OCR Text |
Show APPENDIX POWELL'S METHOD IN 2 DIMENSIONS Powell's direction-set technique was first proposed as a method for minimizing a mathematical function [Powell, 1964]. To explain Powell's method in two dimensions, first consider an objective function, F, which is a function of two variables, Xl and Xb represented by the vector, X. Now consider a search direction, represented by the nonnalized vector, S. Thus, the direction along the Xl coordinate would represented by S = (1, 0), and a search in the direction 45° to the Xl and X2 directions would result if S = (1, 1). Given an initial position, X;, and an initial search direction, S;, a new position, Xi+l is found by maximizing (or minimizing) the objective function along the line defined by the search direction. Computationally, optimization is accomplished in an iterative fashion by finding the (scalar) value of a; that produces a value of . .. " Xi+h which maximizes the objective function, F, according to This is, in fact, the general optimizing strategy for all direction-set methods. The direction-set method chosen for this research is a zero order technique, because it requires evaluation of the objective function (performance index) only. This method is a modified version of a powerful, popular, and well-understood technique known as Powell's method. Figure A 1 is a flowchart describing the modified version of Powell's method employed in this study. In discussions about search processes of this modified type, some clarification should be made regarding numbering convention. A position in the search space is still denoted by the vector, X = (Xl, X2), where Xl == S' and X2 == EA. In the modified approach, each position in the search space has two numbers associated with it, X ;J. The first index, i, corresponds to the search iteration, just as in the discussion of Powell's method. For the initial position, i = O. After maximization along the initial direction, no, the new value of i is 1. And so on. The second index, j, refers to steps within a search direction. Hence, the first position is denoted Xo.o" and the first evaluation of perfonnance is Jo.o. Following the first step along no, the j index is incremented by one, but the i index does not change until the search direction changes. So, if it takes ten steps along the initial search direction to maximize J, then the final position is denoted XO.lO' and the 16 |