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The Robustness of Hybrid Algorithms in Multimodal Functions Optimization. 姓名: 何怡偉 元智工業工程與管理博士班. The characteristic of Nelder-Mead simplex method. A simple direct search technique. Easy to use and does not need the derivatives of the function. - PowerPoint PPT Presentation
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The Robustness of Hybrid Algorithms
in Multimodal Functions Optimization
姓名 : 何怡偉元智工業工程與管理博士班
The characteristic of Nelder-Mead simplex method
A simple direct search technique.
Easy to use and does not need the derivatives of the function.
Very sensitive to the choice of initial points and not guaranteed to attain the global optimum.
The characteristic of evolutionary computation technique
Eventually locate the desired solution.
The high computational cost of the slow convergence rate.
Do not utilize much local information to determine a most promising search direction.
Nelder-Mead Simplex OperationsA
B CD
G
H
E
J
The Structure of hybrid NM-GA
N elites
Modified Simplex
N
1
N+1
Best
Worst
Ranked Population New Population
Selection 100% Crossover 30% Mutation
GA Reproduction
N+1 from
simplex design
N+1 from random
generation
N
1
N+1
The Structure of hybrid NM-PSO
N elites
Modified Simplex
SelectionMutation for global best
Velocity update
Modified PSO Method
Ranked Population
Updated Population
Best
Worst
Initialize Population
N
1
2N
N
1
2N2N
fromrandom
generation
N+1from
simplexdesign
The populations design for the five algorithms
NM GA PSO NM-GA NM-PSO
population size N+1 5N 5N 2N+2 3N+1
The surface plot of the Himmelblau function
The contour plot of the Himmelblau function
-6 -4 -2 0 2 4 6-6
-4
-2
0
2
4
6
global minimum
X1-axis
X2-
axis
0x M E T H O D x
xS
F
FS
ITE
ITES
O P T I M A L 3 , 2 0
N M 2 . 9 9 9 9 , 2 . 0 0 0 1 1 . 8 8 3 7 e - 0 7 1 0 6
G A - 0 . 6 0 9 4 , 1 . 0 1 6 0
( 3 . 2 5 7 7 , 2 . 6 9 7 1 )
3 . 0 1 8 8
( 2 . 7 5 5 4 )
5 1 . 2
( 4 . 8 2 5 9 )
)0 ,0( P S O 2 . 4 7 9 4 , 1 . 7 3 0 7
( 1 . 8 5 9 5 , 1 . 2 9 7 3 )
0 . 4 9 9 1
( 1 . 1 5 1 4 )
7 1 . 3
( 2 . 2 1 3 6 )
N M - G A 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
2 . 6 7 0 0 e - 0 8
( 0 . 0 0 0 0 )
3 6 . 2
( 2 . 8 9 8 3 )
N M - P S O 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
2 . 8 0 4 0 e - 0 8
( 0 . 0 0 0 0 )
3 9 . 3
( 1 1 . 1 5 6 0 )
N M 3 . 0 0 0 0 , 2 . 0 0 0 0 4 . 0 8 0 6 e - 0 8 3 4
G A 3 . 1 7 4 4 , 0 . 8 5 3 8
( 0 . 2 8 0 9 , 1 . 8 4 5 6 )
0 . 4 5 1 3
( 0 . 7 2 6 7 )
5 0 . 6
( 9 . 9 1 3 0 )
)1 ,1( P S O 2 . 4 2 1 3 , 2 . 1 1 2 8
( 1 . 8 3 0 0 , 0 . 3 5 6 8 )
0 . 3 4 8 7
( 1 . 1 0 2 7 )
7 0 . 9
( 4 . 9 3 1 8 )
N M - G A 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
2 . 9 7 8 7 e - 0 8
( 0 . 0 0 0 0 )
3 3 . 7
( 5 . 2 7 1 5 )
N M - P S O 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
2 . 9 0 9 4 e - 0 8
( 0 . 0 0 0 0 )
3 7 . 8
( 5 . 7 1 1 6 )
N M - 3 . 7 6 3 5 , - 3 . 2 6 6 1 7 . 3 6 7 3 3 4
G A 0 . 7 6 2 0 , 0 . 6 6 5 6
( 3 . 2 8 5 4 , 2 . 5 2 8 5 )
2 . 2 3 4 2
( 2 . 3 0 0 5 )
5 2 . 4
( 6 . 0 9 5 5 )
)3 ,3( P S O - 1 . 2 6 7 0 , - 0 . 9 1 4 2
( 3 . 3 7 2 8 , 2 . 8 5 9 9 )
4 . 4 4 4 4
( 2 . 6 4 2 7 )
7 7 . 4
( 4 . 7 1 8 8 )
N M - G A 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
3 . 3 8 4 9 e - 0 8
( 0 . 0 0 0 0 )
3 7
( 4 . 6 4 2 8 )
N M - P S O 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
3 . 6 7 0 0 e - 0 8
( 0 . 0 0 0 0 )
4 5 . 5
( 9 . 0 9 5 2 )
Computational results on the Himmelblau function
0x M E T H O D x
xS
F
FS
ITE
ITES
N M 3 . 5 8 1 5 , - 1 . 8 2 0 8 1 . 5 0 4 4 3 1
G A 1 . 5 5 4 6 , 0 . 4 2 8 1
( 3 . 0 0 5 0 , 2 . 4 0 6 0 )
1 . 7 9 8 3
( 1 . 3 1 0 5 )
4 7 . 5
( 5 . 5 4 2 8 )
)1 ,3( P S O 3 . 2 9 0 7 , 0 . 0 8 9 6
( 0 . 3 0 6 5 , 2 . 0 1 3 7 )
0 . 7 5 2 2
( 0 . 7 9 2 9 )
7 3
( 4 . 8 5 3 4 )
N M - G A 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
4 . 0 5 3 1 e - 0 8
( 0 . 0 0 0 0 )
4 2 . 6
( 1 2 . 2 1 2 9 )
N M - P S O 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
3 . 3 2 6 3 e - 0 8
( 0 . 0 0 0 0 )
3 6 . 9
( 7 . 0 7 8 1 )
N M - 2 . 7 8 7 1 , 3 . 1 2 8 2 3 . 4 8 7 1 3 1
G A 1 . 2 8 2 6 , 0 . 9 3 4 9
( 3 . 0 5 3 3 , 2 . 3 1 2 5 )
1 . 7 3 5 0
( 2 . 4 2 7 2 )
5 0 . 9
( 6 . 6 0 7 2 )
)2 ,2( P S O 2 . 4 2 1 3 , 2 . 1 1 2 8
( 1 . 8 3 0 0 , 0 . 3 5 6 8 )
0 . 3 4 8 7
( 1 . 1 0 2 7 )
7 2 . 4
( 4 . 1 4 1 9 )
N M - G A 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
3 . 5 3 3 6 e - 0 8
( 0 . 0 0 0 0 )
3 9 . 2
( 6 . 6 1 3 1 )
N M - P S O 3 . 0 0 0 0 , 2 . 0 0 0 0
( 0 . 0 0 0 0 , 0 . 0 0 0 0 )
3 . 3 5 3 0 e - 0 8
( 0 . 0 0 0 0 )
3 6 . 2
( 1 2 . 2 5 4 7 )
A surface plot of the peaks function
A contour plot of the peaks function
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
X1-axis
X2-
axis
global maximum
global minimum
0x M E T H O D x
xS
F
FS
ITE
ITES
O P T I M A L - 0 . 0 0 9 3 , 1 . 5 8 1 4 8 . 1 0 6 2
N M - 0 . 0 0 9 3 , 1 . 5 8 1 4 8 . 1 0 6 2 3 6
G A - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
3 3 . 7
( 5 . 6 1 8 4 )
)0,0( P S O - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
6 6 . 1
( 4 . 6 0 5 6 )
N M - G A - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
3 2 . 6
( 3 . 2 0 4 2 )
N M - P S O - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
3 0 . 7
( 2 . 9 0 7 8 )
N M - 0 . 0 0 9 3 , 1 . 5 8 1 4 8 . 1 0 6 2 3 4
G A - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
2 9 . 1
( 4 . 7 9 4 7 )
)1,0( P S O - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
6 7 . 6
( 3 . 1 3 4 0 )
N M - G A - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
2 8
( 1 . 4 9 7 0 )
N M - P S O - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
3 3 . 7
( 8 . 5 3 8 1 )
N M - 0 . 4 6 0 1 , - 0 . 6 2 9 2 3 . 7 7 6 6 2 9
G A - 0 . 0 5 4 5 , 1 . 3 6 0 3
( 0 . 1 4 2 5 , 0 . 6 9 9 0 )
7 . 6 7 3 2
( 1 . 3 6 9 1 )
2 7 . 5
( 4 . 5 2 7 7 )
)1,1( P S O 0 . 0 6 9 4 , 0 . 3 7 9 9
( 0 . 6 7 5 3 , 1 . 0 6 1 6 )
5 . 4 7 1 6
( 2 . 2 6 8 6 )
6 6 . 8
( 4 . 1 4 0 2 )
N M - G A - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
3 5 . 7
( 5 . 6 9 7 0 )
N M - P S O - 0 . 0 0 9 3 , 1 . 5 8 1 4
( 0 . 0 0 0 , 0 . 0 0 0 )
8 . 1 0 6 2
( 0 . 0 0 0 )
3 7 . 9
( 4 . 5 8 1 4 )
Computational Results on the peak function for
searching the global maximum
0x METHOD x
xS
F
FS
ITE
ITES
NM 1.2858, -0.0048 3.5925 32
GA -0.0093, 1.5814
(0.000, 0.000)
8.1062
(0.000)
28.2
(3.4897)
)0,1( PSO 0.5087, 0.9469
(0.6687, 0.8191)
6.3007
(2.3309)
65.2
(7.3606)
NM-GA -0.0093, 1.5814
(0.000, 0.000)
8.1062
(0.000)
32.8
(3.4577)
NM-PSO -0.0093, 1.5814
(0.000, 0.000)
8.1062
(0.000)
34.8
(2.7809)
Summary
The proposed hybrid NM-GA and NM-PSO are indeed effective, reliable, efficient and robust at locating best-practice optimum solutions for multimodal functions.
Stochastic Optimization
,,,, Minimize 21 kxxxfy
![HoME: a Household Multimodal Environmenting learning algorithms [28, 29, 30], while HoME additionally aims to provide a unified platform for multimodal learning in a realistic context](https://img.pdfslide.net/doc/110x75/5ebed1580ed3a37c686dc726/home-a-household-multimodal-environment-ing-learning-algorithms-28-29-30-while.jpg)
