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

Abstract

A very frequent problem in advanced mathematical programming models is the linear approximation of convex and non-convex non-linear functions in either the constraints or the objective 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 constant, piecewise adjacent, piecewise non-adjacent additional and piecewise non-adjacent segmented functions. In each type of modelization we show the problem size and optimization results of using the following techniques: separable programming, mixed integer programming with Special Ordered Sets of type 1, linear programming with Special Ordered Sets of type 2 and mixed integer programming using strategies based on the quasi-integrality of the binary variables.