文章目录
- Nonlinear Program 非线性规划
- KKT condition KKT条件
- Golden section method 黄金分割法
- Newton's method 牛顿法
- Gradient steepest descent/ ascent method 梯度下降/上升法
- Integer Programming 整数规划
- Dynamic programming 动态规划
- Linear programming 线性规划
- Central path 中心路径法(内点法)
- Karmarkar算法(内点法)
- Eliposoid method 椭球法(外点法)
- Transportation problem & unimodular matrix 运输问题与幺模矩阵
- Graph Theory 图论
- 附原题
Nonlinear Program 非线性规划
Problem 29.
Suppose . Show that the function of
with is concave.
Solution
Problem 18.
Let be a subset of and let be a function on . If is a relative minimum point of over , then for any that is a feasible direction at we have
- if , then
Solution
Problem 4.
Find the minimum number of , such that any local optimal solution is a global optimal solution for the following nonlinear program of .
Solution
KKT condition KKT条件
Problem 3.
Show the KKT condition for the nonlinear program of , and try to solve it with the KKT conditions.
Solution
Golden section method 黄金分割法
Problem 16.
For the problem over , we use golden section method to solve it. Please prove the convergence ratio is .
Solution
Newton’s method 牛顿法
Problem 17.
For one dimension nonlinear problem over , we use Newton’s Method to solve it. Please prove the convergence ratio of Newton’s method is at least two.
Solution
Gradient steepest descent/ ascent method 梯度下降/上升法
*Problem 4.
Use the gradient steepest ascent method to find the optimal solution of above nonlinear program, with the start point .
Solution
Integer Programming 整数规划
Branch and bound 分支定界法
Problem 1.
Use branch and bound method to solve .
Note that at each branching node, you can use graphic method to get the (local) upper bound for each node.
Draw the branch and bound tree, and point out the Node Program, GLB and LUB clearly.
Solution
branch and bound tree
Gomory cutting plane algorithm 割平面法/ Column generation 列生成算法
Problem 26.
For above , suppose now we have a basic feasible solution corresponding to basis matrix and Non-basis , i.e., , and let and . Prove:
Solution
Problem 12.
Use the Gomory cutting plane algorithm to solve the following integer linear programming.
Solution
Problem 28.
For above , denote by columns. Let , denote .
Suppose , and is the optimal solution for subproblem and respectively, and objectives of , correspondingly. Let , is the basis related to for .
Please prove: If , then
Solution
Dynamic programming 动态规划
IP 有整数约束
Problem 8.
Use dynamic programming method to solve the following integer programming.
Note that you need to point out recursive formula clearly, and write down calculation steps clearly.
Solution
LP 无整数约束
*Problem 8.
Use dynamic programming method to solve the following linear programming.
Note that you need to point out recursive formula clearly, and write down calculation steps clearly.
Solution
Linear programming 线性规划
Central path 中心路径法(内点法)
Problem 25.
Let be the central path of
Then prove:
(a) The central path point is bounded for and any given .
(b) For ,
Furthermore, if and .
Solution
Problem 11.
Compute the central path for the following linear programming.
Solution
Karmarkar算法(内点法)
Problem 6.
What are the differences between Karmarkar method and Simplex method for linear programming? Please show the logics of them clearly. Possibly you can use figures to show what your idea.
Solution
Eliposoid method 椭球法(外点法)
Problem 7.
Use ellipsoid method solve:
The initial Ellipsoid is taken to be .
You only need to give THREE STEPS
Solution
Transportation problem & unimodular matrix 运输问题与幺模矩阵
Problem 20.
For transportation problem stated as follows.
There are m origins that contain various amounts of a commodity that must be shipped to destinations to meet demand requirements. Specially, origin contains an amount , and destination has a requirement of amount . It is assumed that the system is balanced in the sense that total supply equals total demand. There is unit cost associated with the shipping of the commodity from origin to destination . The problem is to find the shipping pattern between origins and destinations that satisfies all the requirements and minimized the total shipping cost.
(1) build an mixed integer linear programming model for above problem.
(2) If the row and column sums of a transportation problem are integers, then the basic variables in any basic solution are integers. 如果运输问题的行和列的和是整数,证明所有基可行解的基变量为整数。
Problem 21.
A matrix is said to be totally unimodular if the determinant of every square submatrix formed from it has value or . (完全幺模矩阵的各阶子式均为0,1或-1)
(1) Show that the matrix defining the equality constraints of a transportation prolblem is totally unimodular. 证明运输问题的系数矩阵是完全幺模的。
(2) In the system of equations , assume that is totally unimodular and that all elements of and are integers. Show that all basic solutions have integer components.
与Problem20(2)的区别在于:此时是任意矩阵,秩不再是
Solution
Graph Theory 图论
Maximum flow 最大流
Problem 9.
Use the Ford-Fulkerson method to solve decide the maximum flow for the following network problem.
Solution
附原题
来源:CSDN
作者:菲"Q
链接:https://blog.csdn.net/weixin_43761124/article/details/103936166