The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
25 views

finding the value of a node in Pascal’s (a.k.a Yanghui's) triangle [on hold]

Image the Pascal Triangle is on an x-y cartesian plane. so that the values of the nodes, by location are ...
2
votes
1answer
50 views

Proving that $H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$

In the article about the Reduced Homology it's stated that $$H_0(X)=\tilde{H_0}(X)\oplus\mathbb{Z}$$ but I don't know how to prove that. I know $$H_0(X)=\bigoplus_{\alpha\in ...
3
votes
1answer
46 views

$\arg\max$ of an increasing function grows as the region grows.

$f_1,\dots,f_N:\mathbb{R}^+\rightarrow\mathbb{R}^+$ are strictly increasing, bounded functions whose derivatives monotonically decrease to $0$ as their argument increases. (Picture the shape of the ...
0
votes
2answers
28 views

How to check if point $x \in \mathbb{R}^n$ is in a $n$-simplex?

Is there any universal solution how to check if a point $x \in \mathbb{R}^n$ is in $n$-simplex for any number of dimensions (any $n$)? Is it possible to use Barycentric coordinates for any $n$? I ...
2
votes
0answers
14 views

A Particular Decomposition of the Simplex

Suppose I have a simplex $S_n$ with unit side-lengths. Fix a vertex $V$. Let $A_n$ be the convex polytope whose points are contained within the simplex, where the euclidean distance from each point ...
0
votes
0answers
28 views

Radio factory linear program

I need a help with this exercise. I’m supposed to write a liner program for the problem below and then solve it using simplex method, but I’ don’t know how to include all the factors into variables. ...
0
votes
0answers
46 views

How to Solve this maximization Problem?

You are given two s: N and K. Lun the dog is interested in strings that satisfy the following conditions: The string has exactly N characters, each of which is either 'A' or 'B'. The string s ...
0
votes
0answers
24 views

How to Solve this Linear Programming Problem?

$$\max[Z(x,y)=x+y]$$ $$-x+y\le 1$$ $$x\ge 0$$ $$y\ge 0$$ What i have done so far ? I tried simplex method , but i can't stop iterating . It really seems like a live lock . So how can i solve ...
0
votes
1answer
47 views

Every point in a simplex is a convex combination of p and a point in $C^{(p)}$

Let's fix an arbitrary point $p \in \Delta_n = \{(x_1, ..., x_n) \in \mathbb{R}^n \ : \ \sum_{i=1}^n x_i = 1 \}$ Could you help me prove that every point in a simplex $\Delta_n$ can be written as a ...
0
votes
0answers
16 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
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 ...
0
votes
0answers
34 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... ...
0
votes
1answer
52 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
1answer
27 views

Simplex algorithm question with restraints

How to perform simplex algorithm on the following: $$-x_1-2x_2 \rightarrow min \\ 4x_1+4x_2 \le 12 \\ x_1 \le 2 \ , x_2 \le 2 \\ x_1 \ge 0,x_2 \ge0$$ I would appreciate any hints how to solve this ...
4
votes
0answers
71 views

Barycentric coordinates in a 2D simplex

Let's fix a point $P$ inside the triangle $\Delta_3 = \{(p_1, p_2, p_3) | \sum _1^3p_i =1, p_i \ge 0\}$. So if we denote the triangle by $ABC, \ A = (1,0,0), B=(0,1,0), C= (0,0,1)$, we consider ...
0
votes
1answer
32 views

A very simple geometric/visual example of what a simplex looks like

I was trying to understand what a simplex was intuitively by constructing an example. Consider only points in $\mathbb{R}^2$. From wikipedia the definition seems to be: Choose k+1 points $u_0, ..., ...
4
votes
1answer
134 views

Convex hull of canonical basis of $\mathbb{R}^n$, face of a simplex

Let $I_n := \{1,2,...,n \}, \ p \in \Delta_n = \{(p_1, ..., p_n) \ | \ p_i \ge 0, \sum_{i=1}^n =1\}$ $ \text{supp (p)}= \{ i \in I_n \ | \ p_i \neq 0\}$ For a convex set $C$ we define $F$ to be its ...
0
votes
0answers
26 views

simplex algorithm with tableau

in the picture below you can see the tableau representation of a linear programm with a coefficient $b\in \mathbb{R}$. BV stands for basic variable. For which $b$ is $y_1=\frac{1}{2}, ...
0
votes
1answer
18 views

Approximating a binomial sum over a simplex

For partial binomial sums such as $\sum_{k\le\Delta} \binom{n}{k}$ we don't tend to have closed forms. However we still know asymptotic expansions that are easy to work with. Can we do something ...
2
votes
1answer
37 views

How does the Simplex method handle test ratios with zeros?

I've been running into an issue choosing a pivot when there are constraints with an RHS of zero. It appears that sometimes you should include zero test ratios when searching for the minimum test ...
0
votes
1answer
22 views

Can a basic variable be the entering variable in Simplex method?

I have got from my teacher that "the entering variable in a maximization problem is the non-basic variable having the most negative coefficient in the Z- row" I think X1,X2 are non basic and ...
0
votes
1answer
25 views

Trapped volume of (n-1)-sphere inside an n-simplex

Assume that we take an $n$-simplex (with side length of 2 units) and place a unit $(n-1)$-sphere at each vertex. For $n=2$, half of a circle is enclosed inside the $2$-simplex. For $n=3$, solid angle ...
0
votes
1answer
33 views

Explain initial basic feasible solution

Let us write x = (x$_0$, x$_s$)$^T$, where x$_0$ contains the original variables and x$_s$ contains the $m$ slack variables. Then it is obvious that by setting x$_0$ = 0, we have Ix$_s$ = b. ...
1
vote
0answers
89 views

Conversion into linear program

I have an optimisation problem with decision variables that are multiplied with another (a weighted average is calculated). I'd like to convert it into a linear program. I found this link that ...
0
votes
1answer
25 views

Beginner Simplex problem

Good evening, I have started studying the simplex method for some examinations I would like to take, and to be perfectly honest, I am stuck really bad. The basic examples and exercises are simple, ...
0
votes
1answer
27 views

Lexicographic Pivoting Rule: Why does that work?

On purpose, I do not write details about it, because I am interested in the intuition and not the details. But if someone asks for it, I can do that. So my question: Could anyone help me with the ...
1
vote
2answers
53 views

Represent if-else or OR condition in a linear equation (optimisation with simplex algorithm)

I would like to write some linear equations and inequations to state that the sum of all possive x - C is smaller than L. As my ...
0
votes
0answers
32 views

Find all optimal solutions by Simplex

Let "stable operation" be an operation on a simplex tableau such that the entering variable has a reduced cost of 0. Recall that a pivoting operation will not change the objective value if either the ...
0
votes
0answers
28 views

General formula for n-Simplex side-lenghts given n-volume and angles

Given a flat triangle's three angles $\phi_i $, and its area $A$, you can calculate the $i$th sidelenght $s_i$ (using Einstein's sum-convention) like so: $$ s_i=\frac{\sqrt{2A} \sin \left(\phi ...
3
votes
1answer
46 views

Simplexes in $\mathbb R^n$ have at most $n+1$ points

This is an exercise from the book Espaços Métricos (metric spaces) by Elon Lima. I'm translating it (the part of it that I'm having trouble with): Show that if $X\subset\mathbb R^n$ is such that ...
2
votes
1answer
55 views

Two-phase simplex

The upper part of the image attached is the question and lower part is the solution. I did the exact same table as table4 in the picture, yet I put the ratio of [2] as 0.5/0.5=1 instead. Could ...
1
vote
1answer
44 views

Can number of constraints be less than number of variables in Linear Programming?

In standard form of LP we have $n$ variable and $m$ constraint. In simplex algorithm we set all non-basic variable to zero and at most $m$ basic variable have positive value. if $m < n$, then at ...
1
vote
1answer
71 views

solving LP problem : no optimal solution exists?

$$\max[Z(x,y)=3x+2y]$$ $$-x+y\le 1$$ $$-x+2y\le4$$ When I tried to solve the above maximization LP problem using the simplex method, from the first iteration, all basic variables became negative. ...
1
vote
2answers
33 views

Correlation between Binary and N-dimensional simplexes

I found an interesting correlation between binary numbers and $n$-dimensional simplexes and I'm trying to find where I can find more information on the subject. I noticed that binary representations ...
1
vote
1answer
106 views

Linear programming algorithm that minimizes number of non-zero variables?

I have real world problems I'm trying to programmatically solve in the form of $$Z = c_1 x_1 + c_2 x_2 + \cdots + c_n x_n$$ Subject to \begin{align} & a_{11} x_1 + a_{21} x_2 + \cdots + a_{n1} ...
1
vote
0answers
39 views

product of barycentric coordinates over a simplex

Show that: $$\int_K \eta_0^{\alpha_0}(x)\cdots\eta_n(x)^{\alpha_n}dx=\frac{{\alpha_0}!\cdots{\alpha_n}!n!|K|}{(\alpha_0+\cdots+\alpha_n+n)!}$$ where $\eta_i$ barycentric coordinates and $K$ is a ...
2
votes
1answer
57 views

Integral over a simplex

Let $C_k$ be the $k$-simplex. I know that $$\int_{C_k} \prod_{i=1}^k x_i^{\alpha_i-1} dx_i = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^k \alpha_i\right)} \equiv ...
1
vote
0answers
37 views

Computing the Optimal Simplex Tableau for Linear Programming

I am learning in my class about computing the optimal simplex tableau. I learned that, if you have an initial basic feasible solution, you can apply a series of formulas to compute the optimal ...
1
vote
0answers
24 views

Degeneracy in Simplex Algorithm

According to my understanding, Degeneracy in a linear optimization problem, occurs when the same extreme point of a bounded feasible region $X$ can be represented by more than one basis, that is not ...
6
votes
2answers
109 views

How big is a tetrahedron?

Let $T$ be a tetrahedron with volume $vol(T)$ and edge lengths $a,b,c,d,e,f$ and let $sum(T) = a^3 + b^3 + ... + f^3$. We wish to compare $vol(T)$ with $sum(T)$. [ IMO (1961 #2 ) handles the case of ...
0
votes
1answer
64 views

Prove max f(x) = -min -f(x)

How do I prove : $$max f(x)= - min -f(x)$$ I am trying to prove this, and have tried to use my book but I am stuck.
2
votes
1answer
32 views

Singular chain complex for integration - pinching on boundary

Singular chain complex, as far as topology are concerned, is just continuous map from standard simplex, and the choice of using simplex over other shape is immaterial. But for integration on manifold, ...
1
vote
0answers
56 views

Explanations about the volume of a regular simplex

I'm really sorry, this may sound ridiculous but I can't understand the Wikipedia explanation about the volume of regular n-dimensional simplices, here. In particular, these two sentences make no ...
0
votes
1answer
42 views

Pachner moves for graph of 4-valent nodes

For 3-simplices (i.e. tetrahedra), I understand the basic idea behind the Pachner moves 1 $\leftrightarrow$ 4, which takes one tetrahedron and replaces it with four (or vice versa), and 2 ...
0
votes
1answer
88 views

Big-M Simplex Method Returns Unfeasible Solution

Consider the following linear programming problem: "Minimize 2x1 + 3x2 subject to the constraints 2x1 + x2 ≥ 4 2x1 - x2 ≥ -1 x1,x2 ≥ 0" Since there are ≥ signs in both ...
0
votes
1answer
46 views

Solve this linear program using 2 phase simplex

Minimize $2x_1 + 3x_2 + 3x_3 + x_4 − 2x_5$ Subject to $x_1 + 3x_2 + 4x_4 −x_5 = 2$ $x_1 + 2x_2 − 3x_4 +x_5 = 2$ $−x_1 − 4x_2 +3x_3 = 1$ $x_1, x_2, x_3, x_4, x_5 \geq 0$ Im not sure if im doing ...
0
votes
0answers
29 views

Solve equation with simplex method

I have equation below and I'm newbie to this method. Can you help me with tutorial or maybe with steps to solve this equation? I know I can use simplex tables, but I don't know a good explanation of ...
1
vote
0answers
24 views

How is distance between two points defined in barycentric coordinates?

Hope someone can help. I have this 3-d simplex (a tetrahedron) and its vertexes have barycentric coordinates defined as follow: $A_1=(1,0,0,0), A_2=(0,1,0,0), A_3=(0,0,1,0), A_4=(0,0,0,1)$. I am ...
0
votes
1answer
103 views

Linear programming and the simplex method

I am trying to solve this system of equations. My approach would be to introduce slack variables and then somehow use the simplex algorithm to solve this. Can anyone show me how this is done?
1
vote
0answers
112 views

Proof of Simplex Method, Adjacent CPF Solutions

I was looking at justification as to why the simplex method runs and the basic arguments seem to rely on the follow: i)The optimal solution occurs at some vertex of the feasible region (CPF points) ...