《高等运筹学》复习题手写解答 Advanced Operations Research: Final Exam:Review Exercises

元气小坏坏 提交于 2020-01-17 00:05:21

Nonlinear Program 非线性规划

Problem 29.
Suppose p<1,p0p<1,p\neq 0 . Show that the function of
f(x)=(i=1nxip)1/p f(x) ={( \sum_{i=1}^nx_i^p)}^{1/p} with domf=R++ndomf = R^n_{++} is concave.

Solution
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Problem 18.
Let be a subset of EnE^n and let fC2f \in C^2 be a function on . If xx^∗ is a relative minimum point of ff over , then for any dEnd\in E^n that is a feasible direction at xx^∗ we have

  1. f(x)d0\triangledown f(x^*)d\geq 0
  2. if f(x)d=0\triangledown f(x^*)d=0, then dT2f(x)d0d^T\triangledown^2 f(x^*)d\geq 0

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Problem 4.
Find the minimum number of cc, such that any local optimal solution is a global optimal solution for the following nonlinear program of P4P4.
P4)max f(x)=cx12+x1x2+2x112x22s.t.  x12 P4) \max \ f(x) = −c∗x^2_1 + x_1x_2 + 2x_1 − \frac{1}{2}x_2^2 \\ s.t. \ \ x_1 ≤ 2
Solution
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KKT condition KKT条件

Problem 3.
Show the KKT condition for the nonlinear program of P3P3, and try to solve it with the KKT conditions.
P3) max 2x122x1x2x22+10x1+10x2s.t.   x12+x225       3x1+x26 P3) \ \max{ \ −2x^2_1 −2x_1x_2 −x^2_2 + 10x_1 + 10x_2} \\ s.t. \ \ \ x^2_1 + x^2_2 ≤ 5 \\ \ \ \ \ \ \ \ 3x_1 + x_2 ≤ 6
Solution
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Golden section method 黄金分割法

Problem 16.
For the problem minf(x)\min{f(x)} over x[a,b]x\in [a,b], we use golden section method to solve it. Please prove the convergence ratio is 0.6180.618.

Solution
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Newton’s method 牛顿法

Problem 17.
For one dimension nonlinear problem minf(x)\min f(x) over x[a,b]x\in [a,b], we use Newton’s Method to solve it. Please prove the convergence ratio of Newton’s method is at least two.

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Gradient steepest descent/ ascent method 梯度下降/上升法

*Problem 4.
max f(x)=12x12+x1x2+2x112x22 \max \ f(x) = −\frac{1}{2}x^2_1 + x_1x_2 + 2x_1 − \frac{1}{2}x_2^2 \\ Use the gradient steepest ascent method to find the optimal solution of above nonlinear program, with the start point x0=(0,4)Tx_0 = (0,4)^T.

Solution
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Integer Programming 整数规划

Branch and bound 分支定界法

Problem 1.
Use branch and bound method to solve P1)P1).
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.
P1)max   13x1+5x2s.t.  3x1+2x26.52x1+3x28x1,x20,integers P1) \max{\ \ \ 13x_1+5x_2}\\ s.t. \ \ 3x_1+2x_2\leq 6.5\\ -2x_1+3x_2\leq 8\\ x_1,x_2\geq0,integers
Solution
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branch and bound tree
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Gomory cutting plane algorithm 割平面法/ Column generation 列生成算法

Problem 26.
mincTxs.t.Ax=bxinteger \min \textbf{\textit{c}}^T\textbf{\textit{x}}\\ s.t. \textbf{\textit{Ax}}=\textbf{\textit{b}}\\ \textbf{\textit{x}}\in integer For above IPIP, suppose now we have a basic feasible solution corresponding to basis matrix B\textbf{\textit{B}} and Non-basis N\textbf{\textit{N}}, i.e., A=(B,N)\textbf{\textit{A}} = (\textbf{\textit{B}},\textbf{\textit{N}}), and let aij=(B1Aj)ia_{ij} = (\textbf{\textit{B}}^{−1}\textbf{\textit{A}}_j)_iand ai0=(B1b)ia_{i0} = (\textbf{\textit{B}}^{−1}\textbf{\textit{b}})_i. Prove:
xi+jNaijxjai0 x_i+\sum_{j\in \textbf{\textit{N}}}\left \lfloor a_{ij} \right \rfloor x_{j}\leq \left \lfloor a_{i0} \right \rfloor
Solution
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Problem 12.
Use the Gomory cutting plane algorithm to solve the following integer linear programming.
min  x12x2               s.t.  4x1+6x29x1+x24             x1,x20,integers \min \ \ x_1 −2x_2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ s.t. \ \ −4x_1 + 6x_2 ≤ 9 \\ x_1 + x_2 ≤ 4 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ x_1,x_2 ≥ 0, integers
Solution
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Problem 28.
IP)mincTxs.t.  Ax=bx0 andintegers IP)\min{c^Tx} \\ s.t.\ \ Ax = b \\ x ≥ 0 \ and \in integers For above IPIP, denote A=[a1,a2,,an]A = [a_1,a_2,··· ,a_n] by columns. Let k<nk<n, denote Ak=[a1,a2,,ak]A_k = [a_1,a_2,··· ,a_k].
IPR)mincTxs.t.  Ax=b      x0 IPR)\min{c^Tx} \\ s.t.\ \ Ax = b \\ \ \ \ \ \ \ x ≥ 0 IPRk)mincTxs.t.  Akx=b      x0 IPR_k)\min{c^Tx} \\ s.t.\ \ A_kx = b \\ \ \ \ \ \ \ x ≥ 0 Suppose xkx_k^*, and xx^∗ is the optimal solution for subproblem IPRk)IPR_k) and IPR)IPR) respectively, and objectives of OBJIPRkOBJ_{IPR_k},OBJIPROBJ_{IPR} correspondingly. Let S={a1,a2,,an}S = \{a_1,a_2,··· ,a_n\}, BB is the basis related to xkx^∗_k for IPRkIPR_k.
PP)mincacBB1as.t.  aS PP)\min c_a −c_BB^{−1}a\\ s.t.\ \ a \in S Please prove: If OBJPP0OBJ_{PP} ≥ 0, then OBJIPRk=OBJIPROBJ_{IPR_k} = OBJ_{IPR}

Solution
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Dynamic programming 动态规划

IP 有整数约束

Problem 8.
Use dynamic programming method to solve the following integer programming.
max  2x1+3x2+x3+2x4            s.t.   5x1+7x2+6x3+5x414                 x1,x2,x3,x40,integers \max {\ \ 2x_1 + 3x_2 + x_3 + 2x_4}\\ \ \ \ \ \ \ \ \ \ \ \ \ s.t. \ \ \ 5x_1 + 7x_2 + 6x_3 + 5x_4 ≤ 14\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x_1,x_2,x_3,x_4 ≥ 0, integers Note that you need to point out recursive formula clearly, and write down calculation steps clearly.

Solution
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LP 无整数约束

*Problem 8.
Use dynamic programming method to solve the following linear programming.
max  2x1+3x2+x3+2x4            s.t.   5x1+7x2+6x3+5x414    x1,x2,x3,x40, \max {\ \ 2x_1 + 3x_2 + x_3 + 2x_4}\\ \ \ \ \ \ \ \ \ \ \ \ \ s.t. \ \ \ 5x_1 + 7x_2 + 6x_3 + 5x_4 ≤ 14\\ \ \ \ \ x_1,x_2,x_3,x_4 ≥ 0, Note that you need to point out recursive formula clearly, and write down calculation steps clearly.

Solution
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Linear programming 线性规划

Central path 中心路径法(内点法)

Problem 25.
Let (x(μ),y(μ),s(μ))(x(\mu),y(\mu),s(\mu)) be the central path of
xs=μeAx=bATy+s=cx0,s0 x◦s = \mu e\\ Ax = b\\ A^Ty + s = c\\ x ≥ 0,s ≥ 0 Then prove:
(a) The central path point (x(μ),y(μ),s(μ))(x(\mu),y(\mu),s(\mu)) is bounded for 0<μμ00 < \mu ≤ \mu_0 and any given 0<μ<0 < \mu < ∞.
(b) For 0<μ<μ0 < {\mu}'< \mu,
cTx(μ)cTx(μ)  and  bTy(μ)bTy(µ) c^Tx( {\mu}') ≤ c^Tx(\mu)\ \ and\ \ b^Ty( {\mu}') ≥ b^Ty(µ) Furthermore, if x(μ)x(μ)x( {\mu}')\neq x(\mu) and y(μ)y(μ)y( {\mu}')\neq y(\mu).
cTx(μ)<cTx(μ)  and  bTy(μ)>bTy(µ) c^Tx( {\mu}') < c^Tx(\mu)\ \ and\ \ b^Ty( {\mu}') > b^Ty(µ)
Solution
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Problem 11.
Compute the central path for the following linear programming.
min x1                         s.t.  x1+x2+x3=2 \min{\ x_1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ s.t. \ \ x_1 + x_2 + x_3 = 2
Solution
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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.

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Eliposoid method 椭球法(外点法)

Problem 7.
Use ellipsoid method solve:
max  0s.t.  x1+5x27        x1+2x26    x1,x20. \max \ \ 0 \\ s.t. \ \ x_1 + 5x_2 ≤ 7 \\ \ \ \ \ \ \ \ \ x_1 + 2x_2 ≥ 6 \\ \ \ \ \ x_1,x_2 ≥ 0 . The initial Ellipsoid is taken to be E(0,100×I2×2)E(0,100×I_{2×2}).
You only need to give THREE STEPS

Solution
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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 nn destinations to meet demand requirements. Specially, origin ii contains an amount aia_i, and destination jj has a requirement of amount bib_i. It is assumed that the system is balanced in the sense that total supply equals total demand. There is unit cost cijc_{ij} associated with the shipping of the commodity from origin ii to destination jj. 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 A\textbf{\textit{A}} is said to be totally unimodular if the determinant of every square submatrix formed from it has value 0,+10,+1 or 1−1. (完全幺模矩阵的各阶子式均为0,1或-1)
(1) Show that the matrix A\textbf{\textit{A}} 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 Ax=b\textbf{\textit{Ax}}=\textbf{\textit{b}}, assume that A\textbf{\textit{A}} is totally unimodular and that all elements of A\textbf{\textit{A}} and b\textbf{\textit{b}} are integers. Show that all basic solutions have integer components.
与Problem20(2)的区别在于:此时A\textbf{\textit{A}}是任意矩阵,秩不再是m+n1m+n-1

Solution
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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
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附原题

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