Approximations with Respect to ℓ1 and ℓ∞-Norms: An Application of Convex Separable Unconstrained Nondifferentiable Optimization

Abstract

In this chapter, we consider the data fitting problem, that is, the problem of approximating a function of several variables given by tabulated data and the analogous problem for inconsistent systems of linear equations. A traditional approach for solving these two problems is the least squares data fitting which is based on discrete ℓ2-norm. We apply an alternative approach: with each of these problems we associate a nondifferentiable (nonsmooth) unconstrained minimization problem with an objective function based on discrete ℓ1- and ℓ∞-norm, that is, we use these norms as proximity criteria. In other words, we solve the problems under consideration by minimizing the residual using these two norms. Some computational results obtained by an appropriate iterative method are given at the end of the chapter. These results are compared with ones obtained by the iterative gradient method for the corresponding “differentiate” least squares problems.