現今，超啟發式計算在解組合最佳化問題上是越來越重要，對於傳統的方法 (例如 k-means、禁忌搜尋及模擬退火) 已不足以得到更好的解品質。最近，新的超啟發式計算藉由模擬大自然的現象 (如漩渦及低氣壓)，「螺旋最佳化」(spiral optimization, SO) 被提出來應用在函數最佳化問題上。在本論文中，我們介紹一個 SO 的延伸演算法－「分散式螺旋最佳化」(DSO)－來解決組合最佳化問題。所提出的方法與原本的 SO 不同之處在於：(1) 增加 k-means 與擾動運算子來提升DSO 的效能，以及 (2) DSO 將整個群體解切分成數個子群體解來增加搜尋過程的多樣性，以達到提升分群結果的目的。為了探討本論文提出的方法之效能，我們在分群問題上比較 SO 、genetic k-means algorithm 及 DSO 的分群結果。此外，為了瞭解參數對 DSO 的重要性，我們透過傅立葉振幅敏感度檢測 (Furier amplitude sensitivity test, FAST) 來分析參數與模組之間的相依性。從結果來說，本論文所提出的方法是非常具有潛力的。

Nowadays, metaheuristics have become more and more important in solving the combinatorial optimization problems (COPs) for which the traditional methods (such as k-means, tabu search, and simulated annealing) are simply not powerful enough to obtain good solutions. Recently, spiral optimization (SO), a new metaheuristic that emulates the natural phenomena (such as swirl and low pressure) was proposed to solve the function optimization problem. In this thesis, we present an extension of SO, called a distributed spiral optimization (or DSO for short), for solving the COPs. The proposed algorithm differs from the original SO by (1) adding to the latter the k-means and oscillation operators to accelerate its convergence speed and (2) splitting the population into subpopulations so as to increase the diversity of the search, thus improving the quality of the clustering result. To evaluate the performance of the proposed algorithm, we compare it with the original SO and genetic k-means algorithm in solving the clustering problem. Moreover, to understand the impact of the parameter on the performance of DSO, we use Fourier amplitude sensitivity test (FAST) to analyze the dependency between the model and parameters. The results show that the proposed algorithm is quite promising.

論文審定書i
誌謝iii
摘要v
ABSTRACT vi
List of Figures x
List of Tables xii
Chapter 1 簡介 1
1.1 動機 2
1.2 論文貢獻 2
1.3 論文架構 2
Chapter 2 文獻探討 4
2.1 資料分群問題 4
2.1.1 分群問題的分類 4
2.1.1.1 監督式與非監督式 5
2.1.1.2 階層式與分割式 5
2.1.1.3 互斥與非互斥 6
2.1.2 分群問題的定義 7
2.2 超啟發式演算法 7
2.2.1 Genetic k-means algorithm 9
2.2.1.1 選擇運算子 10
2.2.1.2 突變運算子 11
2.2.1.3 k-means運算子 11
2.2.2 螺旋最佳化演算法 12
2.2.2.1 二維空間上的旋轉運算子 14
2.2.2.2 n維空間上的旋轉運算子 15
Chapter 3 分散式螺旋最佳化 18
3.1 分散式螺旋最佳化的理念 18
3.2 分散式螺旋最佳化的流程 19
3.3 旋轉運算子 20
3.4 擾動運算子 21
3.5 k-means運算子 22
3.6 Migration運算子 23
3.7 實作例子 23
Chapter 4 實驗結果 27
4.1 執行環境、資料集介紹、參數設定 27
4.2 螺旋最佳化的參數探討 29
4.2.1 旋轉角度 29
4.2.1.1 模組群體解的收斂行為 29
4.2.1.2 模組結果的影響性 29
4.2.2 收斂比率 32
4.2.2.1 模組群體解的收斂行為 32
4.2.2.2 模組結果的影響性 33
4.3 模擬結果 35
4.4 分析 37
4.4.1 最佳解的搜尋趨勢 37
4.4.2 參數的敏感度分析 40
4.4.2.1 FAST 40
4.4.2.2 eFAST 43
4.5 總結 46
Chapter 5 結論與未來展望 48
5.1 結論 48
5.2 未來展望 48
Bibliography 50
Appendix A 敏感度分析 54
A.1 FAST 54
A.1.1 干擾問題 57
A.1.2 挑選頻率集 57
A.1.3 實作範例 57
A.2 eFAST 60
A.2.1 挑選頻率集 62
A.2.2 實作範例 62

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