Applications of Convex Separable Unconstrained Nondifferentiable Optimization to Approximation Theory

In this chapter, the data fitting problem is considered, 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. Also, the problem of numerical solution of systems of nonlinear algebraic equations with even powers and systems of nonlinear equations, defined by convex differentiable functions, is studied. A traditional approach for solving these problems is the least squares data fittingLeast squares data fitting, which is based on discrete $$\ell _2$$ ℓ 2 -norm. An alternative approach is applied in this chapter: with each of these problems, nondifferentiable (nonsmooth) unconstrained minimization problems are associated, with objective functions based on discrete $$\ell _1$$ ℓ 1 - and $$\ell _\infty $$ ℓ ∞ -norms, respectively, that is, these norms are used as proximity criteria. In other words, the problems under consideration are solved by minimizing the residuals using these two norms. Some computational results, obtained by an appropriate iterative method, are presented at the end of the chapter. These results are compared with the results, obtained by the iterative gradient method for the corresponding “differentiable” least squares problems.