A Hybrid Optimization Approach for Global Exploration

1. A Hybrid Optimization Approachfor Global Exploration 2005年度 713番 日和　悟 Satoru HIWA 知的システムデザイン研究室 Intelligent Systems Design Laboratory

2. Optimization • Optimization problem consists of: • Objective function:we want to minimize or maximize. • Design variables:affect the objective function value. • Constraints:allow the design variables to take on certain values but exclude others. Mathematical discipline that concerns the finding of minima or maxima of functions, subject to constraints Real-world applications • Optimization techniques have been applied to various real-world problems. e.g.) Structural design Electric device design

3. Problem Solving by Optimization • There are many good optimization algorithms. • Each method has its own characteristics. • It is difficult to choose the best method for the optimization problem. • It is important to select and apply the appropriate algorithms according to the complexities of the problems. • It is hard to solve the problem with only one algorithm when the problem is complicated. Hybrid optimization approach, which combines plural optimization algorithms, should be necessary. Purpose of the research: To develop an efficient hybrid optimization algorithm

4. Hybrid Optimization Approach • It provides the high performance which cannot be accomplished with only one algorithm. Hybrid optimization algorithm To develop an efficient hybrid optimization algorithm • We have to determine what kinds of solutions are required. • Desired solutions may vary depending on the user: • One may require the better result within a reasonable time. • The other may want not only the optimum, but also the information of the landscape. • Optimization strategy • First, how the optimization process is performed should be determined.

5. Optimization Strategy • By this, we can obtain not only the optimum point, but also the information of the landscape. • Many optimization algorithms are designed only to derive an optimum. To explore the search space uniformly and equally Why is the strategy needed?

6. Why is the Strategy Needed? • When we solve real-world optimization problems; • Usually, the landscape and the optimum are unknown. • In this case, the obtained results should be reliable. • Genetic Algorithms (GAs) are powerful techniques to obtain the global optimum. • Probabilistic algorithm inspired by evolutionary biology • Example of optimization by GAs: Problem GAs

7. Why is the Strategy Needed? • When we solve real-world optimization problems; • Usually, the landscape and the optimum are unknown. • In this case, the obtained results should be reliable. • Genetic Algorithms (GAs) are powerful techniques to obtain the global optimum. • Probabilistic algorithm inspired by evolutionary biology • Example of optimization by GAs: Problem GAs

8. Why is the Strategy Needed? • When we solve real-world optimization problems; • Usually, the landscape and the optimum are unknown. • In this case, the obtained results should be reliable. • Genetic Algorithms (GAs) are powerful techniques to obtain the global optimum. • Probabilistic algorithm inspired by evolutionary biology • Example of optimization by GAs: Unexplored area exists. Is real optimum in the area? Unknown The result is not reliable. Problem GAs

9. Why is the Strategy Needed? • When we solve real-world optimization problems; • Usually, the landscape and the optimum are unknown. • In this case, the obtained results should be reliable. • Genetic Algorithms (GAs) are powerful techniques to obtain the global optimum. • Probabilistic algorithm inspired by evolutionary biology • Example of optimization by GAs: Unexplored area exists. Is real optimum in the area? Unknown The result is not reliable. Problem GAs The strategy is not achieved only by GAs.

10. Why is the Strategy Needed? • When we solve real-world optimization problems; • Usually, the landscape and the optimum are unknown. • In this case, the obtained results should be reliable. • Genetic Algorithms (GAs) are powerful techniques to obtain the global optimum. • Probabilistic algorithm inspired by evolutionary biology • Example of optimization by GAs: The strategy is achieved. The landscape is grasped. Unknown Reliability can be evaluated. Problem GAs

11. Optimization Algorithms • The strategy is not achieved only by GAs. • Other algorithm, which provides more global search, is needed. • However, the globally-intensified search converges slowly compared to GAs or local search algorithms. • the much time is consumed in exploring the entire search space. • There are tradeoff between the search broadness and the convergence rate. It is necessary to balance the global and local search. Both global and local search algorithms are hybridized. • GAs • DIRECT: explores search space globally. • SQP: is high-convergence local search method.

12. DIRECT • Deterministic, global optimization algorithm • Its name comes from‘DIviding RECTangles’. • Search space is considered to be a hyper-rectangle(box). • Each box is trisected in each dimension. • Center point of each box is sampled as solution. • Boxes to be divided • are mathematically guaranteed to be promising. • are called ‘potentially optimal boxes.’

13. Characteristics of the DIRECT search • Potentially optimal boxes potentially contain a better value than any other box. • DIRECT divides the potentially optimal boxes at each iteration.

14. Characteristics of the DIRECT search • Example: 2-dimensional Schwefel Function • Some Local optima exist far from the global optimum. • DIRECT explores the search space uniformly and equally. • DIRECT also detects the promising area. Global Optimum Local Optima

15. Characteristics of the DIRECT search • Example: 2-dimensional Schwefel Function • Some Local optima exist far from the global optimum. • DIRECT explores the search space uniformly and equally. • DIRECT also detects the promising area. Global Optimum Local Optima

16. Characteristics of the DIRECT search • Example: 2-dimensional Schwefel Function • Some Local optima exist far from the global optimum. • DIRECT explores the search space uniformly and equally. • DIRECT also detects the promising area. Global Optimum Local Optima

17. Genetic Algorithms (GAs) • Heuristic algorithms inspired by evolutionary biology. • Solutions are called ‘individuals’, and genetic operators (Crossover, Selection, Mutation) are applied. • Real-coded GAs • Individuals are represented by real number vector. • Although GAs are global optimization algorithm, the search broadness is inferior to DIRECT. • GAs are used as more locally-intensified search than DIRECT. Parents Children Individuals

18. Sequential Quadratic Programming (SQP) • Gradient-based local search algorithm • The most efficient method in nonlinear programming • By using gradient information, SQP rapidly converges to the optimum. • Advantage • High convergence • Disadvantage • SQP is often trapped to the local optima, for the problem which has many local optima.

19. Hybrid Optimization Algorithm • Perform the DIRECT search. • Execute GAs. • Improve the best solution obtained in GAs search by SQP. Idea of the proposed hybrid optimization approach Global exploration by DIRECT Fine tuning by SQP Locally-intensified search by GAs Procedure of the proposed algorithm

20. Hybrid Optimization Algorithm • Perform the DIRECT search. • Execute GAs. • Improve the best solution obtained in GAs search by SQP. Idea of the proposed hybrid optimization approach Optimum Global exploration by DIRECT Fine tuning by SQP Locally-intensified search by GAs Procedure of the proposed algorithm

21. How to Combine DIRECT and GAs • GAs utilize the center points of the potentially optimal boxes in DIRECT as their individuals. DIRECT stopped. GAs start. • Number of potentially optimal = number of individuals • Number of potentially optimal differs at each iteration. • Number of individuals are determined according to the complexities of the problems. (e.g. In N-dim. space, N×10 individuals are recommended.)

22. How to Combine DIRECT and GAs • GAs utilize the center points of the potentially optimal boxes in DIRECT as their individuals. DIRECT stopped. GAs start. • Number of potentially optimal = number of individuals • Number of potentially optimal differs at each iteration. • Number of individuals are determined according to the complexities of the problems. (e.g. In N-dim. Space, N×10 individuals are recommended.) Number of potentially optimal boxes should be adjusted according to the number of individuals.

23. How to Combine DIRECT and GAs • If the number of potentially optimal is smaller than Ni, randomly generated individuals are added. • If the number of potentially optimal is larger than Ni, a certain number of potentially optimal boxes are selected. Ni: Number of individuals in GAs Box selection rules are proposed and applied.

24. Box Selection Rules for DIRECT • DIRECT sometimes performs an local improvement. • In the hybrid optimization, it is not necessary for DIRECT to perform locally-intensified search. • Proposed rules reduce the crowded boxes. • Distance from the box with best function value is calculated. • A certain number of boxes far from the best point are selected. • The rules are applied at each iteration in DIRECT search. Idea of selecting the boxes to be divided

25. Box Selection Rules for DIRECT • DIRECT sometimes performs an local improvement. • In the hybrid optimization, it is not necessary for DIRECT to perform locally-intensified search. • Proposed rules reduce the crowded boxes. • Distance from the box with best function value is calculated. • A certain number of boxes far from the best point are selected. • The rules are applied at each iteration in DIRECT search. Idea of selecting the boxes to be divided Potentially optimal boxes near the best point are discarded, and locally-biased search is prevented. The number of potentially optimal boxes is reduced without breaking the global search characteristics of DIRECT.

26. Experiments • 10-dimensional Schwefel function • A lot of local optimum exist. • The function value of the global optimum is zero. Verification of effectiveness of the hybrid approach • Numerical example is shown • to verify whether the proposed method achieve the proposed strategy− to explore the search space uniformly and equally. • The proposed hybrid optimization algorithm • is applied to the benchmark problem. • is compared to the search only by GAs. Target problem

27. Results and Discussions Searching ability • Average values of function value and the number of function evaluations are shown. • Proposed hybrid algorithm obtains better function value than that of GAs, with less function evaluations.

28. Results and Discussions To see whether the proposed strategy is achieved… • Search histories of DIRECT and GAs in the hybrid algorithm are checked. • History in 10-dimensional space is projected into 2-dimensional plane. • Although 45 plots exist, 4 typical examples are picked. (x1, x2, …, x10) → (x1, x2), (x1, x3), …

29. Search History of DIRECT (x2, x5) (x1, x2) (x7, x9) (x3, x6)

30. Search History of DIRECT

31. Search Histories of DIRECT and GAs

32. Search Histories of DIRECT and GAs The proposed strategy is achieved.

33. Conclusions • ‘optimization strategy’ is proposed: • To explore the search space uniformly and equally • Optimization algorithms used for the strategy: • DIRECT, GAs, and SQP Hybrid optimization approach is proposed. Modification to DIRECT • Box selection rules are proposed and applied. Hybrid optimization algorithm • It achieved the proposed strategy. • It provided the efficient performance than the search only by GAs.

34. Paper List • Mitsunori Miki, Satoru Hiwa, Tomoyuki Hiroyasu “Simulated Annealing using an Adaptive Search Vector” Proceedings of IEEE International Conference on Cybernetics and Intelligent Systems 2006 (Bangkok, Thailand) Proceeding of International Conference The Science and Engineering Review of Doshisha University • 三木光範，日和 悟，廣安知之 「LEDを用いた調色用照明システムの基礎的検討」 同志社大学理工学研究報告 Vol.46 No.3 pp 9-18，2005 Oral Presentation (in Japan) • 日和 悟，廣安知之，三木光範 「大域的最適化のための複数最適化手法の動的制御法」 日本機械学会 第7回最適化シンポジウム，2006 • 日和 悟，廣安知之，三木光範 「大域的最適化のための複数最適化手法の動的制御法」 日本機械学会 第6回設計工学・システム部門講演会，2006 • 三木光範，日和 悟，廣安知之 「適応的探索ベクトルをもつシミュレーテッドアニーリング」 日本機械学会 第8回計算力学講演会，2005

36. Lipschitzian Optimization [Shubert 1972] It requires the user to specify the Lipschitz constantK • K is used as a prediction of the maximum possible slope of the objective function over the global domain. – K +K . Slope = +K Slope = −K x1 x1 x1 x2 x2 x3 a b a b a b

37. DIRECT (one-dimensional) a b Box 1 Box 1 Box 2 Box 3 Box 3 Box 1 Box 2 Box 5 Box 4

38. DIRECT (one-dimensional) Slope = K a b Box 1 Box 1 Box 2 Box 3 Slope = K1 Slope = K2 Box 3 Box 1 Box 2 Box 2 Box 1 Box 5 Box 4 Box 4 Box 5

39. DIRECT (one-dimensional) • If box i is potentially optimal, then f(ci) <= f(cj) for all boxes that are of the same size as i. • In the largest boxes, the box with the best function value is potentially optimal. Slope = K a b Box 1 Box 1 Box 2 Box 3 Slope = K1 Slope = K2 Box 3 Box 1 Box 2 Box 2 Box 1 Box 5 Box 4 Box 4 Box 5

40. DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. • Potentially optimal boxes are defined by: Identification of potentially optimal boxes A hyper box j is potentially optimal if there exists some such that cj: center point of the box j dj: distance from the center point to vertices

41. DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes Search space

42. dj DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes cj Box j Search space

43. dj DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes cj f (cj) Box j Center - vertex distance (dj) Search space

44. dj DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes cj f (cj) Box j fmin ( 0, fmin -ε| fmin | ) Center - vertex distance (dj)

45. dj DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes Make the convex hull which contains all points. cj f (cj) Box j fmin ( 0, fmin -ε| fmin | ) Center - vertex distance (dj)

46. dj : Potentially optimal DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes Boxeson the lower part of convex hull is selected as potentially optimal. cj f (cj) Box j fmin ( 0, fmin -ε| fmin | ) Center - vertex distance (dj)

47. dj : Potentially optimal DIRECT ーPotentially Optimal Boxes • DIRECT divides all potentially optimal boxes. Identification of potentially optimal boxes Boxeson the lower part of convex hull is selected as potentially optimal. cj f (cj) Box j Center - vertex distance (dj) Search space

48. Genetic Algorithms (GAs) • Global search algorithm inspired by evolutionary biology. • Solutions are called ‘individuals’, and genetic operators (Crossover, Selection, Mutation) are applied. • Real-Coded GAs (RCGAs) • Individuals are represented by real number vector. • Crossover operator significantly affects the searching ability. • Simplex Crossover (SPX) • One of the efficient crossover operator for RCGAs. • Generates offspring in a simplex, which is formed by n+1individuals in n-dimensional space • RCGAs using the SPX operator • has both global and local search characteristics. RCGAs using the SPX operator are used.

49. GAs and SQP • Gradient-based local search algorithm • By using gradient information, SQP rapidly converges to the optimum. GAs (Genetic Algorithms) • Heuristic algorithm inspired by evolutionary biology. • Solutions are called ‘individuals’, and genetic operators (Crossover, Selection, Mutation) are applied. Parents Children Individuals SQP (Sequential Quadratic Programming)

50. Stopping Criterion DIRECT • is terminated when the size of the best potentially optimal box is less than certain value prescribed. • A certain depth of search space exploration is obtained. GAs • are terminated when their individuals converged. • Spread of the individuals in design variable space: xmax–xmin < threshold SQP • continues its search until the improvement of solution becomes a minute value.