All Questions
740 questions
6
votes
1
answer
761
views
Checking if one polytope is contained in another
I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.
At the moment I am ...
- 4,713
4
votes
0
answers
153
views
Perimeters of nested convex spherical polygons
I seek a reference—not a proof—that if $P_1$ and $P_2$
are two convex polygons on a sphere composed of geodesic segments,
contained in a hemisphere, and
$P_1 \subseteq P_2$, then the ...
- 151k
3
votes
3
answers
349
views
Sensitivity analysis in conic optimization
I have a conic optimization of the form:
$$\min_x \langle c, x \rangle,\ \text{s.t.}\ Ax = b,\ x \in K.$$
where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a ...
CommunityBot
- 1
9
votes
1
answer
436
views
The $32$-deg polynomial for the tetrahedron inscribed in the icosahedron?
This MO answer discusses this table involving the maximal side lengths of the five Platonic solids $T,C,O,D,I$ inscribed in the other solids,
This table is also found in Moritz Firsching's paper. I ...
- 10.7k
2
votes
0
answers
80
views
Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector
I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...
- 123
1
vote
0
answers
20
views
Calculating Cost-Optimal 1-Factors in Digraphs
I need to find a cost-optimal 1-factor in a positively weighted, directed, regular graph $G(V,A)$ without antiparallel arcs, i.e. given $$\text{deg}_{\text{in}}(u)=\text{deg}_{\text{in}}(v)=\text{deg}...
CommunityBot
- 1
2
votes
0
answers
105
views
Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
1
vote
1
answer
250
views
Are there two tetrahedrons with the same volume that share their opposite edge lengths and arent the same or a different chirality of the same? [closed]
I have been coming up with an efficient way to decide if two tetrahedrons are similar. I believe that it is enough for a computer to check for the ordered by length list of pairs of opposite edges on ...
- 11
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 ...
- 151k
4
votes
2
answers
1k
views
Dilogarithm, tetrahedrons, and hyperbolic space
The Bloch-Wigner function $D(z)$ gives the volume of an ideal tetrahedron in the hyperbolic space $\mathbb{H}^3$. Here $z$ is the cross-ratio $(z_1,z_2,z_3,z_4)$ parametrizing the tetrahedron in $\...
- 4,713
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
2
votes
1
answer
135
views
Polyhedral structure of functions writable as a finite signed sum of max of linear functions
For any two positive integers $k,n$ consider the space of functions writable as,
$\sum_i \sigma_i \max \{ L_{i1},L_{i2},..,L_{ik} \}$ (a finite sum) where each $L_{*} : \mathbb{R}^n \rightarrow \...
- 4,786
2
votes
0
answers
344
views
Linear programming with an infinite matrix
I would like to solve the following infinite linear system subject to $x_i \ge 0$ that minimizes $x_3$.
The third column contains no additional nonzero values beyond what is shown. Though the first ...
4
votes
1
answer
3k
views
intersection of convex and non-convex polyhedra
I am trying to find the best appropriate way to intersect polyhedra which may be non-convex.
The number of vertices that build the polyhedron is hence always small (up to 20 or so).
The ...
- 151k
6
votes
1
answer
544
views
Isometric embedding a convex cap to render its boundary planar
I would like to know if there is a polyhedral analog to this beautiful
theorem of Hong:
Theorem 11.0.1.
Any smooth positive disk $(\bar{D},g)$ with a positive geodesic
curvature along $\partial ...
- 151k
17
votes
6
answers
2k
views
On the number of Archimedean solids
Does anyone know of any good resources for the proof of the number of Archimedean solids (also known as semiregular polyhedra)?
I have seen a couple of algebraic discussions but no true proof. Also, ...
- 4,713
2
votes
1
answer
148
views
Fast algorithm for large-scale, asymmetric transportation linear program
I have a large-ish instance of a transportation problem that is very asymmetric, say of dimensions $100\times10000$. I am currently solving it with a stock LP solver, but obviously something like the ...
CommunityBot
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1
vote
2
answers
83
views
Algorithm for a linear optimization problem
For the vectors $X=(x_1,\cdots, x_n),~ Y=(y_1,\cdots, y_n)$ and $\alpha=(\alpha_1,\cdots,\alpha_n),~ \beta=(\beta_1,\cdots, \beta_n)\in\mathbb R^n_+$ s.t. $\sum_{k=1}^n\alpha_k~~=~~\sum_{k=1}^n\beta_k~...
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 19.6k
11
votes
2
answers
489
views
Shortest morphing between shapes embedded in $\mathbb{R}^3$
I am interested in what in computer graphics is called
morphing between two topologically equivalent shapes $S_0$
and $S_1$ in 3D.
This is a continuous "path" of shapes $S_t$, each embedded and
all ...
- 151k
1
vote
0
answers
78
views
Family of functions which satisfies $f(\boldsymbol{x}) = 0$ if $\nabla f(\boldsymbol{x})=0$? [closed]
I have a Lagrangian of which I want to find the supremum in the primal variable $\boldsymbol{x}$:
$\mathscr{L}(\boldsymbol{x},\boldsymbol{\lambda})=f(\boldsymbol{x})^T\boldsymbol{a} + \boldsymbol{\...
1
vote
0
answers
47
views
Linear programs [closed]
Can the optimal value of the primal problem of a linear program ever be less then zero?
An example is: minimize $C=2x_1 +3x_2$ Subject to: $3x_1+4x_2 \leq 5$. Obviously, $x_1$ and $x_2$ are free ...
- 993
9
votes
0
answers
543
views
Maximum volume convex body coverable by a unit square
Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body ...
CommunityBot
- 1
1
vote
0
answers
93
views
quick hull algorithm detail
When using quick hull algorithm to find the polytope for half space intersection, we are required to provide an interior point to the solver qhalf.
In other words, providing
$$Ax \le b$$
is not ...
- 867
3
votes
2
answers
1k
views
Equality constraints in mixed-integer optimization
Suppose I have a linear mixed-integer optimization problem of the form
$$MIP: min_{(x,y) \in \mathbb{R}^n \times \mathbb{Z}^m} c^\top x + d^\top y \hspace{0.2cm} \text{s.t.} \hspace{0.1cm} Ax+By \leq ...
- 342
1
vote
0
answers
43
views
a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
- 61
2
votes
1
answer
218
views
Polyhedra names question [closed]
So I've been playing around with polyhedra for my own amusement, but I ended up with some that I couldn't find names for. I have been trying to find them on my own by Googling for polyhedra with these ...
- 151k
3
votes
2
answers
877
views
Polyhedra Classification
The following is inspired by this question. From time to time I search the web for tables of polyhedra, but without much success. Part of the problem is that there are many non-equivalent questions ...
CommunityBot
- 1
4
votes
1
answer
967
views
Solving for Hamiltonian path with constraints on allowable routes through vertices
Suppose you have a complete graph with N vertexes, with a distinguished vertex $n=1$ ("start"), and you wish to find a route traveling exactly once through each vertex so that the distance along the ...
- 151k
0
votes
1
answer
79
views
algorithms and tools available for a particular polytope computation
Let me define each half space i as:
$${H_i}:{c_i}{\bf{x}} \le {b_i}$$
The intersection of all such ${H_i}$ gives a polyhedron (bounded or not). Suppose I am interested in if ${H_i}$ is active (...
- 13.6k
4
votes
1
answer
3k
views
optimization of inverse matrix with constraint on matrix elements
everyone! I have this optimization problem with constraint.
$D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter.
$x$ and $v$ are two known p-dimensional vectors.
The ...
CommunityBot
- 1
24
votes
1
answer
2k
views
Building a genus-$n$ torus from cubes
I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting object ...
- 151k
3
votes
4
answers
4k
views
Existence of nonnegative solutions to an underdetermined system of linear equations
Similar questions have been asked elsewhere, but I think this is sufficiently different to warrant a new post. I have a particular matrix $A$ and would like to know when the system $Ax = 0$ has at ...
1
vote
1
answer
73
views
minimize number of unique elements in a vector
I was wondering if there is a simple or known way to minimize the number of unique elements in a decision variable (vector). Note that I'm not asking for minimization of nonzero elements (rank ...
- 14k
2
votes
2
answers
403
views
Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
27
votes
5
answers
2k
views
Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 43.2k
5
votes
0
answers
233
views
Is there a well-established terminology for polyhedra/polytopes?
I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...
CommunityBot
- 1
1
vote
0
answers
628
views
Totally unimodular Matrices
A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...
9
votes
1
answer
3k
views
Inverse of a totally unimodular matrix
A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...
9
votes
1
answer
6k
views
Proving that a binary matrix is totally unimodular
I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...
1
vote
1
answer
206
views
Show $0-1$ Knapsack is polynomially reducible to this problem
I have already posted this question here but have not received an answer so I am cross-posting with hope to reach a larger amount of mathematicians:
Let $T=\{1,\cdots,n\}$ and consider the ...
CommunityBot
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0
votes
0
answers
258
views
Boundary regularity of the solution of a Poisson equation in a polyhedron
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$f\in L^2(\Lambda,\mathbb R^d)$
$u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...
CommunityBot
- 1
1
vote
1
answer
113
views
Are convex combinations of 0-1 Pareto efficient vectors efficient?
Let $Y$ be any subset of $\{0,1\}^n$ for $n\geq3$. A vector $\alpha\in$ $Y$ is Pareto efficient if there is no $\beta\in$ $Y$ such that $\beta_i$ $\geq$ $\alpha_i$ for each $i\in\{1,...,n\}$ and $\...
- 1,502
6
votes
2
answers
1k
views
Linear programming is continuous
Consider an arbitrary linear program:
$$\max \vec c \cdot \vec x$$
subject to:
$$\textbf{A}\cdot \vec x = 0, \quad \vec a \le \vec x \le \vec b$$
Assume that this program is feasible and bounded. ...
- 884
0
votes
1
answer
81
views
Can convex combinations of indicator functions for pairwise non-disjoint sets unordered by inclusion dominate one another?
Let $N$ be a finite subset of the naturals. Let $P$ be a set of subsets of $N$ such that:
1) $P\neq \varnothing$,
2) $\forall x\in P, |x| >1$,
3) $\forall x,y\in P,$ if $x\neq y$, then $x\not\...
- 13.4k
0
votes
0
answers
68
views
A seemingly easy integer programming question
Let $k, m \in \mathbb{Z}_{ > 1}$. Let $a \in \mathbb{Z}_{> 0}^m$ and $t \in \mathbb{Z}^k$. Let $\varepsilon = (\varepsilon_{i,j})_{1 \leq i \leq m \\1 \leq j \leq k}$ be a matrix with entries in ...
- 501
8
votes
2
answers
246
views
Are sums of 0-1 Pareto efficient vectors Pareto efficient?
Does there exist $m,n\ge1$, an $m \times n$ matrix $A$, and a vector $x \in \mathbb{R}^n$ such that:
The entries of $A$ are $\in \{0, 1\}$.
For all pairs of columns $u, v$ of $A$ the entries of $u - ...
- 11
1
vote
0
answers
62
views
Sampling in a polyhedral complex
Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of ...
- 2,246
1
vote
1
answer
184
views
Do doubly infeasible Linear Programming problems always have doubly infeasible bases?
Consider a Linear Programming problem in dictionary form,
$$\max\Big\{f^\pi+\!\!\!\sum_{j\in D(\pi)}\!\! d^\pi_jx_j~\Big|~\forall~i\!\in\!B(\pi)~~~ b^\pi_i+\!\!\!\sum_{j\in D(\pi)}\!\! G^\pi_{ij}x_j\...
CommunityBot
- 1
2
votes
1
answer
3k
views
max-flow at max-cost
I have a flow network with gains. In practical terms, a gain is the opposite of a cost. So, I interested in finding the maximal gain of a network flow, what could be interpreted as finding a maximum ...
CommunityBot
- 1