Applications of convex separable unconstrained nonsmooth optimization to numerical approximation with respect to l1 - and l∞-norms

Abstract

In this paper, 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 nonsmooth (nondifferentiable) unconstrained minimization problem with an objective function based on discrete l1 - and l∞, 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. The emphasis is on implementation of these techniques. A subgradient method is used to solve the two problems and respective subgradients are calculated. Some computational results are given at the end of the paper. These results are compared with ones obtained by the iterative gradient method for the corresponding “smooth” (“differentiable”) least squares problems.