Linear fitting of non-linear functions in optimization. A case study: Air pollution problems

A problem on mathematical programming is the linear approximation of nonlinearities in the constraints or in the objective function of a linear programming problem. In this paper, we compare the representation of nonlinear functions of a single argument by approximations based on piecewise constant, piecewise adjacent, piecewise non-adjacent additional and piecewise non-adjacent segmented functions. In each modelization we show the problem size and the results of 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. The considerations involved in these alternative modelizations are illustrated using an air pollution abatement model. Also we study the possibilities of the sensitivity analysis in each type of modelization and optimizing strategy.