线性规划、摩尔-彭诺斯伪逆与重力方法Linear programming with Moore-Penrose inverses and the gravitation method
李帮喜,卢睿,赵奕菡,藤森赖明
摘要(Abstract)：
本文通过引入重力方法,探讨了摩尔-彭诺斯(MP)伪逆在线性规划中的应用。这一思想起源于矩形系数矩阵方程组的求解。用MP伪逆表示的第一个解向量可能会包含非正分量。根据重力方法,我们可以一步一步地将解向量向目标函数更大的方向改进。在寻找最优解的算法中,我们应用了MP伪逆。重力方法为优化分配规则提供了一种具体的增量方法。
关键词(KeyWords)： 线性规划;摩尔-彭诺斯伪逆;重力方法;最优化
基金项目(Foundation): 国家社会科学基金项目“数理政治经济学的理论与经验研究”（No.17BJL020）的支持
作者(Author): 李帮喜,卢睿,赵奕菡,藤森赖明
参考文献(References)：
- [1] Ben-Israel, A. and T. Greville,Generalized Inverse:Theory and Applications(New York:Springer, 2003[1974]).
- [2] Campbell, S. L. and C. D. Meyer,Generalized Inverses of Linear Transformations(Philadelphia:Society for Industrial and Applied Mathematics, 2009[1991]).
- [3] Chang, S. Y. and K. G. Murty，“The steepest descent gravitational method for linear programming，”Discrete Applied Mathematics, 1988, 25(3):211-239.
- [4] Fujimori, Y.,Modern Analysis of Value Theory(Springer, 1982).
- [5] Greenberg, H. J.，“Klee-Minty polytope shows exponential time complexity of simplex method，” University of Colorado at Denver, 1997.
- [6] Li, B.,Linear Theory of Fixed Capital and China’s Economy:Marx,Sraffa and Okishio(Springer, 2017).
- [7] Li, B. and Y. Fujimori，“Fixed capital, renewal dynamics and Marx-Sraffa equilibrium，” in Kasamatsu, M.(ed.),Macro-and Microfoundations of Economics(Tokyo:Waseda University Press, 2013).
- [8] Meyer, C. D.,Matrix Analysis and Applied Linear Algebra(Philadelphia:Society for Industrial and Applied Mathematics, 2000).
- [9] Morin, T. L., N. Prabhu and Z. Zhang，“Complexity of the gravitational method for linear programming，”Journal of Optimization Theory and Applications, 2001,108(3):633-658.
- [10] Pyle, L. D.，“A generalized inverseε-algorithm for constructing intersection projection matrices with applications，”Numerische Mathematik, 1967, 10(4):86-102.
- [11] Pyle, L. D.，“The generalized inverse in linear programming basic structure，”SIAM Journal on Applied Mathematics, 1972, 22(3):335-355.
- [12] Pyle, L. D. and R. E. Cline，“The generalized inverse in linear programminginterior gradient projection methods，”SIAM Journal on Applied Mathematics,1973, 24(4):511-534.
- [13] Rao, C. R. and S. K. Mitra,Generalized Inverse of Matrics and Its Applications(London:John Wiley&Sons, 1971).
- [14] van Rossum, G.,Python Tutorial, 3rd ed.(Python Software Foundation,2016).