All Questions
739 questions
0
votes
0
answers
36
views
ILPs with square constraint matrix
Given the Integer Linear Programming ($\text{ILP}$) problem
\begin{array}{ll}
\text{minimize} & c^T x \\
\text{subject to}& \mathbf{A}^T x \ge b \\
\text{where}&c,x,b\in\mathbb{N}_0^n,\\ &...
- 13.2k
11
votes
3
answers
3k
views
polyhedra with equilateral pentagons faces
In page http://loki3.com/poly/isohedra.html around six polyhedra with equilateral pentagons as faces are shown: a pyritohedron, icositetrahedrons... Is there a complete list of this kind of polyhedra? ...
- 2,370
2
votes
0
answers
119
views
Seeking insights on bounded set positive solutions for a set of linear systems in $\mathbb{R}^n$
Before delving into my query, I'd like to provide some context. Consider a continuous function $f:\mathbb{R}^{k}\rightarrow\mathbb{R}^{m}$ and a compact set $\mathcal{B}\subset \mathbb{R}^{k}$ (...
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 ...
0
votes
0
answers
26
views
Monotony of enforced subtour merging
Is it true that for a symmetric TSP instance in the sequence of edges generated by successively:
calculating the optimal 2-factor
adding cardinality constraints on the edgesets of the 2-factor's ...
- 5,429
0
votes
0
answers
171
views
Solve NP-hard type problems with linear programming
I would like to know if there is any way to solve an NP-hard type problem, for example, the TSP, sum of subsets or knapsack problem, by using linear programming and not by brute force.
I ask this ...
- 5,429
0
votes
0
answers
64
views
Alternatives to McCormick Envelope
I have an optimization problem for which I have the optimal solution obtained by the ILP.
However, when I introduced the McCormick Envelope to replace the product of a bi-linear term in its LP ...
- 3,065
0
votes
1
answer
169
views
How to integrate an indicator function/constraint into the cost function of a linear program?
I have a mathematical model $P$ for which I optimize two cost functions say $F_1$ and $F_2$ subject to a set of constraints $C1$–$C10$.
In $F_2$, I want it to be included only when its expression ...
- 5,429
0
votes
0
answers
164
views
Inf-convolution of norm 1 and norm 2 square
The inf-convolution of the functions $f$ and $g$ defined on $\mathbb{R}^n$ is
$$
h(x)=\inf _{y \in \mathbb{R}^n} f(y)+g(x-y) .
$$
We can prove that if $f,g$ are convex functions, then $h$ is convex.
...
- 35.4k
14
votes
12
answers
1k
views
Database of integer edge lengths that can form tetrahedrons
Is there a collection of lists of six integer edge lengths that form a tetrahedron? Is there a computer program for generating such lists? I need to find approximately thirty such tetrahedral ...
0
votes
1
answer
28
views
Calculating vertex potentials from optimal matchings
Question:
can the solution to the dual of a Linear Program be calculated directly from the solution of the primal Linear Program?
If yes, what are known algorithms and their bounds on complexity.
As ...
- 5,429
4
votes
3
answers
1k
views
Is there a pyramid with all four faces being right triangles? [closed]
If such a pyramid exists, could someone provide the coordinates of its vertices?
- 31
1
vote
0
answers
94
views
Linear Program Optimal Value
If $f(A,b,c)$ is the optimal value of a linear program
$\min c.x$
subject to $A.x \leq b ; x \geq 0.$
Does $f(A,b,c)$ have a piecewise polynomial/rational upper bound in $(A,b,c)$ on the domain of ...
1
vote
1
answer
181
views
Linear programming with "nice" matrices
Consider the following linear programming problem
\begin{array}{ll}
\text{minimize} & \mathrm 1^{\top} \mathrm x\\
\text{subject to} & v\le \mathrm A \mathrm x \le u\\
& \mathrm x \geq ...
CommunityBot
- 1
1
vote
0
answers
82
views
A face and all its neighbors: terminology?
Suppose $F$ is a face of a 2-complex, and $F_1,\dotsc,F_n$ are the faces that are adjacent to (i.e., share an edge with) $F$. Is there a standard term for a collection of faces of the form $\{F,F_1,\...
- 12.9k
5
votes
1
answer
190
views
Regular polyhedral spaces
By symmetrically gluing together opposite faces of a dodecahedron together, one of three spaces can be obtained, depending on the angle the faces are rotated by before twisting. In fact, this can be ...
- 20k
2
votes
1
answer
240
views
Is the problem of vertex enumeration from an H-representation of a polytope NP-hard?
According to the Wikipedia page on the issue, the vertex enumeration problem is NP-hard.
However, double description and reverse linear search are algorithms listed to solve the problem. Moreover, ...
- 7,875
10
votes
1
answer
3k
views
Computionally efficient vertex enumeration for (convex) polytopes
Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
- 14k
2
votes
0
answers
306
views
Why is it impossible to create a numerically balanced die with more than 120 sides?
I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
- 5,429
0
votes
0
answers
55
views
Relationship of optimal solutions between the total function and the sub function
This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where $x\in\mathbb{R}^...
- 5,429
0
votes
0
answers
85
views
Show that $\max_{P_X : X\in (0,1) } \left| \frac{\mathbb{E} [ f'(X) ]}{ \mathbb{E} [ f(X) ] } \right|$ is maximized by at most two mass points
Let $f$ be some given well-behaved function. Consider the following optimization problem overall probability distribution on $[0,1]$
\begin{align}
\max_{P_X : X\in [0,1] } \left| \frac{\mathbb{E} [ ...
- 671
9
votes
2
answers
341
views
Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?
A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
- 13.6k
1
vote
1
answer
69
views
$A^*$ algorithm to find shortest path when weights in my graph are the inverse of distance
Given a graph G=(V,E) where the weights on my edges are inverse of Euclidian distance between nodes, I want to know if I can use A* algorithm to find the shortest path. How I need to modify the ...
- 13.2k
1
vote
1
answer
119
views
Optimization on non-convex set
Let $\Omega$ be an open bounded subset of $\mathbb{R}^2$ and $f\in L^2(\Omega)$ be a given function. Consider the optimization problem
$$\mathrm{min} \int_\Omega u(x) f(x) \,dx\,,$$
where a minimum is ...
- 128k
4
votes
3
answers
1k
views
Minimax theorem on a non convex domain
A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$:
$$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in \mathcal{...
0
votes
1
answer
172
views
Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes?
Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs:
The facet complex of any simplicial ...
- 50.8k
1
vote
0
answers
182
views
Does this sequence stop?
Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
- 1,182
0
votes
1
answer
116
views
How can I find the hyperplane passing through a 600-cell
I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...
2
votes
2
answers
2k
views
Mathematical tools appropriate to analyse convex polyhedra
What mathematical tools (means: set of areas of mathematical knowledge) are appropriate to begin with to analyse (to enumerate face vectors associated with polyhedron, to calculate the combinatorial ...
- 63.9k
1
vote
2
answers
121
views
How to solve the optimization problem $\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$?
I am looking for an algorithm to solve the following optimization problem
$$\max_{\mathbf{w}}\sum_i\text{sign}(\mathbf{w}^T \mathbf{x}_i)$$
where $\mathbf{w}$ and each $\mathbf{x}_i\in\mathbb{R}^d$.
...
1
vote
0
answers
57
views
Inside-out dissections of solids
We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there.
How does one inside-out dissect a tetrahedron into ...
- 63.9k
2
votes
3
answers
2k
views
Better tactics for removing redundant constraints than Linear Programming?
After reading:
Detection of Redundant Constraints
It appears that linear-programming is the most commonly known way to remove ALL redundant constraints from a system of inequalities of the form
$$ ...
- 141
13
votes
2
answers
3k
views
How many vertices/edges/faces at most for a convex polyhedron that tiles space?
I wonder if this problem has already been examined before:
Consider a convex polyhedron that tiles $\mathbb R^3$. What is the maximum of vertices/edges/faces that such a polyhedron can have?
...
- 2,821
1
vote
0
answers
41
views
About the number of faces of the conification of a polytope
Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
- 131
5
votes
1
answer
227
views
The bounded complex of a polyhedral decomposition
Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties:
The union ...
- 13.6k
0
votes
1
answer
103
views
Constrained linear optimization problem on $C^1$
I am dealing with a problem of the form ($a<b$)
$$
\displaystyle \max_{v \in C^1([a, b])} \int_a^b v(x)~\mathrm{d}x, \quad \mathrm{s.t.} \int^b_a \big(-o'(x)v(x)-v'(x)o(x)\big)f(x)~\mathrm{d}x \...
1
vote
0
answers
131
views
What is the difference between a simple polyhedron and a triangulated graph?
On a famous website I've seen the following:
The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above. That "illustration above&...
- 13.4k
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
145
views
Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
1
vote
1
answer
119
views
Adding linear constraint to the domain
I don't know if it is a well-known problem, but I have been struggling to come up with an algorithm.
I have a set of linear constraints $Ax\le b$, $b\ge 0$ ($b$ and $A$ are given, $x$ is a variable). ...
- 5,429
0
votes
0
answers
84
views
1-degree SOS proof refutes Linear Programming
I am trying to understand Sums-of-Squares proof systems.
A degree $d$ Sums-of-Squares refutation for a set of polynomial equations $P = \{p_1(x) = 0, ..., p_m(x) = 0\}$ is defined as
$\sum_{i=1}^m g_i(...
96
votes
4
answers
5k
views
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
CommunityBot
- 1
2
votes
1
answer
75
views
Generic infinitesimal rigidity of polyhedra
Let $M$ be a 1-skeleton of a triangulation of a sphere with $V$ vertices and $E$ edges.
Definition 1 A polyhedron is a map $M\to \mathbb R^3$ that is affine on edges (and non-degenerate on faces). The ...
- 276
1
vote
0
answers
96
views
On optimizing a multivariate quadratic function subject to certain conditions
The problem is to maximize $f(x_1,x_2,\cdots,x_n)=\sum\limits_{i=1}^{n}\Big(x_i-k_i\Big)^2$ for $n\ge 3$ subject to the conditions (1) $\sum\limits_{i=1}^{n}x_i=\sum\limits_{i=1}^{n}k_i\le n(n-1)$ ...
- 61
2
votes
0
answers
94
views
Is it possible to deduce Poincaré duality from duality of polytopes?
I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance:
https://math.stackexchange.com/a/14469/454016
Poincaré duality is explained through a duality of ...
1
vote
0
answers
335
views
Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 4,219
1
vote
1
answer
169
views
Best projection on non-convex discrete set with two constraints
I want to compute the projection of a vector $\left( x\right) _{1\leq
i,j\leq n}\in \lbrack 0,1]^{n\times n}$ on the following discrete set
$$
S=\left\{ x\in \{0,1\}^{n\times n}:x_{i,j}+x_{j,i}\leq 1;\...
- 128k
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 $k$ ...
0
votes
1
answer
147
views
Is there a redundant constraint in linear programming? [closed]
From wikipedia:
But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, ...
- 63.9k
16
votes
3
answers
1k
views
If I have zeros at the vertices of an icosahedron, where should the poles go?
I've been tinkering with Newton's method applied to polynomials. E.g., Newton's method for $z^5 - 1 = 0$ gives:
There aren't a lot of symmetric patterns of finite sets of points in the plane, so I ...
- 2,370