The method of steepest descent for non-linear minimization problems

Abstract

The problem is that of minimizing a function G(x 1,..., x n) of n real variables and includes, as a special case, that of solving a set of simultaneous equations f 1(x 1,.., x n)=0 (i = 1,2,..., m) (eqn. 1), since the function G(x 1,..., x m)= Sigma i = 1mf 12 has a minimum at a solution of eqn. (1). The method, which is not new [Abstr. 1945A01647], leads to a practical process for the approximate evaluation of a stationary point of G. An outline of a proof of convergence is given. This is elementary and gives a weak result. Previously, a proof had been given, in the linear case only, by Temple [Abstr. 1939A01498].