f(X )=1.0-4.25+2(1.0)+2(4.25)+18.0625=25.3125

步骤5：因为f(X )< f(X )，并且得到新的顶点

            X = ，X = ，X = 

步骤6：为了收敛，我们计算Q值为

         Q= =26.1

因为Q> ，我们进行下一个迭代。

此过程可以继续进行，直到满足指定的收敛。当收敛满足要求时，最新单形X 的重心可以采取最佳值。

6.7 POWELL'S METHOD

Powell's method is an extension of the basic pattern search method. It is the most widely used direct search method and can be proved to be a method of conjugate  directions [6.7]. A conjugate directions method will minimize a quadratic function in a finite number of steps. Since a general nonlinear function can be approximated reasonably well by a quadratic function near its minimum, a conjugate directions method is expected to speed up the convergence of even general nonlinear objective functions. The definition, a method of generation of conjugate directions and the property of quadratic convergence are presented in this section.

6.7.1 Conjugate Directions

Definition: Conjugate Directions Let A = [A] be an n x n symmetric matrix. A set of n vectors (or directions) {S ,} is said to be conjugate (more accurately A-conjugate) if                            S AS =0 for all   i≠j， i=1,2, …,n，j=1,2, …,n               (6.27)

It can be seen that orthogonal directions are a special case of conjugate directions (obtained with [A] = [I] in Eq. (6.27)).

Definition: Quadratically Convergent Method If a minimization method,using exact arithmetic, can find the minimum point in n steps while minimizing a quadratic function in n variables, the method is called a quadratically convergent method.

Theorem 6.1 Given a quadratic function of n variables and two parallel hyperplanes 1 and 2 of dimension k < n. Let the constrained stationary points of the quadratic function in the hyperplanes be X and X , respectively. Then the line joining X and X  is conjugate to any line parallel to the hyperplanes.

Proof: Let the quadratic function be expressed as

            Q(X)=  X AX+B X+C

The gradient of Q is given by

             Q(X)=AX+B

and hence

             Q(X )- Q(X )=A(X -X )

If S is any vector parallel to the hyperplanes, it must be orthogonal to the gradients Q(X ) and ▽Q(X ).Thus

             S  Q(X )= S A X + S B=0                 (6.28)

             S  Q(X )= S A X + S B=0                (6.29)

By subtracting Eq. (6.29) from Eq. (6.28), we obtain

             S A (X - X )=0                             (6.30)

Hence S and (X - X ) are A — conjugate.

The meaning of this theorem is illustrated in a two-dimensional space in Fig. 6.8.

 If X  and X are the minima of Q obtained by searching along the direction S from two different starting points X  and X , respectively, the line  (X - X )will be conjugate to the search direction S.

Theorem 6.2 If a quadratic function

 Q(X)=  X AX+B X+C                   (6.31)

is minimized sequentially, once along each direction of a set of n mutually conjugate directions, the minimum of the function Q will be found at or before the nth step irrespective of the starting point.