#Lecture 9: Linear Programming and ALM: Cash-Flow Matching
两种解决办法，其中一种叫Simplex
首先，Simplex的中文是单纯形法
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Linear Programming Theory
- Formulation
- Duality
- Solution Methods
- Simplex Method
- Primal-Dual Interior-Point Method
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LP: Asset Liability Manamgement: Cash-Flow Matching
- Basic setup
- Example
- Nonlinear scenario
- Stochastic liabilities
Formulation
其中A为一个m*n矩阵
若A行满秩
则可以找到基矩阵B，并寻找初始基解。
描述线性规划问题的常用和最直观形式是标准型。标准型包括以下三个部分：
一个需要极大化的线性函数，例如
{\displaystyle c_{1}x_{1}+c_{2}x_{2}}
c_1 x_1 + c_2 x_2
以下形式的问题约束，例如：
{\displaystyle a_{11}x_{1}+a_{12}x_{2}\leq b_{1}} a_{11} x_1 + a_{12} x_2 \le b_1
{\displaystyle a_{21}x_{1}+a_{22}x_{2}\leq b_{2}} a_{21} x_1 + a_{22} x_2 \le b_2
{\displaystyle a_{31}x_{1}+a_{32}x_{2}\leq b_{3}} a_{31} x_1 + a_{32} x_2 \le b_3
和非负变量，例如：
{\displaystyle x_{1}\geq 0} x_1 \ge 0
{\displaystyle x_{2}\geq 0} x_2 \ge 0
Simpler than any other optimization problem
With plenty of application (even in Finance)
Serve as a good starting point
Duality
对偶规划问题
- 对偶规划中：y 为人任意值
- 假设 B 是 (P) 的一个可行基，对应的可行解 x_B
Solution Methods
Simplex (Dantzig 1947)
Ellipsoid (Kachian 1979, the first algorithm known to be in polynomial time)
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Interior Point (Karmakar 1984, the first practical polynomial time algorithm)
- Projection method (Karmakar 1984)
- Affine Method (Dikin 1967)
- Log-Barrier Method (many ...)
Interior Point method has been extended to NLP problems, has been the focus of research for optimization in the last few decades.
Although asymptotically superior, there is no clear winner between Simplex and Interior Point for LP problems: very much depends on the problem
Algorithm (Simplex Method)
相关链接：
Wiki - 线性规划