THE PROBLEM OF MINIMIZING NONLINEAR FUNCTIONALS: I. LEAST SQUARES

Abstract

This chapter discusses the problem of minimizing nonlinear functionals. It presents some theoretical and practical aspects of the problem of finding a minimum point of a nonlinear functional F: En→E, n > 1. It is assumed that the functional has continuous second partial derivatives on those portions of En that are of interest. The problem of minimizing functionals subject to auxiliary constraints is considered only in so far as the functionals discussed may be regarded as already reflecting the constraining conditions. Of particular interest is the case in which the functional is a weighted sum of squares, which has been singled out for special consideration. There are two different types of problems to be considered that require different definitions of equivalent functional evaluations. In the first type, the functional is time-consuming to evaluate but its first derivatives are trivial to evaluate by comparison and are simple enough that they may be written out explicitly without serious chance of error. For these problems the number of equivalent functional evaluations is simply the number of times the functional is evaluated in the flowchart summed over all iterations.