The estimation of parameters in semiempirical models is essential in numerous areas of engineering and applied science. In many cases these models are represented by a set of nonlinear differential-algebraic equations. This introduces difficulties from both a numerical and an optimization perspective. One such difficulty, which has not been adequately addressed, is the existence of multiple local minima. In this paper, two novel global optimization methods will be presented which offer a theoretical guarantee of convergence to the global minimum for a wide range of problems. The first is based on converting the dynamic system of equations into a set of algebraic constraints through the use of collocation methods. The reformulated problem has interesting mathematical properties which allow for the development of a deterministic branch and bound global optimization approach. The second method is based on the use of integration to solve the dynamic system of equations. Both methods will be applied to the problem of estimating parameters in differential-algebraic models through the error-in-variables approach. The mathematical properties of the formulation which lead to specialization of the algorithms will be discussed. Then, the computational aspects of both approaches will be presented and compared through their application to several problems involving reaction kinetics.

The estimation of parameters in nonlinear algebraic models through the error-in-variables method has been widely studied from a computational standpoint. The method involves the minimization of a weighted sum of squared errors subject to the model equations. Due to the nonlinear nature of the models used, the resulting formulation is nonconvex and may contain several local minima in the region of interest. Current methods tailored for this formulation, although computationally efficient, can only attain convergence to a local solution. In this paper, a global optimization approach based on a branch and bound framework and convexification techniques for general twice differentiable nonlinear optimization problems is proposed for the parameter estimation of nonlinear algebraic models. The proposed convexification techniques exploit the mathematical properties of the formulation. Classical nonlinear estimation problems were solved and will be used to illustrate the various theoretical and computational aspects of the proposed approach.

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