Newton's method for convex programming and Tchebycheff approximation

Abstract

w 1. Introduction. The rationale of Newton's method is exploited here in order to develop effective algorithms for solving the following general problem: given a convex continuous function F defined on a closed convex subset K of E,~, obtain (if such exists) a point x of K such that F(x)_<_F(y) for all y in K. The manifestation of Newton's method occurs when, in the course of computation, convex hypersurIaces are replaced by their support planes. The problems of infinite systems of linear inequalities and of infinite linear programming are subsumed by the above problem, as are certain Tchebycheff approximation problems for continuous functions on a metric compactum. In regard to the latter, special attention is devoted in w167 27--30 to the feasibility of replacing a continuum by a finite subset in such a way that a discrete approximation becomes an accurate substitute for the continuous approximation.