u20131218. 10.1016\/j.compchemeng.2003.11.006","journal-title":"Comput & Chem Eng"},{"key":"5552_CR32","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1007\/BF01582889","volume":"52","author":"P Hansen","year":"1991","unstructured":"Hansen P, Jaumard B, Lu S: An analytical approach to global optimization. Math Programming 1991, 52: 227\u2013254. 10.1007\/BF01582889","journal-title":"Math Programming"},{"issue":"2","key":"5552_CR33","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1006\/abio.1996.0238","volume":"237","author":"P Kuzmic","year":"1996","unstructured":"Kuzmic P: Program DYNAFIT for the analysis of enzyme kinetic data: application to HIV proteinase. Anal Biochem 1996, 237(2):260\u2013273. 10.1006\/abio.1996.0238","journal-title":"Anal Biochem"},{"issue":"10","key":"5552_CR34","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1093\/bioinformatics\/14.10.869","volume":"14","author":"P Mendes","year":"1998","unstructured":"Mendes P, Kell D: Non-linear optimization of biochemical pathways: applications to metabolic engineering and parameter estimation. Bioinformatics 1998, 14(10):869. 10.1093\/bioinformatics\/14.10.869","journal-title":"Bioinformatics"}],"container-title":["BMC Bioinformatics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/1471-2105-13-90.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T21:12:02Z","timestamp":1630530722000},"score":1,"resource":{"primary":{"URL":"https:\/\/bmcbioinformatics.biomedcentral.com\/articles\/10.1186\/1471-2105-13-90"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,5,10]]},"references-count":34,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2012,12]]}},"alternative-id":["5552"],"URL":"https:\/\/doi.org\/10.1186\/1471-2105-13-90","relation":{},"ISSN":["1471-2105"],"issn-type":[{"value":"1471-2105","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,5,10]]},"assertion":[{"value":"7 November 2011","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 2012","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 2012","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"90"}}