A Lagrangian Method for Multivariate Continuous Chebyshev Approximation Problems

Abstract

Let X ⊂ R^N be a Cartesian product of closed intervals, and let f: X × Rⁿ → R be a twice continuously differentiable function of its parameters. We are concerned here with the numerical solution of the problem

$$\text{find}\,\text{a} \in \text{R}^\text{n} \,\text{to}\,\text{minimize}||\text{f}( \bullet ,\text{a})||$$

(1.1)

where the norm is the Chebyshev norm on X. This problem, particularly in the case when f is an affine function of a, has received a great deal of attention, and a number of algorithms have been proposed for its solution. The ascent algorithms of Remes are perhaps the best known, and variants of these for linear problems have been proposed by a number of authors, see e.g. [2] [3] [4], [13]. Methods of descent type have also been suggested (e.g. [11], [12]), but arguably the best currently available methods for both general linear and nonlinear problems are those of two-phase type: the second phase is a fast locally convergent method, usually Newton’s method applied to the first order necessary conditions for a local solution; the first phase supplies a good initial approximation, for example through the solution of a discretized problem or by the application of a robust slowly convergent method. Methods of this type have been given, for example, in [1], [6], [7], [8], [14].