A comparative analysis of linear fitting for non-linear functions on optimizationA case study: air pollution problems

Resumen

A very frequent problem on advanced mathematical programming models is the linear approximation of convex and non-convex non linear functions in either the constraints of the objetive function of an otherwise linear programming problem. In this paper, based on a model that has been developed for the evaluation and selection of pollutant emission control policies and standards, we shall study several ways of representing non-linear functions of a single argument in mixed integer, separable and related programming terms. Thus we shall study the approximations based on piecewise non-adjacent segmented functions. In each type of modelization we show the optimization results and problem size of using the following techniques: separable programming, mixed integer programming with Special Order Sets of type 1, linear programming with Special Order Sets of type 2 and mixed integer programming using strategies based on the quasi integrality of the binary variables. Some of the detailed considerations involved in setting up these alternative modelizations are illustrated using an air pollution abatement model. In this problem we consider the planning of pollutant emissions reductions to be imposed on each influence emitter grid square, so that the probability that the real pollutant concentration exceeds a set level is not greater than a given maximum for each polluted receptor grid square. In this paper we shall study different possibilities of sensitivity analysis in each binomial type of modelization and type of optimizing strategy.