最优化理论与方法 试题2007e

Final Examination of Optimal Theory and Methods

Name

Score

Notice ： Please write all the answers on the answer sheet. 1. Answer the following questions （10 points ）

（1）List four methods to solve one dimensional optimal problems. （2）List four methods to solve non-constrained multi-dimensional optimal problems.

（3）List four methods to solve constrained multi-dimensional optimal problems.

（4）List two methods to solve optimal problems by using the gradient. （5）List one method to solve optimal problems by using Hessian array. 2. Use simplex method to solve the following LP problem. （10 points ）

()321336max x x x f +−=x

s.t. ⎪⎪

⎩

⎪⎪⎨⎧=≥≤+−≤+−−≤+3 2, 1, j 0182143248

232132121j x x x x x x x x x

3. Use 0.618 method to find the minimum point with a object function of

()2

1

2−

−=x x x f . List the results of first four steps. The region is [ 0, 1.2 ]. （10 points ）

4. Use Newton method to find the minimum value of object function

()2

22

125x x f +=x . （5 points ）

( Note: take x 0＝[2，2]T ，and perform iteration once)

5. Use interior-point penalty function method to solve the following problem.

min f (x )＝ax s.t.

g (x )＝b －x ≤0

（Note ：take k r ＝10－k ）

）

（5 points ）

6. Use Kuhn-Tucker criteria to judge whether point [2，1]T and point [0，0]T are the extreme points of the following problem. （10 points ）

()()()2

22

123min −+−=x x f x

s.t. ()()()()⎪⎪⎩⎪⎪

⎨⎧≤−=≤−=≤+=≤+=0

4

2524

13

21222211x g x g x x g x x g x x x x 7．Briefly explain A * algorithm. In Fig.1, the start point is S and the end point is E of an 8 number problem. Using Misplaced(n ) as a heuristic function ，form a A * algorithm searching diagram.（10 points ）

⎥⎥⎥⎦

⎤

⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=56748321 45768132E S

Fig. 1

8．By using binary encoding genetic algorithm to solve the following optimal problem.

min f (x )＝x 1＋x 2 s.t. 8≤x 1≤12

3≤x 2≤7

It has been known that the three initial bodies （x 1, x 2）

are （10, 5）、（12, 6）

and （9, 7）, and their binary codes are （1 0 1 0， 0 1 0 1），（1 1 0 0， 0 1 1 0），（1 0 0 1， 0 1 1 1）. Please optimize the problem by using crossover (交叉)，mutation （变异） and etc. Only the first two steps should be written.

（10 points ）

9．The status function of a given system is as follows.

u x x

⎥⎦

⎤⎢⎣⎡+⎥⎦⎤⎢⎣⎡=100010

If the beginning boundary conditions are ：

0x =)0(，； the ending boundary conditions are ： x 1(1) ＋ x 2(1)－1＝0, find optimal control u *(t ) and the optimal routine x *

(t ) to have the performance functional ：()∫=1

2

dt t u J

to

be a minimum.（10 points ）

10．If there are 5 cities 1, 2, 3, 4 and 5. The distances between them are shown in Fig.2. Please use function space iteration method (函数空间迭代

法) or strategy space iteration method (策略空间迭代法) to find the

shortest routes and the shortest distances of each city to City 5.（8 points ）