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

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4
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41 views

Parameterizing equilateral polygons

I'm not exactly sure how to describe what I want, so if I butcher terms, please forgive me :) I want to "parameterize" the space of simple irregular equilateral polygons with n sides, or at least a ...
0
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1answer
28 views

How to write this to a linear programming problem?

A procedure of animal feed makes two food products: F1 and F2. The products contain three major ingredients: M1, M2, and M3. Each ton of F1 requires 200 pounds of M1, 100 pounds of M2, and 100 pounds ...
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0answers
11 views

Linear Programming Duality with Big M

I wanted to check of my proof for the following is correct. I am least sure of step 3. Given a linear program $LP1$. $$\text{minimize}\left\{\sum_{i\in I}c_iy_i\right\}\\ \text{subject to, }\\ ...
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1answer
23 views

Question about problem linear programming math modeling

Consider points $A(4.7,−4.1,−1.5)$,$B(−0.4,−2.4,1.9)$,$C(−0.3,−2.1,−6.5)$ and $D(2.7,−3.6,4.0)$. How to discover if segment $AB$ has intersection different of zero with the segment $CD$? Formulate ...
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0answers
28 views

Mixed strategies as LP problem

A row player is playing against a column player and his yield table is -, C1, C2, C3 R1, -3, 2, -1 R2, 0, -2, 1 R3, -1, 3, -5 Is it then correct to ...
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0answers
23 views

How do I convert this to a linear programming problem?

It takes a tailoring 2 hours of cutting and 4 hours of sewing to make a knit suit. To make a worsted suit, it takes 4 hours of cutting and 2 hours of sewing. At most 20 hours per day are available for ...
0
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1answer
36 views

How do I convert this into a linear programming problem?

A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of \$50 and each acre of barley yields a profit of \$70. To sow the crop, two machines, a tractor and tiller, are ...
-2
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1answer
42 views

Modelling Problem in Linear Programming Standard Form

I'm having a hard time setting this up, so that's what I need help with. The solving I understand. We’re making a drink with the following requirements: at least 500 calories, at least 20 mg. of ...
0
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1answer
60 views

convertion into integer linear program

I am trying to model the Ising spin state problem into Integer linear program and find the optimal ground state using lp_solve. (This is just a miniature version of Ising state problem) $$ maximise: ...
2
votes
1answer
38 views

How to remove fields from sudoku puzzle in such way to assure there's still only 1 solution?

I'm trying to create a Sudoku puzzle (programatically, if that matters). Here's how I do it. STEP 1: Creating an initial set, with unique solution: 123456789 456789123 789123456 ...etc... STEP 2: ...
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0answers
23 views

finding the dual

I am supposed to find the dual of max $c^Tx$ subject to $a \le Ax \le b$ $l \le x \le u$. In order to find the dual I think I have to write it in standard form, the standard form is: max $Ax'$ ...
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0answers
47 views

Tucker's theorem from Farkas lemma

I am trying to understand the proof of Tucker's theorem using Farkas lemma but there are some points that are not clear to me. The proof I am following is in this paper at page 16. What I do not ...
3
votes
1answer
142 views

Can I know all the elements of a matrix given that I know its sum along one dimension and the fact that it is axisymmetric?

For this discussion I will assume a 9x9 matrix but my question is for a general nxn matrix. I have a matrix which is not only symmetric along the vertical and the horizontal axis, but is axisymmetric ...
2
votes
2answers
72 views

How to maximize the sum of vectors in target direction.

Given a number of vectors, and an unknown variable for each vector, say for example: $v_1, v_2, v_3,\dots,v_n$ and $x_1, x_2, x_3,\dots,x_n$ and a target vector $v_t$ I am trying to create an ...
1
vote
1answer
67 views

Can you generate math problems that are solveable?

If you take Linear Programming, it problems are formulated like this: You know that Cabinet X costs 10 cents per unit, requires 6 square feet of floor space, and holds 8 cubic feet of files. ...
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2answers
37 views

Linear equations - how to find the solution over the boolean field closest to zero

I want to solve a system of linear equations over the field of $F_2$, in a way such that the solution vector is as close to the zero vector as possible. For example, suppose I have a system of ...
3
votes
2answers
88 views

In a linear program, how to add a conditional bound to x?

I am working with a standard linear program: $$\text{min}\:\:f'x$$ $$s.t.\:\:Ax = b$$ $$x ≥ 0$$ Goal: I want to enforce all nonzero solutions $x_i\in$ x to be greater than or equal to a certain ...
0
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0answers
38 views

Simplex method: tableau at some stage, finding objective row

How do I find the objective row for the tableau if all I am given is the tableau values at the certain stage (without RHS)? Here is the tableau $T$ without the objective row: $$ \begin{bmatrix} 0 ...
1
vote
1answer
30 views

Simplex updates for the inequality LP

Consider the task of minimizing $c^Tx$ subject to the constraint that $Ax \leq b$. I had a couple of questions in relation to the simplex algorithm (applied to this problem): How does one ...
1
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0answers
22 views

Find the vertices of the polygon given by $|f_1(x,y)|+…+|f_n(x,y)| \le C$

Given functions $f_1(x,y),...,f_n(x,y)$, we know that the locus of points $(x,y)$ satisfying $|f_1(x,y)|+...+|f_n(x,y)| \le C$ for some real constants $C$ is the interior of a polygon. How do I find ...
0
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1answer
72 views

Explicit solution for a linear program with two constraints

This is not a homework problem, although it wouldn't surprise me if it happens to exist in a textbook somewhere. Is there an explicit solution for the linear program $$\max_x c^Tx ~~ s.t. \\ d^Tx = q ...
2
votes
2answers
29 views

Formulation of Linear Programming problem?

I want to maximise the function: $$l(\beta,\sigma,\alpha) = -n\log(\sigma) - \frac{1}{\sigma} A(\alpha)\vert{\bf y}-{\bf X}\beta\vert,$$ where $\vert \cdot \vert $ represents the entry-wise absolute ...
1
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0answers
54 views

How to solve this using computer.?

Given $B = \begin{pmatrix} 0.3 & 0 \\ 0 & 0.4 \\ \end{pmatrix}$, and $\pi = \begin{pmatrix}0.4\\0.6\end{pmatrix} $, I need to find the elements of the stochastic matrix (the rows sum to ...
2
votes
1answer
37 views

Simplex Method: simplifying constraints

In my Computer Science class we've been exploring the Simplex Method and the applications it has with discovering optimal solutions. I've loved the challenge how much easier it makes finding solutions ...
1
vote
1answer
23 views

Changing a linear map such that given properties are satisfied

We are given $\{v_1, \dots, v_s\} \subseteq \mathbb{R}^n$, all with the same euclidean norm, say $\|v_i\| = \sqrt{(v_i^{(1)})^2 + \dots + (v_i^{(n)})^2} = 1$. Let's also assume $v_i \notin ...
0
votes
1answer
17 views

Optimalization, plan comparision

Let's say there are two tariff plan options of a provider offering internet access and landline telephony. Option 1: DSL flatrate, landline flatrate : 29,95 \$ Option 2: DSL flatrate: 24,95 \$ , ...
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0answers
30 views

Transforming into a convex program

$\max c^Tx$ $s.t. xy = a, \quad x \le b, \quad L \le y \le H$ Is there a way to transform this problem into a convex problem? $a,b,L,H$ are constants. $x,y$ are optimization variables.
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0answers
23 views

Dantzig-Wolfe Decomposition

While reading revised simplex method, I came to know about Datnzig-Wolfe Decompostion. Can you please explain whats the connection here ?
0
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0answers
28 views

Linear Programming $\boldsymbol{c}^T \boldsymbol{x}$ s.t. $\boldsymbol{Ax} = \boldsymbol{b}$

Prove for the linear programming \begin{equation} \left\{ \begin{array}{cc} min & \boldsymbol{c}^T \boldsymbol{x} \\ s.t. & \boldsymbol{Ax} = \boldsymbol{b} \end{array} \right. ...
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0answers
25 views

Linear program of 0-1 knapsack problem and proof of integer

I have some questions about the knapsack problem. How can the 0-1 knapsack problem described as a linear program? How to proof that the solution of the 0-1 knapsack problem are integer? (I'm ...
1
vote
1answer
38 views

Linear optimization problem with additional constant cost for non-zero variables

I have a linear optimization problem with integer variables of the form minimize $a_1 x_1 + ... + a_n x_n$ under a set of constraints Bounds for each variable $a_i \le constant_i$ Bounds for ...
0
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0answers
30 views

Feasible set for linear system with linear constraints

I have a linear underdetermined system $Ax = b$ with constraints $0 \le x \le 1$. Matrix $A \in \mathbf{R}^{n \times m}$ with $n < m$, elements of which are either $0$ or $1$, and sum of each ...
0
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0answers
37 views

how to solve a simplex with n variables

I don't know how to resolve a simplex with n variables I have this primal problem \begin{cases} \text{min}& z=-x_1 - x_2 -... - x_n\\ &a_1x_1 + a_2x_2 +... + a_nx_n \le 1\\ &x_1... ...
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0answers
33 views

A System of Inequalites arising from the Divisors of a Number, Showing Its Non-Solvability

Let $n$ be a natural number. Denote by $d(n)$ the number of divisors of $n$, i.e. with the notation from Wikipedia:Divisor Function we have $d(n) = \sigma_0(n)$. Suppose we have the $d(n) - 2$ ...
0
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1answer
176 views

How to test if a feasible solution is optimal - Complementary Slackness Theorem - Linear Programming

I have this linear program $$\begin{cases} \text{max }z=&5x_1+7x_2-3x_3\\ &2x_1+4x_2-2x_3&\le8\\ &-x_1+x_2+2x_3&\le10\\ &x_1+2x_2-x_3&\le6\\ &x_1,\,x_2,\,x_3\ge0 ...
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0answers
21 views

Interpretation of the dual of shortest path problem

I am trying to find an interpretation of the dual of shortest path problem of the form: $ maximze \; y_s - y_t$ subject to $y_i - y_j \leq c_{ij}$ $y_i$ are unrestricted, $c_{ij}$ being the cost ...
0
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0answers
17 views

Am I understanding this question correctly? Linear programing

A company produces and sells two products A and B. Let x denote the number of items of product A and let y denote the number of products of product B. The profit is DKK 30 per unit of product A and ...
0
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1answer
60 views

If we start with a feasible tableau in simplex method, are we basically generating a different feasible point in every pivot step?

This is a true and false question. The actual question reads: "In solving a linear program by the simplex method, starting with a feasible tableau, a different feasible point is generated after every ...
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0answers
9 views

How to test if a set of underdetermined equations have solution in a particular region?

For a underdetermined system $A\cdot x = b$ where $A$ is a $m \times n$ matrix with $m<n$, how to test if it has a solution within a specific region $\{ x | lb<x_i<ub \}$? Basically I have ...
0
votes
1answer
24 views

finding a plane in R3 given 5 points

I have multiple vertices that I need to create a plane out of. What is a formula/method for having a plane that contains 5 points? I believe I could use just 3 and the other two will be included
2
votes
1answer
26 views

Matrix that doesn't lose zeros

I have a vector $\vec x$ with some entries that are zero, and I apply a linear transformation so that for some matrix $A$, $A \vec x$ = $\vec b$. I would like to find a matrix $A$ (or better yet, ...
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0answers
13 views

Give a solution set of a dodecahedron [duplicate]

Using basic inequalities, find a set of inequalities so that the set of all solutions is a dodecahedron with twelve pentagons for sides.
3
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1answer
18 views

linear programming and flow network

Here is the problem: I have hard time understanding the problem , what does it mean by "conservation factors" and how to approach the problem using linear programming. For what I understand, if a ...
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0answers
36 views

How can I find multiple solutions for a system of equations?

I'm writing a program for CheckIO.org that is supposed to return an array, $$ \begin{bmatrix} x\\ y\\ z \end {bmatrix} $$ , that satisfies the System of Equations $$ A \begin{bmatrix} x\\ y\\ z \end ...
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0answers
42 views

The set of all vectors satisfying $Ax\ge\vec{0}$

Consider a rectangle matrix $A\in\mathbb{R}^{m\times n}$ with $m\ge n$, and the set of all vectors $x\in\mathbb{R}^n$ satisfying $Ax\ge\vec{0}$. I note this set is closed under multiplication by a ...
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0answers
20 views

Uniqueness of Solution in infinite linear programming

I would like to ask about a sufficient condition under which a solution for an infinite linear programming is unique. In standard finite dimensional linear programmings, like $\min_x p\cdot x$ ...
0
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2answers
57 views

Setting up a linear programming word problem

Problem: A metalworking shop needs to cut at least 32 large disks and 219 small ones. There are three cutting patterns for the standard size metal rectangle. One cutting pattern produces two large ...
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0answers
32 views

How is the pivot chosen for the symbolic weights for the Cassowary algorithm?

I am trying to understand The Cassowary Linear Arithmetic Constraint Solving Algorithm, and I am having trouble understanding symbolic weights, starting in section 2.3. Working through the example, ...
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1answer
76 views

Linear Programming - deriving the Dual of the Primal

I've the following linear programming problem: This is the LP representation of the uncapacitated facility location problem. This is the dual representation of this problem: My question is how ...
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0answers
7 views

Linear Programming Neighborhood Problem 2

Suppose we are given a set of 2n integers and we wish to partition into 2 sets $S_1$ and $S_2$ so that |$S_1$|=|$S_2$| and so that sum of number in $S_1$ is as close to sum of numbers in $S_2$.Let ...