All Questions
19 questions
4
votes
0
answers
46
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Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
1
vote
0
answers
36
views
Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?
Suppose that $P$ is a polyhedron represented by
$$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$
and $P$ contains interior points. Moreover, the ...
- 33
0
votes
0
answers
165
views
Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
- 235
0
votes
1
answer
131
views
How hard is a linear programming with a bounded constraint?
Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...
0
votes
0
answers
93
views
Number of vertices in a polyhedron
Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...
- 13.9k
1
vote
0
answers
172
views
continuity of linear programming
I have the following conjecture:
Given a closed convex set $S \subseteq \mathbb{R}^n$ and one of its exposed face $F=\{x \in S \mid \pi x = \pi_0\}$, where $\pi x =\pi_0$ is the supporting hyperplane ...
- 37
1
vote
1
answer
1k
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Check if a point is in the interior of the convex hull of some other points in high dimensions, and lower-bounding the largest enclosed ball [closed]
Given $m$ points $P=\{p_0, p_1, ..., p_m\}$ in high dimensions (e.g. 100), it is known that computing (or even representing) their convex hull $\text{conv}(P)$ is generally intractable due to the ...
- 11
1
vote
0
answers
68
views
Fundamental regions in convex programming
In linear programming, the fundamental regions are polyhedra, since those are the intersection of half-spaces defined by linear inequalities. In semidefinite programming, the fundamental regions are ...
- 13.9k
3
votes
1
answer
152
views
A question about polytopes related to linear programming
Given linear functions $f_1({\bf x}),\dots,f_n({\bf x})$ on ${\bf R}^m$, let $K = \{(a_1,\dots,a_n) \in {\bf R}^n:$ the $n$ halfspaces $\{{\bf x}: f_i({\bf x}) \leq a_i\} $ have nonempty intersection$\...
- 19.7k
0
votes
0
answers
368
views
Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
5
votes
0
answers
162
views
Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
- 1,150
-1
votes
1
answer
137
views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
- 19
0
votes
0
answers
68
views
A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
- 501
1
vote
1
answer
390
views
Efficiently Generating the Convex Hulls of Two Polytopes and Counting Faces
Suppose you have two polytopes $P_1, P_2 \in \Bbb{R}^n$ given by
$$ P_1 = \lbrace x: A_1 x \le b_1\rbrace$$
$$ P_2 = \lbrace x: A_2 x \le b_2\rbrace $$
I wish to find their convex hull, that is a ...
- 2,689
2
votes
0
answers
210
views
Finding optimal linear transformation for intersection of convex polytopes
I previously posted this on MathSE and am now trying here.
I have the following situation, as shown in the following diagram:
$W=\{w_i\}_{i=1..|W|}$ is a set of vertices (show in diagram in blue) ...
- 695
4
votes
1
answer
3k
views
Find the minimum distance between two convex hulls
We work over $\mathbb{R}^N$. Let $\mathbf{P}_1$ denote the hyperplane constructed using $N$ points, each of which is on a different axis (there are $N$ axes). We denote by $\mathbf{P}_2$ the convex ...
- 233
0
votes
0
answers
194
views
A linear program related question
Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, \alpha_2^...
- 769
17
votes
3
answers
6k
views
The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
- 1,513
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
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