The literature abounds with the application of optimization methods for estimating model parameters in equation systems. The utility of these methods is frequently demonstrated on pathological examples using simulated data generated from a known model with a random error component and a known statistical distribution. Unfortunately, parameter estimation problems encountered in practice do not have this advantage. The true model is frequently not known. In fact, one is faced with choosing among various candidate models, all of which may be wrong. Moreover, the error structure is generally unknown and must be estimated from the data. Finally, a great deal of mathematical expertise is required to transform the model and select meaningful starting guesses before parameter estimation can be successful.
In order to demonstrate the difficulties of parameter estimation in the industrial environment and the limitations of existing methods, a parameter estimation problem formulated by the Dow Chemical Company is presented and solved. This test problem consists of a stiff differential/algebraic (DAE) model that describes complex kinetics and requires the estimation of nine parameters from batch reactor data. Here the model was inadequate to describe the data, the error structure was not specified and the starting guesses led to a nontrivial optimization problem.
The Dow parameter estimation problem was distributed in 1981 to 165 researchers as a followup to the 1980 FOCAPD conference. Of those researchers, eleven agreed to apply their methodologies and expertise to this problem. However, only five acceptable solutions were finally submitted. Here we present and compare these results. Each solution was obtained using different strategies. In most cases the form of the model was also changed to accommodate the algorithms used and to ease the solution procedure. Therefore, while this case study does not present a direct numerical comparison of algorithms, it does offer guidelines and insight towards the solution of difficult parameter estimation problems.