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    題名: 線性規劃求解方法之研究—個案分析
    Search Direction Analysis in Linear Programming -- Case Study
    作者: 莊文華
    貢獻者: 陸行
    莊文華
    關鍵詞: 線性規劃
    求解方法
    投影法
    Linear programming
    Search direction
    Projection
    日期: 2001
    上傳時間: 2016-04-15 16:02:55 (UTC+8)
    摘要: 線性規劃的求解方法依幾何觀點可區分為單體法(Simplex Method)和起源於Karmarkar的內點法(Interior Point Method)，無論在理論上或應用上這兩者的許多變體仍然不斷地在改進中。Dantzig的單體法和它變體的演算法是從邊上角點走到另一角點直至到達最佳點，內點法則是透過這個可行地區的內部的可行點走到可行點的方法。眾所週知，當遭遇最壞案例時單體法求解速度的函數表示成指數形式；而內點法卻成多項式形式。就一般案例而言，實驗證據提議，當問題大小為小型時，以單體法求解速度較佳。而內點法僅僅在很大規模的線性規劃時，其解法優於以單體法為基礎的方法。
    封面頁
    證明書
    致謝詞
    論文摘要
    目錄
    表格目錄
    圖例目錄
    1 序論
    1.1 研究動機
    1.2 研究目的
    1.3 研究架構及範圍
    2 線性規劃的搜尋方向之探討
    2.1 線性規劃發展簡介
    2.2 文獻回顧
    2.2.1 仿射演算法
    2.2.2 有效搜尋法
    2.2.3 鄰近頂點上的單體演算法
    2.2.4 最陡峭邊單體演算法
    3 搜尋方向與投影法
    3.1 搜尋方向
    3.2 投影法
    4 投影矩陣的數值計算
    5 新演算法
    6 實證研究
    6.1 新演算法求解過程
    6.2 退化解問題
    6.3 起始點的影響
    7結果與討論
    8未來研究工作
    參考書目
    附錄
    附錄一
    附錄二
    附錄三
    附錄四
    附錄五
    參考文獻: Arbel, A., Exploring Interior-Point Linear Programming. The MIT Press, (1993)
    Barnes, E. R., A Variation on Algorithm for Sloving Linear Programming Problems. Mathematical Programming 36, 174-182, (1986).
    Bazaraa, M. S. and C. M. Shetty, Nonlinear Programming Theory and Algorithms. John Wiley and Sons publishers(1979)
    Bixby, R. E. and J. W. Gregory, I. J. Lustig, R. E. Marsten and D. F. Shanno, Very large-scale linear programming: A case study in combining interior point and simplex methods. Operations Research 40, 885-897, (1992).
    Dantzig, G. B., Linear Programming and Extensions. Princeton University Press, Princeton N.J., (1963).
    Dikin, I. I., Iterative Solution of Problems of Linear and Quadratic Programming. Soviet Mathematics Doklady, 8, 674-675, (1967)
    Dikin, I. I., O Skhodimosti Odnogo Iteratsionnogo Protsessa.(in Russian), Upravlyaemye Sistemy, 12, 54-60, (1794)
    Fang , S. C. and S. Puthenpura., Linear Optimization and Extensions. Prentice-Hall, Englewood Cliffs New Jersey, USA,(1993).
    Forrest, J. J. and Donald Goldfarb, Steepest-Edge Simplex Algorithms for Linear Programming. Mathematical Programming 57, 341-374(1992).
    Goldfarb, D. and J. K. Reid, A Practicable Steepest-Edge Simplex Algorithm. Mathematical Programming 12, 361-371(1977).
    Huard, P. and A. Auslender, Point-to-set maps and mathematical programming. Amsterdam ; New York : North-Holland Pub. Co. (1979)
    Hurley, W., Adjacent Vertex Algorithms: More Experimental Results on Random Problems. Computers and Operations Research 21, 535-541, (1994).
    Karmarkar, N. K., A new polynomial time algorithm for linear programming. Combinatorica 4, 373-395, (1984).
    Khachiyan, L. G., A Polynomial Algorithm in Linear Programming., Doklady Akademiia Nauk SSSR 244:s, 1093-1096, (1979), translated in Soviet Mathematics Doklady 20:1, 191-194,(1979)
    Klee, V. and G. J. Minty, How Good is the Simplex Algorithm? In O. Shisha(Ed.) Inequalities 3. Academic Press, New York,(1972).
    Kuhn, H. W. and R. E. Quandt, An Experimentical Study of Simplex Method. In Proceedings of the Symposia in Applied Mathematics(Edited by Metropolis et al.) XV. Providence, R.I.(1963).
    Luh, H. and Ray Tsaih, An Efficient Search Direction for Linear Programming Problems. Computers and Operations Research forthcoming. Terlaky, T., Interior Point Methods of Mathematical Programming. Kluwer Academic publishers, (1996)
    Trefethen, L. N. and David Bau, Numerical Linear Algebra., The Society for Industrial and Applied Mathematics, (1997)
    Vanderbei, R. J., Affine-Scaling for Linear Programs with Free Variables, Mathematical Programming 43, 31-44, (1989).
    Vanderbei, R. J., M. S. Meketon and B. A. Freedman., A Modiffication of Karmarkar`s Linear Programming Algorithm. Algorithmica 1, 395-407, (1986).
    Ye, Y., Interior Point Algorithms Theory and Analysis., John Wiley and sons, (1997)
    Winston, W. L., Operations Research Applitions and Algorithms., Duxbury Press(1994)
    Wolfe, P. and L. Culter, Experiments in linear programming. In Recent Advances in Mathematical Programming(Edited by Graves and Wolfe P.). Mcgraw-Hill, New York(1963).
    描述: 碩士
    國立政治大學
    應用數學系
    資料來源: http://thesis.lib.nccu.edu.tw/record/#A2002001141
    資料類型: thesis
    顯示於類別:[應用數學系] 學位論文

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