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Title Page

Contents

ABSTRACT 11

1. Introduction 13

1.1. Research background 13

1.2. Research Objectives 17

1.3. Thesis organization 18

2. Previous Studies of the SAGO 20

2.1. Kriging-based SAGO 20

2.1.1. Kriging metamodel 20

2.1.2. Expected Improvement 22

2.1.3. Generalized Expected Improvement 23

2.1.4. Bootstrapped Expected Improvement 24

2.2. RBF-based SAGO 25

2.2.1. Radial Basis Function metamodel 25

2.2.2. Gutmann-RBF 26

2.2.3. Bjorkman-RBF 27

2.2.4. CORS-RBF 28

2.2.5. Weighted EI 29

2.2.6. CORS-RBF-Restart 30

3. Proposed Adaptive Sequential Approximate Global Optimization 31

3.1. Infill Sampling Criterion using the weighted minimum distance 31

3.2. Adaptive balancing technique 34

3.3. Numerical Procedure 42

3.3.1. Initial DOE 43

3.3.2. Metamodel generation/update 43

3.3.3. Weight selection using an adaptive balancing technique 44

3.3.4. Optimization of ISC 45

3.3.5. Stopping Criterion 45

3.4. Effect of the weighted minimum distance 46

3.5. Effect of the adaptive balancing technique 53

4. Numerical results and discussion 62

4.1. Test problems 62

4.2. Numerical test settings 63

4.3. Comparison with the previous studies 67

4.4. Effect of the number of initial sample points 72

4.5. Effect of user-defined parameter 76

4.6. Scalability 81

5. Extension to the constrained problem 83

5.1. Weighted minimum distance for constrained problem 83

5.2. ISC for constrained optimization 85

5.3. Numerical test settings for constrained optimization 85

5.4. Numerical test results 86

6. Conclusions 89

Appendix 91

Appendix A. Literature survey on the DOE 91

Appendix B. Literature survey on the metamodel 92

Appendix C. Mathematical test examples for unconstrained optimization 95

Appendix D. Scalable mathematical examples 100

Appendix E. Mathematical test examples for constrained optimization 101

Reference 106

국문요지 113

List of Tables

Table 4.1. Summary of the mathematical test examples 62

Table 4.2. Parameters on the Kriging metamodel 66

Table 4.3. Parameters on the adaptive balancing technique 66

Table 4.4. Parameters on the particle swarm optimization 66

Table 4.5. Average number of function evaluations 68

Table 4.6. Standard deviation of the number of function evaluations 69

Table 4.7. Success rate 70

Table 4.8. Effect of the number of initial sample points on the performance of the ASAGO 73

Table 4.9. Percentage of the standard deviation against the average of the... 75

Table 4.10. Combination of user-defined parameters 76

Table 4.11. Effect of the user-defined parameters 77

Table 4.12. Summary of the scalable mathematical examples 81

Table 4.13. Scalability check for griewank and schwefel examples 82

Table 4.14. Scalability check for rosenbrock example 82

Table 5.1. Numerical test examples for constrained optimization 86

Table 5.2. Numerical test results for constrained problem 86

List of Figures

Fig. 2.1. Graphical illustration of the probability distribution of true response 23

Fig. 3.1. Comparison of distance and weighted minimum distance 33

Fig. 3.2. Pseudo-code for counting consecutive iterations 36

Fig. 3.3. A case when the standard deviation of the nearest neighbor point is... 38

Fig. 3.4. A case when the standard deviation of the nearest neighbor points is... 39

Fig. 3.5. A case when the standard deviation of the nearest neighbor points is... 40

Fig. 3.6. Pseudo-code for counting consecutive iterations 41

Fig. 3.7. Flowchart of an adaptive balancing technique 42

Fig. 3.8. Flow chart of the proposed ASAGO 43

Fig. 3.9. Re-fitting process of the Kriging metamodel 44

Fig. 3.10. Contour plot of Branin example and initial sample points 47

Fig. 3.11. Contour of true response and positions of ISPs at 4th iteration(이미지참조) 48

Fig. 3.12. Contour of true response and positions of ISPs at 8th iteration(이미지참조) 49

Fig. 3.13. Contour of true response and positions of ISPs at 11th iteration(이미지참조) 50

Fig. 3.14. Contour of true response and positions of ISPs of the ASAGOm 51

Fig. 3.15. Convergence history of the objective function 52

Fig. 3.16. Weight history 52

Fig. 3.17. Contour of the Branin example and initial sample points 53

Fig. 3.18. Positions of ISPs and weight history of the ASAGO 54

Fig. 3.19. Positions of ISPs and weight history of the simple SAGO 55

Fig. 3.20. Positions of ISPs of the EI 56

Fig. 3.21. Positions of ISPs of the GEI 57

Fig. 3.22. Positions of ISPs of the WEI 58

Fig. 3.23. Positions of ISPs of the CORS-RBF-Restart 59

Fig. 3.24. Positions of the ISPs until find the global optimum 60

Fig. 3.25. Convergence history of the SAGO algorithms 61

Fig. 4.1. Contour plot of the SC example and positions of the global optimums 64

Fig. 4.2. Positions of ISPs of the ASAGO 65

Fig. 4.3. Average number of function evaluations 67

Fig. 4.4. Standard deviation of the number of function evaluations 71

Fig. 4.5. Success rate 72

Fig. 4.1. Effect of the number of initial sample points on the average number... 74

Fig. 4.2. Effect of the number of initial sample points on the standard... 74

Fig. 4.3. Effect of the number of initial sample points on the success rate 75

Fig. 4.9. Effect of user-defined parameter on the efficiency 79

Fig. 4.10. Effect of user-defined parameter on the robustness 79

Fig. 4.11. Effect of user-defined parameter on the reliability 80

Fig. 5.1. Examples of calculation of weight factor for constraint violation 84

Fig. 5.2. Comparison of average function evaluations for constrained problems 87

Fig. 5.3. Comparison of standard deviation of function evaluations for... 87

Fig. 5.4. Comparison of success rate for constrained problems 87

초록보기

 본 연구에서는 전역 탐색을 위한 향상된 방법인 weighted minimum distance와 국부 탐색과 전역 탐색의 비중을 적응적으로 조절하는 적응적 균형 조절 기법 (adaptive balancing technique)을 제안하였고 이들을 결합하여 Adaptive Sequential Approximate Global Optimization (ASAGO)를 개발하였다.

순차적 근사 전역 최적화 (Sequential Approximate Global Optimization; SAGO)란 초기 메타모델을 생성한 후 순차적으로 실험점을 추가해가며 전역 최적해를 찾아나가는 방법이다. 이러한 SAGO에서는 아직 탐색하지 않은 설계영역을 탐색해 나가는 전역탐색(global search)과 알고 있는 최선해를 개선시켜 나가는 국부탐색(local search)을 조합해 전역 최적해(global optimum)를 탐색한다.

기존의 순차적 근사 최적화 방법들은 이러한 전역탐색과 국부탐색의 중요도를 고정하거나 사전에 정의된 패턴을 반복하기 때문에 설계 문제에 따른 적응성의 부재로 인해 수렴성의 기복이 심한 단점이 있다. 이를 극복하기 위한 방안으로 수렴이력과 실험점에서의 정보를 이용해 순차적 최적화 단계 별로 전역탐색과 국부탐색의 비중을 조절해 주는 adaptive balancing technique을 제안하였다.

또한 기존의 전역 탐색을 위한 방법으로는 단순히 기존 실험점과의 거리나 메타모델의 오차가 사용되었는데 실험점에서의 응답값의 크기에 따른 가중치를 부여해 보다 전역 최적해의 부근을 빠르게 탐색할 수 있는 전역 탐색 방법인 weighted minimum distance를 제안하였다.

본 연구에서는 weighted minimum distance와 adaptive balancing technique을 결합하여 Adaptive Sequential Approximate Global Optimization(ASAGO)를 개발하였다. 그리고 제안된 ASAGO의 성능에 초기 실험점수와 사용자 지정 파라미터들이 주는 영향을 10개의 수학적 예제들에 대해 테스트하였으며 그 결과로부터 적절한 값들을 추천하였다. 또한 기존 순차적 전역 최적화 방법들과 비교하여 제안된 ASAGO의 우수한 성능을 입증하였다.

참고문헌 (69건) : 자료제공( 네이버학술정보 )

참고문헌 목록에 대한 테이블로 번호, 참고문헌, 국회도서관 소장유무로 구성되어 있습니다.
번호 참고문헌 국회도서관 소장유무
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20 Infill sampling criteria for surrogate-based optimization with constraint handling 네이버 미소장
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44 Augmented D-Optimal Design for Effective Response Surface Modeling and Optimization 소장
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66 Multiple Surrogate Modeling for Axial Compressor Blade Shape Optimization 네이버 미소장
67 Toward an optimal ensemble of kernel-based approximations with engineering applications 네이버 미소장
68 Ensemble of metamodels with optimized weight factors 네이버 미소장
69 Pointwise ensemble of meta-models using v nearest points cross-validation 네이버 미소장

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