Global Optimization for the Parameter Estimation of Differential-Algebraic Systems

Abstract

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.

Parameter Estimation in Biochemical Pathways: A Comparison of Global Optimization Methods

“…Global optimization methods can be roughly classified as deterministic (Horst and Tuy 1990;Grossmann 1996;Pinter 1996;Esposito and Floudas 2000) and stochastic strategies (Guus et al 1995;Ali et al 1997;Törn et al 1999).…”

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

“…The control parameterization at the start of the algorithm being rather coarse, we find that two lifting operations are applied during the first two branch-and-lift iterations; that is, the lifting condition (19) happens to be satisfied for M = 0 and M = 1 without branching. As the parameterization order is increased to M = 2, however, the algorithm performs a number of branching operations and the fathoming test successfully excludes a number of suboptimal control regions.…”

“…The evaluation of the objective and constraint functions in the discretized NLP is via the numerical integration of the differential equations. This approach was originally introduced in a local optimization context [4,15,16], but it has more recently been extended to global optimization, see for example [17][18][19][20][21][22][23]. Note that all of these approaches have in common that they rely on branch-and-bound search [24] to solve the resulting NLP problem to guaranteed global optimality.…”

Global optimization in the 21st century: Advances and challenges

“…Barton, Banga, and Galan (2000) studied the optimization of hybrid discrete/continuous dynamic systems, presented a framework based on hybrid optimal control, investigated existence and sensitivity results, introduced a modified stochastic search approach, and presented computational results for a tank changeover problem. Esposito and Floudas (2001) pointed out the theoretical rigor and advantages of the proposed global optimization methods by Esposito and Floudas (2000a) and the differences between local search approaches and global optimization methods. Esposito and Floudas (2002) studied the isothermal reactor network synthesis problem, formulated it as nonconvex NLP with differential-algebraic constraints, introduced a global optimization framework based on the integration approach and the ␣BB, investigated alternative types of reformulations, and reported extensive computational studies for complex reaction/reactor networks.…”