# Example 1[24]

m a x x ≥ 0 F ( x , y )

where y solves m a x y ≥ 0 f ( x , y )

subject to x 2y x 2y 2x y x + 2y x + 2y

y_{2 }

+y_{3 }≤ 1

2y_{2 }

+0.5y_{3 }≤ 1

2 y

0.5y_{3 }≤ 1

4y_{3 }

=

x + 3y

x≥0

max F (x, y)

=

x

3y

y≥0

where y solves max f(x, y)

=

8x_{1 }

4x_{2 }+ 4y_{1 }

40y_{2 }

=

x_{1 }+ 2_{2 }+ y_{1 }+ y_{2 }+ 2_{y}3

# Example 2[13]

≤ ≤6 ≤ 21 ≤ 38 ≤ 18 10

subject to y_{1}+ 2x_{1 }y_{1}+ 2x_{2 }+ 2y_{1 }

# Example 3[4]

x≥0

max F (x, y)

=

y≥0

subject to x_{1}+ max f(x, y)

2x_{2 }=

subject to y_{1}+ 4x_{1 }2y_{1}+ 4x_{2 }+ 4y_{1 }

y_{2 }4y_{2 }2y

2

8x_{1 }+ 4x_{2 }

y_{3 }≤ 1.3

2y_{1 }

y_{2 }

+y_{3 }≤ 1 +y_{3 }≤ 2 y_{3 }≤ 2

4y_{1 }+ 40y_{2 }+ 4y_{3 }

2y_{3 }

The compare of the results thorough 500 generations by the algorithms in the paper and the results in the references is as follows:

c m MP P

(x, y)

F

f

(x, y)

F

1

50

0.7

0.15

, (15.959 10.972)

48.875

16.957

, (16 11)

49

2

100

0.6

0.2

(0, 0.899, 0, 0.6, 0.394)

29.172

3.186

(0, 0.9, 0, 0.6, 0.4)

29.2

3

100

0.6

0.3

(0.533, 0.8, 0, 0.197, 0.8)

18.544

1.797

(0.2, 0.8, 0, 0.2, 0.8)

18.4

parameters

result in the paper

result in the references

mutation program-

# Notes:

# M

denotes

the

population

scale,

P c

denotes

crossover

probability,

P m

denotes

lower-level

and

upper-level

the

of

value

function

probability. F and f are the ming problem, respectively.

objective

From the numerical result, the results by the method in this paper accord with the results in the references. So the method is very efficient. In addition, the experiments show: the scale of the population and the rate of the mutation have some influence on the efficiency and convergence rate of the method, but crossover operator has little influence on them. For example, the rate of convergence will slow down if the size of the population is larger, and it is benefit for the rate of the convergence to choose the lager rate of mutation.

6

f 17 3.2 1.8