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第6回進化計算学会研究会

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Tokyo conference

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  13. 13. Single-Objective Test Function • Sphere 𝑓𝑆𝑝ℎ𝑒𝑟𝑒 𝑥 = 𝑖=1 20 𝑥𝑖 2 . −5.12 ≤ 𝑥𝑖 ≤ 5.12, 1 ≤ 𝑖 ≤ 20 • Exact solution is 0 • Rastrigin 𝑓𝑅𝑎𝑠𝑡𝑟𝑖𝑔𝑖𝑛 𝑥 = 200 + 𝑖=1 20 {𝑥𝑖 2 −10cos(2𝜋𝑥𝑖)}. −5.12 ≤ 𝑥𝑖 ≤ 5.12, 1 ≤ 𝑖 ≤ 20 • Exact solution is 0 13
  14. 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. 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. 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. 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
  18. 18. NACA 4 digit Airfoil • Equation for a symmetrical 4-digit NACA airfoil 𝑦𝑡 = 𝑡 0.2 𝑐 0.2969 𝑥 𝑐 − 0.1260 𝑥 𝑐 − 0.3516 𝑥 𝑐 2 + 0.2843 𝑥 𝑐 3 − 0.1015 𝑥 𝑐 4 • Equation for a cambered 4-digit NACA airfoil • 𝑦𝑐 = 𝑚 𝑥 𝑝2 2𝑝 − 𝑥 𝑐 , 0 ≤ 𝑥 ≤ 𝑝𝑐 𝑚 𝑐−𝑥 (1−𝑝)2 1 + 𝑥 𝑐 − 2𝑝 , 𝑝𝑐 ≤ 𝑥 ≤ 𝑐 • The example of airfoil when use t=0.12, m=0.04 and p=0.4 18 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 1.2
  19. 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. 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
  21. 21. Results-Sphere 21 • This graph show the design parameter of the best solution in each generation that can show the convergence rate -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 100 200 300 400 500 600 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 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 100 200 300 400 500 600 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 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 100 200 300 400 500 600 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
  22. 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. 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. 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
  25. 25. Results-Rastrigin 25 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 100 200 300 400 500 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 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 100 200 300 400 500 600 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 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 0 100 200 300 400 500 600 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 design parameter of the best solution in each generation that can show the convergence rate
  26. 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. 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
  28. 28. Results-Rosenbrock Problem • This graph show the average solution in each generation 28
  29. 29. Results-Rosenbrock 29 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 100 200 300 400 500 600 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 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 100 200 300 400 500 600 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 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 100 200 300 400 500 600 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 design parameter of the best solution in each generation that can show the convergence rate
  30. 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. 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. 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. 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. 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. 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
  36. 36. Results-Airfoil Optimization Problem • Metrices of Airfoil Optimization Problem 36 Hypervolume Maximum Spread Spacing
  37. 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. 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. 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

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