Sequential Minimization for Sum of Squares of Unknown Functions

摘要

In practical situations is a frequent need to find an optimal point in a given domain such that the optimal point minimizes an objective function which is in the form as a sum of squares of unknown functions. Assume that the objective function is denoted by F(chi), where chi = (chi sub 1 , chi sub 2 ,..., chi/sub n/)/sup T/ with T being the symbol indicating the transpose of a vector or a matrix. Then F(x) is the sum of squares of functions f/sub i/(x), i=1,2,...,m. Each function f/sub i/(x) actually represents a special characteristic. The analytical expression of f/sub i/(x) is not known, and its value is normally numerically calculated. The goal is to find chi = chi sub 0 such that F(chi sub 0 ) attains the minimum of F(chi) for chi in a predetermined domain D. A sequential scheme is presented in this work with some numerical tests provided to illustrate the feasibility of the sequential scheme. To find each f/sub i/(x) value for given x is very costly in actual cases. Therefore our goodness criteria for a scheme is to have the number of the sequentially selected points as less as possible under the condition that the end point of the sequence is within certain accuracy limit to chi sub 0 . (ERA citation 10:009423)