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

learn more… | top users | synonyms

0
votes
1answer
18 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
27 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: ...
0
votes
0answers
21 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 $Ax'$ ...
0
votes
0answers
21 views

Chebfun arbitrary constant [on hold]

How do you have a constant which you can change in chebfun? For example: f=chebfun ('c./(x-1-1i)',[-100,100]); And, I would like to pick different values of ...
1
vote
0answers
24 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
131 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 ...
1
vote
0answers
65 views

An application of Motzkin's theorem

I don't know an argument in a lecture of my teacher about Farkas' lemma, Gordan's theorem and Motzkin's theorem. Let $a_1, \ldots, a_m$ be $m$ nonzero vectors in $\mathbb R^n$. The argument is: The ...
2
votes
2answers
45 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 ...
0
votes
0answers
26 views

Linear Programming - Finding the objective function

So I have this problem and I can't find the answer, only about half of it.. They give a graphic of the feasable region, with important points: $A(0,5), B(1,3), C(3,1)$ and $D(7,0).$ The point of ...
1
vote
1answer
51 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. ...
1
vote
2answers
25 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 ...
2
votes
2answers
48 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
votes
0answers
15 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 ...
0
votes
0answers
23 views

solving integer programming by simplex [closed]

In the LP formulation of Transportation problem, will the solution always be integers? what are the conditions under which an IP can be solved by simplex only? thanks
1
vote
1answer
29 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
vote
0answers
21 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
votes
1answer
42 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
25 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
vote
0answers
50 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
22 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
21 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 \$ , ...
0
votes
0answers
25 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.
1
vote
0answers
13 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
votes
0answers
26 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. ...
1
vote
0answers
14 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
23 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
votes
0answers
24 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
votes
0answers
33 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... ...
1
vote
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
votes
1answer
34 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 ...
0
votes
0answers
15 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
votes
0answers
13 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
votes
1answer
39 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 ...
0
votes
0answers
8 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
17 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, ...
0
votes
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.
0
votes
0answers
26 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 ...
0
votes
0answers
36 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 ...
0
votes
0answers
16 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
votes
2answers
41 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 ...
0
votes
0answers
16 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, ...
0
votes
1answer
30 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 ...
0
votes
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 ...
0
votes
0answers
17 views

Linear Programming Neighbourhood problem

In the n-city TSP what is the cardinality of $N_2(t)$ , the neighborhood of tour t determined by 2 change? What is the cardinality of $N_3(t)$?
0
votes
1answer
23 views

Primal feasible solution implies Dual optimal solution?

Every feasible solution of P puts an upper(or lower, depending on whether it is a maximization or minimization problem) bound on the optimal solution of D(assuming of course that D has a feasible ...
0
votes
1answer
38 views

Mixed Integer Linear Programming: Construction Rods

I have an interesting problem involving linear programming. The problem is the following, I have 4 different kinds of rods (rod sized found in the local market): 9m rod 11m rod 12m rod 15m rod ...
0
votes
1answer
16 views

Conditions for a LP to be integral

What are the conditions to be met by a LP that allows to infer that its optimum solution will be integral? Is total unimodularity a necessary and/or sufficient condition? What if all variables are ...
0
votes
1answer
17 views

LP Word Problem Construction

I am having difficulty constructing the constraints on a word problem as follows: The Brite-lite Company receives an order for 78 floor lamps, 198 dresser lamps, and 214 table lamps from condoski ...