All Questions
16 questions
7
votes
2
answers
242
views
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$
\langle x_k, \theta_k \rangle &...
2
votes
1
answer
61
views
Counting the number of pair of d-uplets with upper bounded distance
Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
- 55
1
vote
0
answers
920
views
Maximizing a piecewise-linear convex function
Crossposted on Operations Research SE.
I am working on an optimization problem where some of the terms of the objective function to maximize are expressed as a piecewise linear function of variables:
...
- 111
2
votes
1
answer
237
views
Sampling algorithms on convex polytopes
Let $f=\mathbf{c}\cdot\mathbf{x}$ be the optimization objective function whose parameter vector $\mathbf{x}\in\mathbb{R}^n$ is subject to the following constraints in the very well-known linear-...
1
vote
1
answer
126
views
a linear programming problem
Recently I have a conjecture on decomposing a linear program into smaller ones. I have tested it in Mathmatica by a lot of examples. However, I cannot prove it. I will appreciate if someone can give ...
- 467
1
vote
1
answer
111
views
Optimal "Generalization" of Polylines
This question is inspired by a lossy compression technique for polylines, namely to identify a subset of the points of polyline $\mathcal{P}$, whose removal yields a polyline $\mathcal{Q}$ within a ...
- 13.2k
0
votes
1
answer
81
views
Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 3
16
votes
3
answers
1k
views
Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?
Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...
- 163
5
votes
1
answer
146
views
How does one go from convexity to submodularity?
If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...
- 1,893
3
votes
1
answer
553
views
Calculate the discrete set of points B which are in the convex hull of the set of points A
This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...
- 151
2
votes
1
answer
134
views
Integer point in a non-empty polytope
I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...
- 243
1
vote
1
answer
296
views
Deducing Linear Inequalities
Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...
- 185
2
votes
1
answer
2k
views
Finding a point farthest away from $k$ points in a polygon
There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized.
...
- 57
3
votes
1
answer
347
views
Grading a non-graded poset as squeezed as possible
Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage).
Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...
- 33.8k
7
votes
1
answer
360
views
Does the Hirsch conjecture hold for $n < 2d$?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
- 7,857
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015