1. for Solving Hierarchical Decomposable Functions Dept. of Computer Engineering, Chulalongkorn Univ., Bangkok, Thailand Simultaneity Matrix Assoc. Prof. Prabhas Chongstitvatana

2. 1. Building blocks 2. Simple Genetic Algorithm 3. Test Functions 3.1 Additively Decomposable Functions (ADFs) 3.2 Hierarchically Decomposable Functions (HDFs) 4. Probabilistic Model Building Genetic Algorithms 5. Simultaneity Matrix 6. Simultaneity-Matrix-Construction (SMC) Algorithm 7. Partitioning (PAR) Algorithm 8. Experiment Setting 9. A Comparison to the Bayesian Optimization Algorithm 10. Conclusions and Recent Work A G E N D A

3. x = 11100f(x) = 28 x = 11011f(x) = 27 x = 10111f(x) = 23 x = 10100f(x) = 20 --------------------------- x = 01011f(x) = 11 x = 01010f(x) = 10 x = 00111f(x) = 7 x = 00000f(x) = 0 Induction 1 * * * * (Building Block)

4. x = 11111f(x) = 31 x = 11110f(x) = 30 x = 11101f(x) = 29 x = 10110f(x) = 22 --------------------------- x = 10101f(x) = 21 x = 10100f(x) = 20 x = 10010f(x) = 18 x = 01101f(x) = 13 1 * * * * (Building Block) Reproduction

5. x = 11111f(x) = 31 x = 11110f(x) = 30 x = 11101f(x) = 29 x = 10110f(x) = 22 --------------------------- x = 10101f(x) = 21 x = 10100f(x) = 20 x = 10010f(x) = 18 x = 01101f(x) = 13 Induction 1 * 1 * * (Building Blocks) 1 1 * * *

7. F 5 : {0,1} 5  {0,1,2,3,4,5} F 5 (11110) = 0 F 5 (11100) = 1 F 5 (11000) = 2 F 5 (10000) = 3 F 5 (00000) = 4 F 5 (11111) = 5 Trap Functions

8. 11110 1110011000 10000 0123 00000 4 11111 5+++++ Fitness = 0 + 1 + 2 + 3 + 4 + 5 = 15 F 6x5 : {0,1} 30  {0,…,30} Additively Decomposable Functions (ADFs)

9. Hierarchically Decomposable Functions (HDFs)

10. x = 11100f(x) = 28 x = 11011f(x) = 27 x = 10111f(x) = 23 x = 10100f(x) = 20 --------------------------- x = 01011f(x) = 11 x = 01010f(x) = 10 x = 00111f(x) = 7 x = 00000f(x) = 0

11. x = 11111f(x) = 31 x = 11110f(x) = 30 x = 11101f(x) = 29 x = 10110f(x) = 22 --------------------------- x = 10101f(x) = 21 x = 10100f(x) = 20 x = 10010f(x) = 18 x = 01101f(x) = 13

12. Bayesian Optimization Algorithm (BOA)

13. Searching for a network structure that maximizes the scoring metric is NP-hard.

14. {{0,1,2},{3,4,5},{6,7,8},{9,10,11},{12,13,14}}

15. 0011  0110 0110  0011 0001  0100 0010  0011 0100  0001 0111  0111

16. 00…… 01…… 01…… 10…… 10…… 10…… 11…… 11…… 11…… 11…… m 0,1 = (1 x 4) + (2 x 3) = 4 + 6 = 10

17. Time Complexity = O(l 2 n)

18. P = {{0,1,2}, …} P = {{0,1,2,3,4,5},…}

19. H max, H min L min

20. Time Complexity = O(l 4 )

21. {{0,1,2},{3,4,5},{6,7,8},{9,10,11},{12,13,14}}

22. Random Solutions Selection Solution Recombination Building-block Identification

28. Conclusions 1. Building Bayesian network is time-consuming. 2. Simultaneity Matrix can be computed in O(l 2 n). 3. Partitioning can be done in O(l 4 ). 4. Scalability (ADFs and HDFs). 1. Chi-square Matrix (CSM). Recent work