Parameter estimation in nonlinear algebraic models via global optimization

The estimation of parameters in semi-empirical nonlinear models through the error-in-variables method has been widely studied from a computational standpoint. This method involves the minimization of a quadratic objective function subject to the model equations being satisfied. Due to the nonlinear nature of these models, the resulting formulation is nonconvex in nature. The approaches to solve this problem presented so far in the literature, although computationally efficient, only offer convergence to a local solution of a model which may contain multiple minima. In this paper a global optimization approach based on a branch-and-bound framework will be presented to solve the error-in-variables formulation. Various estimation problems were solved and will be presented to illustrate the theoretical and computational aspects of the proposed method.

[1]  Thomas F. Edgar,et al.  Robust error‐in‐variables estimation using nonlinear programming techniques , 1990 .

[2]  F. Al-Khayyal.  Jointly constrained bilinear programs and related problems: an overview , 1990 .

[3]  Christodoulos A. Floudas,et al.  Rigorous convex underestimators for general twice-differentiable problems , 1996, J. Glob. Optim..

[4]  Garth P. McCormick,et al.  Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems , 1976, Math. Program..

[5]  H. Schwetlick,et al.  Numerical Methods for Estimating Parameters in Nonlinear Models With Errors in the Variables , 1985 .

[6]  Warren E. Stewart,et al.  Parameter estimation from multiresponse data , 1992 .

[7]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[8]  N. L. Ricker.  Comparison of methods for non-linear parameter estimation , 1984 .

[9]  Yonathan Bard,et al.  Nonlinear parameter estimation , 1974 .

[10]  Jean-Francois Fabries,et al.  Method of evaluation and reduction of vapor‐liquid equilibrium data of binary mixtures , 1975 .

[11]  O. Paladino,et al.  Fitting of experimental data to implicit models using a constrained variation algorithm , 1989 .

[12]  Edward A. Grens,et al.  Evaluation of parameters for nonlinear thermodynamic models , 1978 .

[13]  Christodoulos A. Floudas,et al.  αBB: A global optimization method for general constrained nonconvex problems , 1995, J. Glob. Optim..

[14]  Christodoulos A. Floudas,et al.  A global optimization method, αBB, for process design , 1996 .

[15]  S. Vajda,et al.  An extended Marquardt-type procedure for fitting error-in-variables models , 1987 .

[16]  Park M. Reilly,et al.  Statistical estimation of parameters in vapor‐liquid equilibrium , 1982 .

[17]  Michael A. Saunders,et al.  MINOS 5. 0 user's guide , 1983 .

[18]  Christodoulos A Floudas,et al.  Global minimum potential energy conformations of small molecules , 1994, J. Glob. Optim..

[19]  Christodoulos A. Floudas,et al.  Finding all solutions of nonlinearly constrained systems of equations , 1995, J. Glob. Optim..

[20]  Response to comments on "Comparison of methods for nonlinear parameter estimation" , 1985 .

[21]  R. H. Fariss,et al.  An efficient computational technique for generalized application of maximum likelihood to improve correlation of experimental data , 1979 .

[22]  Y. Bard,et al.  Kinetics Analysis by Digital Parameter Estimation , 1969 .

[23]  T. Brubaker,et al.  Nonlinear Parameter Estimation , 1979 .

[24]  Vladimír Rod,et al.  Iterative estimation of model parameters when measurements of all variables are subject to error , 1980 .

[25]  Lorenz T. Biegler,et al.  Reduced successive quadratic programming strategy for errors-in-variables estimation , 1992 .

[26]  W. H. Southwell,et al.  Fitting experimental data , 1969 .

[27]  C. Adjiman,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results , 1998 .

[28]  William R. Esposito,et al.  Global Optimization in Parameter Estimation of Nonlinear Algebraic Models via the Error-in-Variables Approach , 1998 .

[29]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..