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Global optimization for parameter estimation of differential-algebraic systems

Michal Čižniar, Marián Podmajerský, Tomáš Hirmajer, Miroslav Fikar, and Abderrazak M. Latifi

Institute of Information Engineering, Automation and Mathematics, Faculty of Chemical and Food Technology, Slovak University of Technology, Radlinského 9, 81237 Bratislava, Slovakia



Received: 19 June 2008  Revised: 8 September 2008  Accepted: 23 September 2008

Abstract: The estimation of parameters in semi-empirical models is essential in numerous areas of engineering and applied science. In many cases, these models are described by a set of ordinary-differential equations or by a set of differential-algebraic equations. Due to the presence of non-convexities of functions participating in these equations, current gradient-based optimization methods can guarantee only locally optimal solutions. This deficiency can have a marked impact on the operation of chemical processes from the economical, environmental and safety points of view and it thus motivates the development of global optimization algorithms. This paper presents a global optimization method which guarantees ɛ-convergence to the global solution. The approach consists in the transformation of the dynamic optimization problem into a nonlinear programming problem (NLP) using the method of orthogonal collocation on finite elements. Rigorous convex underestimators of the nonconvex NLP problem are employed within the spatial branch-and-bound method and solved to global optimality. The proposed method was applied to two example problems dealing with parameter estimation from time series data.

Keywords: parameter estimation - orthogonal collocation - dynamic optimization - global optimization

Full paper is available at

DOI: 10.2478/s11696-009-0017-7


Chemical Papers 63 (3) 274–283 (2009)

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