# [Bil09] Estimation of spatial influence models using mixed-integer programming

**Revue Internationale avec comité de lecture : **
*Journal Int. J. of Environmental Informatics*,

vol. 14(1),

pp. 31-40,
2009, (

doi:

10.3808/jei.200900151)

**Mots clés: ** spatial influence model, nonlinear least-squares estimation, mixed-integer programming, computational experiments

**Résumé: **
Estimation of ecosystem models is an important task and many studies have been carried out on the problem. However, estimating some models may be difficult. Here, we want to estimate two nonlinear spatial influence models by using the classical least-squares method, and that requires the solution of difficult nonlinear optimization problems. The aim of this paper is to show that both models can be efficiently estimated using mathematical programming. The estimation problems are first formulated as nonlinear optimization problems which are then transformed into convex quadratic mixed-integer programs. The transformation is based on the discretization of some variables and on the linearization of the product of a Boolean variable with a real variable. The approach which allows to find the best estimation with a certain precision and in the least-squares framework, is interesting for several reasons: the definition of the mathematical programs is relatively simple, they are easy to implement using a mathematical programming language together with a quadratic mixed-integer programming software, and computational experiments carried out on large sets of simulated data show the excellent performance of the approach. Moreover, the ideas underlying the method can be used for other difficult least-squares estimations. These results suggest that mixed-integer programming may be an efficient tool for practitioners and researchers in environmental modeling.
Keywords: spatial influence model, nonlinear least-squares estimation, mixed-integer programming, computational experiments