On the Application of Iterative Methods of Nondifferentiable Optimization to Some Problems of Approximation Theory

We consider the data fitting problem, that is, the problem of approximating a function of several variables, given by tabulated data, and the corresponding problem for inconsistent (overdetermined) systems of linear algebraic equations. Such problems, connected with measurement of physical quantities, arise, for example, in physics, engineering, and so forth. A traditional approach for solving these two problems is the discrete least squares data fitting method, which is based on discrete l 2 -norm. In this paper, an alternative approach is proposed: with each of these problems, we associate a nondifferentiable (nonsmooth) unconstrained minimization problem with an objective function, based on discrete l 1 - and/or l ∞ -norm, respectively; that is, these two norms are used as proximity criteria. In other words, the problems under consideration are solved by minimizing the residual using these two norms. Respective subgradients are calculated, and a subgradient method is used for solving these two problems. The emphasis is on implementation of the proposed approach. Some computational results, obtained by an appropriate iterative method, are given at the end of the paper. These results are compared with the results, obtained by the iterative gradient method for the corresponding “differentiable” discrete least squares problems, that is, approximation problems based on discrete l 2 -norm.