0
votes
1answer
23 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 $X_i\geq 0$, ...
1
vote
0answers
62 views

Area of a 2D convex polytope made of halfspaces

For a computer program I am attempting to solve the area of a convex polytope defined by a finite number of halfspaces. I understand that this forms a polygon and given the vertices of a polygon I am ...
4
votes
2answers
56 views

Very symmetric convex polytope

Let $C_n$ be the convex polytope in ${\mathbb R}^n$ defined by the inequalities (in $n$ variables $x_1,x_2, \ldots ,x_n$) : $$ x_i \geq 0, x_i+x_j \leq 1 $$ (for any indices $i<j$). Denote by ...
2
votes
1answer
301 views

Polytopes and matrices

How can one show that the vertices of a polytope are the matrices contains 2 ones on each row and col? and if $M \in P$ is not a $Z_2$ matrix then $M$ is a derived ...
3
votes
1answer
182 views

Find the vertices of the polytope

Let $x,n$ be 2 integers with $x<n$. I need to find the vertices of the polytope $P$ of $2 \times n$ nonnegative matrices $A$ such that: The first row in $A$ is summed to $x$. $$\sum_{j=1}^n ...
1
vote
2answers
743 views

Explain `All polyhedrons are convex sets´

My teacher in course in Mat-2.3140 of Aalto University claims that 'All polyhedrons are convex sets' here. This premise was in a false-or-not-problem 'The feasible set of linear integer problem is ...
3
votes
0answers
61 views

relation between solution of a linear program and its perturbation

I have a linear program over a finite set of points $(x_1, x_2,\ldots, x_m)\in\mathbb{R}^n$: $$ \max_j c' x_j $$ Suppose the solution of this LP is obtained at a point $x_{j_1}$, which is a vertex ...