All Questions
608 questions
5
votes
1
answer
280
views
Classification of vertex-transitive zonotopes
Zonotopes are convex polytopes that can be defined in several equivalent ways:
parallel projections of cubes,
Minkowsi sums of line segments,
only centrally symmetric faces,
...
I wonder whether ...
- 13.6k
4
votes
2
answers
172
views
4-polytopes with only one kind of regular facet
Is there a neat way to show (or a reference that already proves) that
the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
the 24-cell is the only convex 4-polytope in ...
- 13.6k
3
votes
1
answer
152
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
- 13.6k
1
vote
0
answers
163
views
Can we reduce the maximization of this integral to the maximization of the integrand?
I would like to know whether we are able to reduce the following optimization problem to the pointwise optimization of the integrand (or how we can solve it otherwise): Maximize $$\sum_{i\in I}\sum_{j\...
- 167
0
votes
1
answer
99
views
Finding dual of a scheduling LP formulation
Suppose I have an LP formulation as such:
$\min\ \ \sum\limits_{i,j,t}\ w_{ij}x_{ijt} (\frac{t-r_j}{p_{ij}}+0.5)$
$\sum\limits_{i,t}\frac{x_{ijt}}{p_{ij}}=1\,\forall\ j$
$\sum\limits_{j}x_{ijt}\leq ...
- 103
0
votes
1
answer
113
views
How do I solve this integer programming problem with non convex constraints?
I am not sure if this is the right place to post this question, please point me to the correct forum if I posted in a wrong place.
I have an optimization problem like this
...
- 103
2
votes
0
answers
159
views
The topological complexity of polytopes
Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be ...
- 3,871
0
votes
0
answers
101
views
How can we analytically solve this max-sum-min problem?
Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
- 167
0
votes
1
answer
61
views
Variant of the linear programming problem
Good afternoon, my experience in mathematical programming is low. I would like to know if there is any general method to address the following problem:
$$\text{Minimize }\sum_{i=1}^n d_i(x_j)$$
$$s.a....
- 193
2
votes
0
answers
148
views
Generalization of Farkas' Lemma to Hermitian Matrices
I recently stumbled upon a well-known version of Farkas' Lemma which, roughly speaking, I would like to generalize from real vectors to hermitian matrices, as it seems promising for something else I ...
0
votes
0
answers
35
views
Converting a vector in a cone statement to inequality constraints
I would like to convert the following condition for $x$
\begin{align}
x = N \lambda, \lambda \geq 0
\end{align}
to a pure linear inequality form, i.e. find an $L$ and eliminate $\lambda$
\begin{...
- 1
1
vote
0
answers
25
views
Weird subspace/equality-constrained LP problem/variant of change-making problem
Assume that we have a set, $\mathscr{R}$ containing $m$-dimensional vectors. Solve
$$\sum_{i=1}^n c_i\leq\delta$$
$$\text{subject to } \sum_{i=1}^n r_i c_i=x^\prime \text{ for all }x^\prime$$
where
$0\...
- 11
6
votes
1
answer
861
views
Is Binary Integer Linear Programming solvable in polynomial time?
The paper Solving the Binary Linear Programming Model in Polynomial Time claims that Binary Integer Linear Programming is in P. However, it seems that no subsequent literature in the mainstream has ...
- 221
8
votes
3
answers
390
views
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...
- 13.6k
3
votes
1
answer
1k
views
Finding the closest special orthogonal matrix in Frobenius norm sense
Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes
\begin{equation}
\|R-M\|_F
\end{equation}
then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are ...
- 33
1
vote
0
answers
49
views
Realizing 0/1-polytopes with shortest possible edge lengths
Has there been something written about the following question?
Question: Given a 0/1-polytope, what is the shortest edge lengths with which this polytope can be realized as a 0/1-polytope.
The ...
- 13.6k
3
votes
0
answers
63
views
Matroids which are transitive on minimal basis exchanges
I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that.
Consider a finite matroid $M$. Define a ...
- 13.6k
6
votes
1
answer
337
views
Edges of the contact polytope of the Leech lattice
Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$.
Question: What are the edges of $P$?
Let'...
- 13.6k
1
vote
1
answer
475
views
Sufficient conditions for a system of linear inequalities to admit a solution
I am looking for sufficient conditions such that a system of linear inequalities of the type $A x >0$ admits a non-negative solution $x \in \mathbb{R}^n_+$. I know a few properties of the $m \times ...
- 355
3
votes
1
answer
275
views
Uniqueness of l1 minimization
Let $A \in \mathbb{R}^{m \times n}$.
Is it true that $$\min \limits_{Q \in \mathbb{R}^{n \times m}}|I - QA|_{\infty} < \frac{1}{2}$$ is criteria for the uniqueness of the 1-sparse solution to
$\...
- 33
1
vote
0
answers
177
views
Prove that these linear programming problems are bounded by $O(k^{1/2})$ [closed]
The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems:
$$\begin{array}{ll} \text{minimize} &...
- 1,183
2
votes
2
answers
242
views
Products of polytopes and the normals of their facets
I need to compute the normals of the facets of certain polytopes that can be represented as products of polytopes in smaller dimensions.
Searching the bibliography I found that the facets of the ...
- 21
0
votes
0
answers
232
views
What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
- 1,183
3
votes
0
answers
178
views
Uniqueness of projection under spectral norm
I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
- 131
5
votes
1
answer
315
views
On optimal dual solutions for the minimum weight perfect matching problems in the case of metric weights
Following Lovasz-Plummer (Matching theory, North-Holland 1986, Theorem 9.2.1),
the minimum weight perfect matching problem on a complete graph
$G$ with even number of vertices and weight $w:E(G)\to
\...
- 4,878
3
votes
1
answer
244
views
What importance does the Hirsch conjecture have to Simplex Complexity?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
- 1,826
3
votes
2
answers
179
views
Number of bitangents to convex polytopes
Let me state my question prior to defining terms:
Q. Let $P_1$ and $P_2$ be disjoint convex polytopes
in $\mathbb{R}^d$ of $n$ vertices each.
What is the maximum number of distinct bitangent
...
- 151k
1
vote
0
answers
36
views
Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
- 19
2
votes
1
answer
243
views
Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
- 1,826
2
votes
0
answers
29
views
When do projection maps of polyhedra factor?
Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...
- 1,826
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
3
votes
1
answer
289
views
On decomposition of polytopes
Given $m$ number of convex polytopes each with $v$ vertices and described by $h$ hyperplane inequalities in $\mathbb R^t$ are there operations on these polytopes that combine then to give an $v^{\...
- 1,826
1
vote
0
answers
126
views
Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
- 1,826
1
vote
0
answers
65
views
Covering a simplex efficiently by efficiently describable polytopes?
Take a standard simplex or cube in $\mathbb R^n$.
Is there a way to cover it with $O(poly(\log n))$ convex polytopes each describable by only $O(poly(\log n))$ half-plane inequalities?
If not what ...
- 1,826
1
vote
0
answers
129
views
Is there a method to cut a hypercube into disjoint cubes [closed]
Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$
so we can cut a hypercube into at least $n+1$ disjoint parts.
Is there a method how can one do that?
- 139
1
vote
1
answer
125
views
Projections of particular simplex yielding boundary of a regular polygon?
What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
- 1,826
0
votes
1
answer
39
views
Gluing simplices through a common point/ realisation of a convex simplicial polytope
Given $m≥d+1$
a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that
1) they all share the common vertex M
2) the ...
- 1,435
1
vote
1
answer
126
views
Quantifier elimination and where is this quantified convex program in the polynomial hierarchy?
I have a quantified convex program of the form that I need to solve
$$\exists(x_{1,1},\dots,x_{1,n})\in\mathbb R^n\quad\forall(x_{2,1},\dots,x_{2,n})\in\mathbb R^n$$
$$\vdots$$
$$\exists(x_{2t-1,1},\...
- 1,826
7
votes
2
answers
909
views
Formula for volume of a convex polytope
So I've been searching around the internet for some answers to this, but I currently have a set of linear constraints: $Ax = b, Cx \le d$ for matrices $A \in \mathbb{R}^{n \times m}$, $b\in \mathbb{R}^...
- 81
1
vote
0
answers
24
views
Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
- 4,049
0
votes
0
answers
89
views
Why there isn't lexicographically smallest solution to a bounded linear program?
I am learning computational geometry when I run into this confusion. "A bounded 2D linear program may not have a lexicographically smallest solution", as the book says. I wonder why? I think we can ...
- 1
1
vote
0
answers
37
views
Fast certficate of negativity for objective value of mixed-integer linear program
Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
2
votes
1
answer
270
views
What optimization problems have solutions with few nonzeros?
Consider the following optimization problem, with $n$ variables and $m$ linear constraints:
\begin{align}
\text{maximize} && c_1 x_1 + \cdots + c_n x_n &
\\
\text{subject to} && a_{...
- 3,549
3
votes
2
answers
331
views
Program to solve Optimization Problem
I have an optimization problem, this problem has linear constraints and nonlinear constraints. I solved the linear part by MATLAB but the nonlinear constraints I could not solve it. I downloaded ...
- 31
13
votes
3
answers
834
views
Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
7
votes
1
answer
374
views
Sampling uniformly from the vertices of a polytope
I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...
- 15.4k
9
votes
2
answers
843
views
How did they come up with the MRRW bound?
Among the good asymptotic bounds in coding theory in the MRRW bound. It is obtained by using the linear programming problem of Delsarte's and providing a solution. The LP problem is
Suppose $C \...
- 885
1
vote
1
answer
330
views
How good is the LP relaxation?
Consider the optimization problem
\begin{align}
\max_{x\in\mathbb{R}^n}~c^Tx~, \text{ s.t. } Ax=b,~x_i\in\{0,1\}~\forall i
\end{align}where $c,b\in\mathbb{R}_{+}^n$ and $A\in\mathbb{R}_{+}^{n\times n}$...
- 1,421
0
votes
2
answers
102
views
Does Max Flow produce uniform results? [closed]
I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have ...
- 11
1
vote
3
answers
138
views
Literature request: Function that depends on a linear optimization problem [closed]
my question is about functions of the following form:
$$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$
where $\mathbf{x},\mathbf{b}, $ ...
