Questions on linear programming, the optimization of a linear function subject to linear constraints.

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1answer
105 views

Reference request: optimal solution for linear program is rational

From various lecture notes and such that are floating around, I get the impression that if a linear program has rational coefficients and has a finite optimal solution, then that solution will be ...
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0answers
1k views

What are canonical vectors?

I just begun with linear programming. Given an objective function $z$ and certain restrictions defined by $Ax = b$, we got to find the values necessary to maximise or minimise that function's ...
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75 views

Minimising waste in a cutting problem.

I have three possible board sizes: $8$, $10$ and $12$ feet long. I want to make some number of cuts to these, say, $3, 2,1,1,1,6,5,3,4,2,1$ feet cuts and I want to minimize waste. I've done a quick ...
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1answer
121 views

Linear Programming - Simplex Method

Let, $\begin{array}{lcl}Min: z =10x_{1}+5x_{2} \\ \\ 20x_{1}+50x_{2} \geq 200 \\ 50x_{1}+10x_{2} \geq 150 \\30x_{1}+30x_{2} \geq 210 \\x_{1},x_{2} \geq 0\end{array}$ a linear programming problem. ...
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1answer
162 views

can I get help in solving this equation using simplex method big-M method

Objective: $\max Z= 100x_1+300x_2+400x_3$ s.t. $10x_1+20x_2+30x_3≤1600$ $\;\,\quad10x_1+15x_2+20x_3≤1500$ $\;\,\quad x_2+x_3≤50$ $\;\,\quad x_1+x_2+x_3=70$ $\;\,\quad x_1,x_2,x_3≥0$
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1answer
307 views

Cycling in Simplex Method - Smallest Subscript Rule

Could someone explain to me how using the smallest subscript rule causes a cycling LP to terminate? At the moment it looks to me that a program would use it to determine whether the matrix from the ...
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0answers
448 views

How do I convert max min problem into a linear programming problem?

Let $A$ be a given $m \times n$ matrix, $c$ a given $n$-vector, and $b$ a given $m$-vector. Show that this problem $$\max \min (c^T x - y^T Ax + b^Ty) \text{ such that } x,y \ge 0$$ can be reduced ...
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0answers
522 views

Transportation problem: optimal solution

So I have an issue with finding the optimal solution (the lowest costs) to a transportation problem. Given the following problem, with $A$ the depots, $B$ the destinations and $C$ the $(i,j)$ matrix ...
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0answers
143 views

Post-optimality analysis: Change in one of the constraints

Consider the LP: \begin{array}{rccc} \max& \quad -3x_1&-x_2& & \\ \text{s.t.}& \quad 2x_1&+x_2 &\leq 3 \\ & -x_1&+x_2 &\geq 1 \\ &&x_1,x_2 &\geq ...
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2answers
105 views

How do I solve max min (x − y) and min max (x − y) such that y≥0 and x≥0?

solve max min (x − y) and min max (x − y) such that y≥0 and x≥0 I don't have a clue where to start.
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0answers
38 views

Linear programming problem - do we have enough data here?

I am to solve a following problem, but it seems to me that it is ill-formulated, i.e. there's not enough data. Am I right? If not what would be the mathematical model for it? Every coffee table ...
1
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1answer
32 views

Linear program that is taking a maximum over $n$ linear programs?

Suppose I have feasible linear programming problems $P_1, \dots, P_n$. Say $f$ assigns a feasible linear program its optimum value. How can I find a linear program $P'$, such that $f(P') = ...
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0answers
38 views

Is leq sufficient for the existence of a slack variable based bfs in a simplex?

I'm attempting to write a MPS to custom format converter for a generalized simplex algorithm and I am running into a couple of difficulties. According to this tutorial on the Big-M method for ...
3
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1answer
139 views

Runs in Rummikub simulation in AMPL

I'm taking a class in linear programming and the project involves modelling a Rummikub game. I have made the simplifying assumptions (for now) that there is no joker and only one piece of each ...
1
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1answer
71 views

Solving a minimization of the minimum problem

Let ${\bf c}_{1}$, ${\bf c}_{2}\in \mathbb{R}^{n}$, ${\bf A}\in\mathbb{R}^{m\times n}$ and ${\bf b}\in\mathbb{R}^{m}$. Show how one can solve the optimization problem: min ...
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1answer
29 views

Notation of a linear inequality system.

Sorry to bother with this rather trivial question, but nowhere in my lectures or books can I quite find out what the topmost line means. Maybe I'm forgetting something. Anyway: Line 2 and 3 are ...
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0answers
23 views

Determining maximum number of groups - maybe Linear Programming

Given a set D dogs, C cats, and B birds, for each dog d in D, there is a set c(d) which indicates the set of cats that dog d likes and a set b(d) birds that dog d likes. How do I find the maximum ...
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1answer
185 views

Underdetermined system with inequality constraints

I have an underdetermined system of equations of the form \begin{equation} Ax = b, \end{equation} where $A \in \mathbf{R}^{m \times n}$ with $m < n$, subject to \begin{equation}0 \preceq x ...
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1answer
74 views

A question about rational number.

Denote $M$ as a $m\times n$ matrix whose components are all nonnegative integers (actually 0 or 1) and $1$ as the $m$ dimensional vector $(1,1,\cdots,1)$. Show that: There is a vector $x_0$ ...
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1answer
71 views

Why optimization problems cannot be solved by simple derivative?

Let $f(\cdot)$ be a linear function. $f:\mathbb{R}^n\rightarrow\mathbb{R}$ $\;\quad\;\mathbf{x}\;\rightarrow f(\mathbf{x})$. Let $\mathbf{A}$ be a matrix in $\mathbb{R}^{m\times ...
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0answers
41 views

Optimizing over a set of optimization problems

This is my first time asking an optimization question on here, so I am looking forward to see what will happen here. In the lack of a better title, I wrote it as it is. At a high-level, I can perhaps ...
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1answer
29 views

making a function non-linear using a Lagrangian function

How Is this formula a Lagrangian function ? And how can a non-linear element be added to a function using this "Lagrangian function" This is where i got this In order to improve the performance ...
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1answer
25 views

Duality and Optimality Conditions

I have seen the solution and it involves adding a $x_5$ and $x_6$ to the inequalities. I really do not understand why this happens? I have not seen any questions like this yet. Any pointers would ...
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1answer
47 views

Is the polar of the polar set the original set?

For each $Q \subset \Bbb R^n$, denote $Q^*:=\{z \in \Bbb R^n:z\cdot x \leq 1,\;\;\text{for all}\; x \in Q\}$. Let $P:=\{x \in \Bbb R^n: Ax \leq b\}$, for the matrix $A$ and the vector $b$. It is ...
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0answers
114 views

Linear Programming, Optimal Solutions

I posted the whole question to give some context, but my problem lies with (iv). I think you're meant to use a formula for the generalization of the optimal solution, but I'm not really sure what ...
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1answer
118 views

Enquiry to network flow

Could anyone advise me on how to find a feasible flow to the following graph so that the edges $(2,5), (4,5), (6,5),(6,7)$ are saturated? This means, I have to formulate the network flow as a linear ...
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1answer
101 views

condition for having a positive solution to a linear equation.

Let $Y$ be a member of $\mathbb{R}^m$. I need a necessary and sufficient condition on a $n\times m$ binary matrix $A$ for having a solution to the linear equation: $$AX=Y$$ Such that $X_i\geq 0$, ...
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1answer
109 views

Min-Cost-Flow Problem

Given a directed graph $G = (V,E)$ with a cost function $\gamma: E \to \Bbb R_{\geq 0}$ and two vertices $u,v \in V$. How to reduce the problem of finding a directed path from $u$ to $v$ with minimum ...
0
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1answer
145 views

Linear programming with non-convex quadratic constraint

Could anyone let me know if the following linear programming problem can be solved in polynomial time or should be NP-hard? $\min c^Tx$ s.t. $x^TQx\geq C^2, x\in [0,1]^n,c\in ...
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1answer
74 views

Reduced Cost in Network Simplex Algorithm

On page 5 of the slide, [T]he reduced cost of a non-basic arc $(i, j)$ is the sum of the costs of the arcs forming a cycle with $(i, j)$ in the current tree solution. Why is that the case?
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3answers
841 views

Linear Programming to find the loan plan to minimize the interest payment

Assume that it is the first of July and you are running a small shop. The sales revenue and the amount of bills you have to pay for the next six months are estimated as following: In short, you ...
0
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1answer
19 views

Formulation of linear problem

I'd like to ask you how to formulate this problem as linear problem (equations)? Marie wants to buy oranges and apples. She has to buy at least 5 oranges and the number of oranges has to be less ...
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1answer
21 views

Finding matching weight for two differing types of cat food

Attempting to figure out how much cat food to give my cat I came across a problem which I am unsure of any way other than iteration to solve. The problem I have is that I have been advised to feed my ...
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1answer
41 views

Linear programming with reuse of services

I came across some questions of this style and was not sure what the minimization function would be. A hotel requires a known number of hand towels for guests to be given during the week and the ...
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1answer
66 views

Linear Inequalities - Allocation Problem

The problem at hand can be summarized as follows: we have to allocate a ressource to $n$ production units. The allocation to production unit $i$ is $x_i$. Each of the production unit will produce at ...
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1answer
24 views

Where did i go wrong in this linear inequation evaluation?

We are currently studying Linear Programming in school and while going through it i seem to of come across a ridiculous error. Problem is, i can't seem to find it. Essentially there is an equation 8 ...
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1answer
157 views

orthogonal triangular decomposition and ordinary least squares

I have just come across orthogonal triangular decomposition whilst looking at ordinary least squares regression. I'm not quite sure how this is being used though to find a solution. In my example I ...
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1answer
25 views

Gradient Question-Linear Regression

When discussing linear regression, we discuss the error of the out of sample data prediction. That is, $$ E_{\operatorname{out}} = \frac{1}{N} \sum_{n=1}^{N} ...
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1answer
79 views

Is it possible to get the dual solution “quickly” once the optimal primal solution is found?

With the primal objective value, I know the dual objective value. I also know the right hand sides of the original program. However, I don't know the values of the dual variables at the optimal point. ...
2
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0answers
76 views

Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
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2answers
163 views

Minimize the minimum - Linear programming

Consider an optimization problem with variables $x_1, x_2, \dots, x_n \in \mathbb{R}$ (maybe subject to some linear constraints), and linear functions $\{f_i(x_1, \dots, x_n)\}_{1\leq i\leq m}$. We ...
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1answer
106 views

Can the search space of a solvable linear optimization problem be discontinuous?

Background Say you have a traditional linear-optimization problem, there is a linear cost function, $\vec{c}\cdot\vec{x}$ and a set of linear constraints, $A_1\vec{x} \geq b_1 $ $A_2\vec{x} \leq ...
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1answer
502 views

How to convert a linear optimization problem into a normal form?

The following linear optimization problem is given: $$ \begin{eqnarray} x_1 + 2x_2 -7x_3 \leq 1\\ |3x_1-5x_2-20| \leq 4 \\ x \geq 0 \\ 6x_1+5x_2-3x_3 \rightarrow min \end{eqnarray} $$ And it is my ...
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1answer
282 views

Get reduced costs from simplex tableau

This is probably a dumb question... but I'm trying to find how to calculate the reduced cost for a particular variable based on the information in a simplex tableau after I've minimized a linear ...
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3answers
526 views

Linear Programming and differentiation, why can't we differentiate to find the optimum solution?

I do understand that differentiating a linear function (for a maximization) subject to some linear restriction (such as the problem $p=ax+by$ s.t. $cx+dy \leq m$) won't necessarily give me the right ...
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0answers
66 views

Online convex programming: Projection followed by normalization

I have the following projected gradient descent online linear programming problem which has been well studied in www.cs.cmu.edu/~maz/publications/techconvex.pdf‎ $\mathbf{y}_{t+1}=\mathbf{w}_t - ...
2
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0answers
62 views

Assigning jobs to minimize cost - Linear programming

I'm stuck trying to solve this linear programming question. You want to make a website with a list of features F, which are n elements long. Each feature has a corresponding value for how long it'll ...
0
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1answer
116 views

Existence of Integer solution for a set of linear equations

Can anyone give a proof sketch of the following claim: If a system of homogeneous linear equations with integer coefficients has a positive real solution, then it also has a positive integer solution? ...
4
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1answer
312 views

Writing a linear program in standard form

Usually I have been asked to write problems in standard form that have inequalities involved. However, this problem has none and I was wondering if anyone had insight on how to go about solving it. ...
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117 views

Linear Programming with Matrix Game

It seems from an easy google of "learning linear programming" that a common way of learning it is to work with Matrices that represent "games" for two players. Here is one I have stumbled across. We ...