Nonlinear Chebyshev approximation subject to constraints

Abstract

The theoretical and practical aspects of Chebyshev approximation problems where the unknown parameters are required to satisfy additional inequality or equality constraints have received a great deal of attention in recent years. Surveys of the work of various authors on this class of problem have been given by Taylor [lo] and Lewis [9]. Characterization theorems for fairly general linear problems of this type are given by Laurent [8] and Andreassen [I], while certain classes of nonlinear problems have been treated by Hoffman [6, 71 and Gislason and Loeb 151. It is the purpose of this paper to investigate the extent to which the charac- terization results for the general linear case can be extended to the nonlinear case, while imposing a minimum of restrictions on the problem. Necessary conditions and sufficient conditions of “zero in the convex hull” type are presented for local best approximations, as defined below. The theorems also generalize similar results for the nonlinear problem without constraints (see, for example, [ll]). We remark that, although the results are fomulated for approximation in a finite interval [a, b] of the single real variable X, no use is made of this restriction in the proofs, and hence the theorems are valid for multivariate approximation. Let f, $(-, u) E C[a, b] be given functions, where u = (01~ , 01~ ,..., ol,)r, and let Sz C lFP be given. Then the basic approximation problem with which we will be concerned can be stated: find a E Q to minimize where