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研究生:翁偉泰
研究生(外文):Weng, Wei-Tai
論文名稱:應用內點演算法求解線性二階規劃問題
論文名稱(外文):Interiro Point Approach for Solving Linear Bilevel Programming Problems
指導教授:溫于平溫于平引用關係
指導教授(外文):Wen, Ue-Pyng
學位類別:博士
校院名稱:國立清華大學
系所名稱:工業工程與工程管理學系
學門:工程學門
學類:工業工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:94
中文關鍵詞:線性二階規劃問題理性反應解集合多目標規劃問題有效解集合有效解集合線性最佳化問題內點演算法仿射比例演算法非線性最佳化問題
外文關鍵詞:linear bilevel programming problemrational reactional setmultiple objective linear programming problemefficient setlinear optimization over the efficient set probleminterior point methodaffine scaling algorithmnonconvex optimization
相關次數:
  • 被引用被引用:1
  • 點閱點閱:167
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  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:2
線性二階規劃(bilevel linear programming)問題是多階規劃問題的特例之一,它是處理分權式決策問題的一種數學模式。在線性二階規劃問題中,雖然高階與低階目標函數都是線性函數,且限制式解集合是凸集合,
然而一般而言,它的理性反應解集合(亦稱為可行解集合)是非凸集合,
因此,線性二階規劃問題是歸類為非線性最佳化問題。
在此論文中,我們以仿射比例內點主演算法(primal affine scaling algorithm)及主對偶(primal-dual)演算法為基礎,建立了求解線性二階規劃問題的內點演算法。我們進而求解隨機產生之線性二階規劃問題,以驗證演算法的正確性與有效性。測試結果顯示,與其它演算法比較,我們所建立之內點演算法非常適合求解較大型的線性二階規劃問題。
另一方面,只要些許的修訂,我們所建立的內點演算法也可以用來求解多目標規劃問題中的有效解集合線性最佳化(linear optimization over the efficient set)問題。經由求解隨機產生之有效解集合線性最佳化問題,結果顯示我們所建立的內點演算法不但正確、有效,而且比其它演算法更適合求解較大型的問題。
Linear bilevel programming (BLP) problem is a special case of multilevel programming problem, which is a mathematical programming model for decentralized decision problems. In a linear BLP problem, the higher and the lower levels objective functions are both linear functions and the constraints region is a convex set, however, the rational reaction set (or, the feasible set) is generally nonconvex. Hence the linear BLP problem is mathematically classified as a nonlinear optimization problem.
In this dissertation, based on primal and primal-dual affine scaling algorithms, we develop interior point algorithms for solving linear BLP problems. We test the accuracy and efficiency of the proposed algorithms by solving randomly generated problems. The computational results show that the interior point method is suitable for solving relatively larger scale linear BLP problems.
On the other hand, after modification, the proposed interior point algorithm also provides an effective and accurate approach to solve linear optimization over the efficient set (OES) problems. The computational results also show that the modified algorithm can solve relatively larger scale linear OES problems.
1 Introduction 1
1.1 Problem Definition . . . . . . . . . . . . . . . . . . . 1
1.2 General Properties and Complexity . . . . . . . . . . . 3
1.3 Applications and Existing Algorithms . . . . . . . . . . 5
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The Related Problems 11
2.1 Multiple Objective Linear Programming . . . . . . . . . 11
2.2 Linear Optimization over the Efficient Set. . . . . . . 12
2.3 The Equivalency Between Linear BLP Problem and Linear
OES Problem . . . . . . . . . . . . . . . . . . . . . . 16
3 Affine Scaling Algorithms 19
3.1 Primal Affine Scaling Algorithm . . . . . . . . . . . . 19
3.1.1 Ideas of Primal Affine Scaling Algorithm. . . . . . 20
3.1.2 Starting and Terminating the Algorithm. . . . . . . 22
3.1.3 Algorithmic Procedure . . . . . . . . . . . . . . . 24
3.2 Primal-Dual Affine Scaling Algorithm. . . . . . . . . . 25
3.2.1 Ideas of Primal-Dual Affine Scaling Algorithm . . . 26
3.2.2 Algorithmic Procedure . . . . . . . . . . . . . . . 29
4 A Primal Affine Scaling Algorithm for Solving Linear BLP
Problems 30
4.1 Characteristics of the Reformulated Problems. . . . . . 30
4.2 Ideas of the Proposed Algorithm . . . . . . . . . . . . 32
4.3 The Proposed Algorithm. . . . . . . . . . . . . . . . . 37
4.4 A Numerical Example . . . . . . . . . . . . . . . . . . 39
4.5 Computational Experience. . . . . . . . . . . . . . . . 43
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . 46
5 A Primal-Dual Affine Scaling Algorithm for Solving Linear
BLP Problems 48
5.1 Ideas of the Proposed Algorithm . . . . . . . . . . . . 48
5.2 The Proposed Algorithm. . . . . . . . . . . . . . . . . 50
5.3 An Illustrative Example . . . . . . . . . . . . . . . . 53
5.4 Computational Experience. . . . . . . . . . . . . . . . 57
5.5 Discussion. . . . . . . . . . . . . . . . . . . . . . . 61
6 A Modified Algorithm for Solving Linear OES Problems 63
6.1 Ideas of the Modified Algorithm . . . . . . . . . . . . 63
6.2 The Modified Algorithm. . . . . . . . . . . . . . . . . 65
6.3 A Numerical Example . . . . . . . . . . . . . . . . . . 68
6.4 Computational Experience. . . . . . . . . . . . . . . . 72
6.5 Discussion. . . . . . . . . . . . . . . . . . . . . . . 75
7 Conclusion 77
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 Future Research . . . . . . . . . . . . . . . . . . . . 78
Bibliography 80
Appendix 89
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