1. New Multi-parent Crossover
based on Crossing Two
Segments Bounded by Selected
Parents
Atthaphon Ariyarit, Kanazaki Masahiro
Department of Aerospace Engineering,
Graduate School of System Design, Tokyo Metropolitan University
第6回進化計算学会研究会 2014/03/07
2. Outline
• Introduction
• Objective
• Overview of Popular Crossover Operators
• Blended Crossover (BLX)
• Unimodal Normal Distribution Crossover (UNDX)
• New Multi-parent Crossover Method
• Test Problems
• Single-objective optimization problems
• Multi-objective optimization problems
• Multi-objective airfoil optimization problem
• Results
• Conclusions
2
3. Introduction
• The Genetic Algorithms(GA) is popular optimization method to solve
single-objective and multi-objective optimization problem
• GA have three main operator, such as, Selection, Crossover and
mutation
• Crossover operator is the main operator for GA performance
• The popular Crossover operator is BLX and UNDX Algorithm[1]
• The GA is popular in aerospace engineering, such as, to increase the
efficiency of aircraft, to used for the navigation of aircraft, or to
reduce the weight of the aircraft weight, etc.
[1] Hajime K, Isao O and Shigenobu K. Theoretical Analysis of the Unimodal Normal Distribution Crossover for Real-coded
Genetic Algorithms. Transactions of the Society of Instrument & Control Engineers, 2002, 2(1), 187-194
3
4. Introduction
• Procedure of Genetic Algorithms
• Genetic operator such as selection, crossover and
mutation are applied to the parent to create the
offspring
• BLX and UNDX cannot always maintain
high diversity
• For real world problem, the algorithm that
can maintain higher diversity, good
convergence rate while it shows
4
5. Non-dominated Sorting Genetic Algorithm (NSGA-II)
• Step 1:
• Each individual is compared with another
randomly selected individual(niche comparison)
• The copy of the winner is placed in the mating
pool
• Step 2:
• Apply crossover rate for each individual in a
mating pool and select a parent
• Step 3:
• All non-dominated fronts of Pt and Qt are copied
to the parent population rank by rank
• Step 4:
• Stop adding the individuals in the rank when the
size of parent population is larger than the
population size
5
6. Non-dominated Sorting Genetic Algorithm (NSGA-II)
• Crowding distance assignment
• Individuals are sorted in each objective domain
• The first individual and the last individual in the rank are assigned
the crowding distance equal to infinity
• For the other individuals, the crowding distance is calculated by
𝑑𝑖 =
𝑚=1
𝑀
𝑓𝑚
𝐼𝑖+1
𝑚
− 𝑓𝑚
𝐼𝑖−1
𝑚
𝑓𝑚
𝑚𝑎𝑥
− 𝑓𝑚
𝑚𝑖𝑛
𝑖 ∈ [2, 𝑙 − 1]
• Niche comparison
𝑖 < 𝑛 𝑗 𝑖𝑓 𝑖 𝑟𝑎𝑛𝑘 < 𝑗 𝑟𝑎𝑛𝑘
𝑜𝑟 ((𝑖 𝑟𝑎𝑛𝑘 = 𝑗 𝑟𝑎𝑛𝑘) 𝑎𝑛𝑑 (𝑖 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 > 𝑗 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒))
• Between two solution with differing non-domination ranks the
solution with the better rank is preferred
• If both solutions belong to the same front, the solution which is
located in lesser crowed region is preferred.
6
7. Objective
• Development efficient crossover operator for real world problem
• Investigation of the proposed crossover operator by solving the
single-objective optimization problem, multi-objective optimization
problem and for the real world problem
7
8. BLX Operator
• BLX is a crossover operator for real code GA
• BLX created two offspring from two parents by use following equation
𝑥𝑖
(1,𝑡+1)
= 𝛾𝑖 𝑥𝑖
1,𝑡
+ 1 − 𝛾𝑖 𝑥𝑖
(2,𝑡)
𝑥𝑖
(2,𝑡+1)
= 1 − 𝛾𝑖 𝑥𝑖
1,𝑡
+ 𝛾𝑖 𝑥𝑖
(2,𝑡)
where 𝛾𝑖 = 1 + 2α 𝑢𝑖 − 𝛼 and α = 0.5
• This is called BLX-0.5
Center pointOffspring1 Offspring2Parent1 Parent2
Possible crossover region
8
dα d α d
9. UNDX operator
• UNDX crossover is a crossover operator for real code Genetic
Algorithms
• UNDX is a multi-parent crossover operator
• UNDX is operator based on the normal distribution
9
10. Algorithm of the UNDX
𝑥 𝑐
𝑥2𝑥1
𝑥3
10
• Select Parent1 (𝑥1), Parent2 (𝑥2), Parent3(𝑥3)• Find the midpoint of Parent1 and Parent2
𝑥 𝑝 =
1
2
(𝑥1 + 𝑥2)
𝑥 𝑝
• Find difference vector of these parents
d = 𝑥2 − 𝑥1
d
• Next, find the distance between the third and forth parent and the line
connecting 𝑥1 and 𝑥2 by use the following equation
D = 𝑥3 − 𝑥1 × 1 −
𝑥3 − 𝑥1
𝑇
𝑥2 − 𝑥1
𝑥3 − 𝑥1 𝑥2 − 𝑥1
D
• The childs𝑥 𝑐 is yielded by the equation
𝑥 𝑐 = 𝑥 𝑝 + 𝜀𝑑 +
𝑖=1
𝑛−1
𝜂𝑖 𝑒𝑖 𝐷
11. New Multi-parent Crossover Operator
• The disadvantage of the UNDX is very hard to find the
optimal solution close to the boundary and low diversity
• The advantage of the UNDX is good convergence rate
• Maintenance of diversity by the UNDX result is required
• Definition of proper discover area can maintain high diversity
11
12. Algorithm of the New Multi-parent Crossover
• Find the midpoint of Parent1 and Parent2
𝑥 𝑝1
=
1
2
(𝑥1 + 𝑥2)
• Find the midpoint of Parent3 and Parent4
𝑥 𝑝2
=
1
2
(𝑥3 + 𝑥4)
12
• Find difference vector of these parents
𝑑1 = 𝑥2 − 𝑥1
𝑑2 = 𝑥4 − 𝑥3
• Next, find the distance between the third and forth parent and the
line connecting 𝑥1and 𝑥2by use the following equation
𝐷1 = 𝑥3 − 𝑥1 × 1 −
𝑥3 − 𝑥1
𝑇 𝑥2 − 𝑥1
𝑥3 − 𝑥1 𝑥2 − 𝑥1
𝐷2 = |𝑥4 − 𝑥1| × (1 − (
𝑥4 − 𝑥1
𝑇(𝑥2 − 𝑥1)
𝑥4 − 𝑥1 |𝑥2 − 𝑥1|
))
• Then, find the distance between the first and second parent and the
line connecting 𝑥3and 𝑥4by use the following equation
𝐷3 = 𝑥1 − 𝑥3 × 1 −
𝑥1 − 𝑥3
𝑇 𝑥4 − 𝑥3
𝑥1 − 𝑥3 𝑥4 − 𝑥3
𝐷4 = |𝑥2 − 𝑥3| × (1 − (
𝑥2 − 𝑥3
𝑇(𝑥4 − 𝑥3)
𝑥2 − 𝑥3 |𝑥4 − 𝑥3|
))
• The childs𝑥 𝑐1
and 𝑥 𝑐2
are yielded by the equation
𝑥 𝑐1
= 𝑥 𝑝1
+ 𝜀𝑑 +
𝑖=1
𝑛−1
𝜂𝑖 𝑒𝑖 𝐷
𝑥 𝑐2
= 𝑥 𝑝1
+ 𝜀𝑑 +
𝑖=1
𝑛−1
𝜂𝑖 𝑒𝑖 𝐷
• The childs𝑥 𝑐3
and 𝑥 𝑐4
are yielded by the equation
𝑥 𝑐3
= 𝑥 𝑝2
+ 𝜀𝑑 +
𝑖=1
𝑛−1
𝜂𝑖 𝑒𝑖 𝐷
𝑥 𝑐4
= 𝑥 𝑝2
+ 𝜀𝑑 +
𝑖=1
𝑛−1
𝜂𝑖 𝑒𝑖 𝐷
• Select Parent1 (𝑥1), Parent2 (𝑥2), Parent3 (𝑥3) and Parent4 (𝑥4)
𝑥1
𝑥3
𝑥2
𝑥4
𝑥 𝑝1
𝑥 𝑝2
𝑑1
𝐷2
𝐷1
𝐷3
𝐷4
𝑥 𝑐3
𝑥 𝑐4
𝑥 𝑐2
𝑥 𝑐1
14. Single-Objective Test Function
• Rosenbrock
𝑓𝑅𝑜𝑠𝑒𝑛𝑏𝑟𝑜𝑐𝑘 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑥
=
𝑖=2
20
{100 𝑥1 − 𝑥𝑖
2 2
+ (𝑥𝑖 − 1)2
}.
−2.048 ≤ 𝑥𝑖 ≤ 2.048, 1 ≤ 𝑖 ≤ 20
• Exact solution is 0
• For Single-Objective optimization use 300
population and 500 generation
14
15. Multi-objective Test Function
• ZDT1
• convex
Minimize: 𝑓1(x) = 𝑥1
Minimize: 𝑓2 x = 1 −
𝑥1
𝑔 𝑥
g x = 1 + 9
𝑖=2
𝑛
𝑥𝑖
𝑛 − 1
0 ≤ 𝑥𝑖 ≤ 1, 1 ≤ 𝑖 ≤ 30
15
16. Multi-objective Test Function
• ZDT2
• Non-convex
Minimize: 𝑓1(x) = 𝑥1
Minimize: 𝑓2(x) = 1 −
𝑥1
𝑔(𝑥)
2
g x = 1 + 9
𝑖=2
𝑛
𝑥𝑖
𝑛 − 1
0 ≤ 𝑥𝑖 ≤ 1, 1 ≤ 𝑖 ≤ 30
16
• For Multi-Objective optimization use 300 population and 200 generation
17. Airfoil Optimization Problem
• Created airfoil my NACA 4 digit equation
• Maximum: Cl
• Minimize: Cd
• Subject to:
• 0.05 ≤ y1 ≤ 0.13; the thickness in upper side of airfoil (dv1)
• 0.05 ≤ y2 ≤ 0.13; the thickness in lower side of airfoil (dv2)
• 0 ≤ y3 ≤ 0.04; the maximum camber of airfoil (dv3)
• 0.2 ≤ y4 ≤ 0.6; the location of the maximum camber (dv4)
• Subject to:
• 0.6 ≤ cl ≤ 1.8 (dv5)
• Use 100 population and 70 generation
• Use Reynolds’ Number 106
17
19. Results-Sphere Problem
• Sphere Problem
• This graph show the best solution in each generation
• The results show proposed method is better solution than BLX and
UNDX
19
20. Results-Sphere Problem
• This graph show the average solution in each generation
• This graph can explained the proposed method can maintain higher
diversity than BLX and UNDX
20
22. Results-Sphere
22
• This graph show the close up view (from generation 1 to 200) of
design parameter of the best solution in each generation that can
show the convergence rate
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
New multi-parent crossover method
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
UNDX
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
BLX
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
23. Results-Rastrigin Problem
• Rastrigin Problem
• This graph show the best solution in each generation
• The results show proposed method is better solution than BLX and
UNDX
23
24. Results-Rastrigin Problem
• This graph show the average solution in each generation
• This graph can explained the proposed method can maintain higher
diversity than BLX and UNDX
24
26. Results-Rastrigin
26
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
New multi-parent crossover method
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
UNDX
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
BLX
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
• This graph show the close up view (from generation 1 to 200) of
design parameter of the best solution in each generation that can
show the convergence rate
27. Results-Rosenbrock Problem
• Rosenbrock Problem
• This graph show the best solution in each generation
• The results show proposed method is better solution than BLX and
UNDX
27
30. Results-Rosenbrock
30
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
New multi-parent crossover method
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
UNDX
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200
BLX
x_1 x_2 x_3 x_4 x_5
x_6 x_7 x_8 x_9 x_10
x_11 x_12 x_13 x_14 x_15
x_16 x_17 x_18 x_19 x_20
• This graph show the close up view (from generation 1 to 200) of
design parameter of the best solution in each generation that can
show the convergence rate
31. Results-ZDT1
• This graph show the Pareto solution of ZDT1 problem
• This Pareto solution show the proposed method can get better
solution than BLX and UNDX
31
32. Results-ZDT1
• Metrices of ZDT1
• These graphs show the proposed method have good convergence
rate for ZDT1 problem
32
Hypervolume Maximum Spread Spacing
33. Results-ZDT2
• This graph show the Pareto solution of ZDT2 problem
• This Pareto solution show the proposed method can get better
solution than BLX and UNDX
33
34. Results-ZDT2
• Metrices of ZDT2
• These graphs show the proposed method have good convergence
rate for ZDT1 problem
34
Hypervolume Maximum Spread Spacing
35. Results-Airfoil Optimization Problem
• This graph show the Pareto solution of airfoil optimization problem
• These Pareto solution shows the proposed method can find the
optimal solution similar to the BLX algorithm and better than UNDX
35
30th generation 70th generation
37. Results-Airfoil Optimization Problem
37
• PCP plot of the Pareto solution at 70th generation
• The plot shown the propose method have the solution similar to BLX
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
dv1 dv2 dv3 dv4 dv5 Cd Cl
BLX
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
dv1 dv2 dv3 dv4 dv5 Cd Cl
UNDX
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
dv1 dv2 dv3 dv4 dv5 cd cl
Proposed method
38. Results-Airfoil Optimization Problem
• Left figure show the airfoil comparison and the right figure show the
Cp(Pressure Distribution) comparison at design point（Cl=0.8 at 70th
generation）
• The plot shown the propose method have the solution similar to BLX
38
39. Conclusion
• Successful to development of new multi-parent crossover
method
• The proposed method successfully find out the better
solution of every test function compared with existent
crossover method
• The multi-objective test function are better compared with
other crossover method
• For the airfoil optimization problem the searching areas are
similar because the parameterizations is relatively smile so
the investigation by more complex problem is need
• This method can maintain higher diversity
39
