Simultaneous parameter estimation and model structure determination in FTIR spectroscopy by global MINLP optimization Anastasia Vaia, Nikolaos V. Sahinidis * Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, 600 South Mathews Avenue, Urbana, IL 61801, USA Received 2 July 2002; received in revised form 23 September 2002; accepted 13 November 2002 Abstract We address the problem of simultaneous model structure determination and parameter estimation in infrared spectroscopy. For given measurements of concentrations (C ) and absorbances (A ), we seek to find the constant of analogy (U) in reverse Beer’s law (C /UA ). Two approaches are described and compared in this paper. Both utilize Akaike’s information criterion (AIC) to obtain an estimate of the constant. The first method is frequently used in practice and requires the iterative solution of mixed-integer convex quadratic optimization problems. The second method is a novel one that requires the solution of a single mixed-integer nonconvex nonlinear program for which we develop a global optimization algorithm. Computational results demonstrate that the latter approach provides better solutions for all of the eleven problems solved in this paper. Our computational experiments also reveal the importance of bounding the errors and number of model parameters when minimizing AIC. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: FTIR spectroscopy; Parameter estimation; Global optimization; MINLP; Akaike information criterion 1. Introduction The accurate determination of the amount of every component of a chemical mixture is a key requirement for solving many problems in process monitoring, analysis, and optimization. This has motivated the development of a great variety of experimental and mathematical techniques for quantitative mixture as- sessment. One of the most common methods for identifying the components of a mixture and measuring their concen- tration is infrared spectroscopy. The approach begins by collecting infrared spectra for mixtures of known composition. These measurements are then used to build a model that relates component concentrations to levels of absorbance when infrared radiation passes through the mixture. The model thus developed is subsequently used to predict unknown compositions from absorbance measurements. Factor analysis (cf. Lawley & Maxwel, 1963) and the Bouguer  /Beer  /Lambert law (cf. Griffiths, 1975) have been used in the literature to relate absorbance to concentration. Beer’s law, in particular, provides a linear relation between the intensity of spectral bands and the concentration of each component in a mixture when infrared radiation passes through it. The classical representation of Beer’s law is: A KC; where A  /R W S is the total absorbance of the solution, C /R N S is the concentration matrix, and K  /R W N is a proportionality matrix. Here, N is the number of components in the mixture, S is the number of collected spectra (number of experiments run), and W is the number of wavenumbers at which absorbances are collected. This representation has the disadvantage that all interfering chemical components must be known and included in the calibration. For this reason, an alternative representation known as reverse Beer’s law has been proposed and extensively used in the literature (Sternberg, Stillo & Schwendeman, 1960; Barnett & Bartoli, 1960; Haaland & Thomas, 1988; Kisner, Brown * Corresponding author. Tel.:  /1-217-244-1304; fax:  /1-217-333- 5052. E-mail address: nikos@uiuc.edu (N.V. Sahinidis). Computers and Chemical Engineering 27 (2003) 763  /779 www.elsevier.com/locate/compchemeng 0098-1354/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0098-1354(02)00262-4