TAILIEUCHUNG - Numerical Methods in Engineering with Python Phần 10

Python, trái lại, có một cách rất đặc biệt để tạo khối lệnh, đó là thụt các câu lệnh trong khối vào sâu hơn (về bên phải) so với các câu lệnh của khối lệnh cha chứa nó. | 385 Powell s Method a b Figure . The method of Powell Figure a illustrates one typical cycle of the method in a two-dimensional design space n 2 . We start with point x0 and vectors v1 and v2. Then we find the distance s1 that minimizes F x0 sv1 finishing up at point x1 X0 s1v1. Next we determine s2 that minimizes F x1 sv2 which takes us to x2 X1 s2v2. The last search direction is v3 x2 x0. After finding s3 by minimizing F x0 sv3 we get to x3 x0 s3v3 completing the cycle. Figure b shows the moves carried out in two cycles superimposed on the contour map of a quadratic surface. As explained before the first cycle starts at point P0 and ends up at P3. The second cycle takes us to P6 which is the optimal point. The directions P0 P3 and P3 P6 are mutually conjugate. Powell s method does have a major flaw that has to be remedied - if F x is not a quadratic the algorithm tends to produce search directions that gradually become linearly dependent thereby ruining the progress toward the minimum. The source of the problem is the automatic discarding of v1 at the end of each cycle. It has been suggested that it is better to throw out the direction that resulted in the largest decrease of F x a policy that we adopt. It seems counterintuitive to discard the best direction but it is likely to be close to the direction added in the next cycle thereby contributing to linear dependence. As a result of the change the search directions cease to be mutually conjugate so that a quadratic form is not minimized in n cycles any more. This is not a significant loss because in practice F x is seldom a quadratic. Powell suggested a few other refinements to speed up convergence. Because they complicate the book keeping considerably we did not implement them. 386 Introduction to Optimization powell The algorithm for Powell s method is listed here. It utilizes two arrays df contains the decreases of the merit function in the first n moves of a cycle and the matrix u stores .

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