All Questions
2,368 questions
1
vote
0
answers
256
views
Equal maximum and minimum in a large-scale linear programming
For a linear optimization of an integral (with integral constraints), I perform a linear programming for the equivalent series. Maximum and minimum of the LP problem tend to be equal as I increase the ...
- 111
1
vote
0
answers
57
views
Covering the annulus of symmetric convex body
Consider a symmetric convex body $A$ in $\mathbb{R}^d$. Now, we draw another object, $A'$, concentric and translated with respect to $A$ and having radius slightly greater than twice to the radius of ...
- 285
1
vote
0
answers
1k
views
Robust optimization in matlab using fmincon [closed]
I am trying to implement the following optimization (from this paper) in Matlab using fmincon:
$\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$
where $\Sigma$ is a positive definite ...
1
vote
0
answers
203
views
Spectrum of Combinatorial Laplacian
The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
- 11
1
vote
0
answers
126
views
Matrix Minimax problem
I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The $M_k$...
- 4,829
1
vote
0
answers
193
views
Lattice-point enumeration question involving linear combinations of matrices
I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, \...
- 170
1
vote
0
answers
371
views
Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k
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 ...
- 11
1
vote
1
answer
241
views
Covering max flow arcs by arc disjoint paths
Let $(N,A,s,t,u)$ be a network with node set $N$, arc set $A$, source $s\in N$, sink $t\in N$ and capacity vector $u\in\{1,2,\ldots,T\}^A$, and let $x=(x_a)_{a\in A}$ be a maximum $(s,t)$-flow. Is it ...
- 2,966
1
vote
0
answers
146
views
Traversing perfect quadratic forms
I am wondering whether there is an efficient algorithm to traverse all the $N\times N$ perfect quadratic forms $Q$ inside the polyhedron $e_j^T Q e_j \geq 1$, $j = 1\ldots m$, where $e_j$ are some ...
- 149
1
vote
0
answers
1k
views
How to solve simple bilinear equations under extra linear constraints
Hello,
This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy:
$\lambda_i^T u_{...
- 53
1
vote
0
answers
450
views
When is the conical hull of a finite set of vectors a subset of the space? (and tilings)
Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...
- 1,314
1
vote
0
answers
513
views
Maximal disjoint hyperplanes
Assume a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ is given occupying a codimension $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these ...
- 800
1
vote
0
answers
263
views
Any 2-coloring of the scope of equilateral triangle must create a monochromatic rectangular triangle ? [closed]
Hello All,
I'm trying to prove something that we know that is correct but can't find how to do it or any example to show how to solve it.
Any 2-coloring of the scope of equilateral triangle must ...
- 11
1
vote
2
answers
169
views
How does one map regression depth to undirected depth of a point?
The regression depth of a line is the minimum number of points it has to cross to take it from its initial position to vertical. The undirected depth of a point is the minimum number of lines a ray ...
- 123
1
vote
0
answers
1k
views
Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
0
votes
1
answer
1k
views
For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?
Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...
- 135
0
votes
1
answer
389
views
Tetris in 3D with 5 units [closed]
Background: There are 7 "bricks" used in the game of Tetris. These are the 7 unique combinations of 4 unit squares in which every square shares at least one edge with another square. ("unique" in this ...
- 29
0
votes
1
answer
319
views
Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
- 21
0
votes
1
answer
229
views
Is this bounded?
May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...
- 1
0
votes
1
answer
340
views
[Matrices over Z] - An algorithm for calculating the diagonal with elementary operations
Dear mathoverflow,
Let
$
\left(
\begin{array}{cc}
a & b \newline
c & d
\end{array}
\right)
$
be a matrix with $a, b, c, d \in \mathbb{Z}$, $\gcd(a,b,c,d) = 1$ and $ad - bc = \pm N$, with $N &...
- 11
0
votes
1
answer
538
views
Method for (binary) optimization under constraints
I would like to know if there is a method to solve the Problem.
Problem:
Maximize the following function: $$f(p_{1,i},p_{2,i},\dotsc,p_{m,i})=\sum_{i=1}^{n}\begin{bmatrix}p_{1,i} & p_{2,i} & \...
- 3
0
votes
1
answer
226
views
Fractional values in linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
0
votes
1
answer
144
views
Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
0
votes
1
answer
145
views
Is Kakutani's Theorem true for the Euclidean plane? [closed]
The following analogue of Kakutani's theorem has been proved by F. J. Dyson (MR44620)
Theorem (F. J. Dyson 1951): Let $\Bbb S^2$ be the surface of a sphere with center $Z$ in Euclidean $3$-space $...
- 4,195
0
votes
1
answer
183
views
7 Points in $\mathbb{R}^3$ that don't lie on the union of 3 orthogonal hyperplanes
Let $1\leq \ell \leq \binom{k}{2}$. It seems that there should exist some set $\mathcal{P}$ of $kd-\ell+1$ points in $\mathbb{R}^d$ which do not lie on the union of any $k$ hyperplanes $H_1,\ldots, ...
- 439
0
votes
3
answers
512
views
The symmetry group of $\mathbb Z^d$
Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm.
I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
- 8,502
0
votes
1
answer
91
views
When do there exist locally regular embeddings of regular graphs?
I don't know how one can tackle the following kind of question, so any hint is welcome. I formulate a precise question in order to fix ideas, but it is to be considered as an example out of a more ...
- 2,041
0
votes
1
answer
98
views
Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent
Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent?
Remark: Every trapezoid is affine equivalent to ...
- 5,979
0
votes
1
answer
36
views
Benefit of adding a trivial constraint to ILPs
let ILP be an integer linear program with constraints-matrix $\boldsymbol{\mathrm{M}}\in\mathbb{Z}^{m\times n}$ and cost vector $\boldsymbol{\mathrm{c}}\in\mathbb{Z}^n$,
${\boldsymbol{\mathrm{x}}^*}\...
- 13.2k
0
votes
1
answer
93
views
How quickly can this IQP or its MILP relaxation be solved
Let $A\in\{0,1\}^{(n,n)}$ be a $n$ by $n$ boolean matrix (in particular think of an adjacency matrix of a graph), and consider the following optimization problem:
$$\begin{align*}&&\max_{P\in\{...
- 71
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
1
answer
214
views
How do you call a linear programming problem when the solution should be "constrained" to a norm?
(apologies for the n00b question)
Let's say we have a vector of length $n$, with to-be-determined values: $a_1, a_2, ...,a_n$.
And we have information that partial sums of these elements are equal to ...
- 143
0
votes
1
answer
320
views
Sub optimal algorithm for linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $c$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,421
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
0
votes
1
answer
332
views
Finding the minimum-area parallelogram containing all vertices of a cuboid projected to a plane
Description edited after comments:
Let's say I have a 3d cuboid that I projected onto a 2d plane. Now I want to find the minimum-area parallelogram that contains all those projected vertices. How do ...
- 121
0
votes
1
answer
212
views
How to find out if a polytope contains a sphere?
Given a polytope described by linear inequalities $Ax \le b, x \in \mathbb R^n$, how do you find out if there exist a (non degenerate) sphere of dimension $n-1$ contained in the polytope?
Thanks!
- 225
0
votes
1
answer
130
views
Cascading minimization problems
Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. ...
- 57
0
votes
2
answers
891
views
Find both maximum and minimum values in linear programming problem
Hi all. I have a linear programming problem where I need to find both maximum and minimum values of the objective function. The optimal points are not relevant.
Is there an efficient way to do so?
- 57
0
votes
2
answers
4k
views
Linear programming piecewise linear objective
I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:
max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$
$s.t. $
$|\...
- 3
0
votes
1
answer
98
views
Chromatic tiling complexity and the chromatic number conjecture
Let $T$ be a finite set of tiles in $\mathbb{R}^d$. A tiling of $\mathbb{R}^d$ by $T$ is a collection of disjoint translates of tiles in $T$ whose union is $\mathbb{R}^d$. A tiling is $k$-chromatic if ...
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 ...
- 3
0
votes
1
answer
126
views
On 'special' points on uniform planar convex regions defined in terms of moment of inertia
The following can be easily proved using perpendicular axes theorem and intermediate value theorem:
Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...
- 5,979
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 ...
- 13.2k
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 ...
0
votes
1
answer
49
views
Constructing a polygon from another with collinearity constraints
Let $P$ be a closed polygon defined by the sequence
$p_0,\,\dots,\,p_{n-1},p_0$ of points.
Question:
how can one construct, with straightedge and compass alone, another sequence of points $q_0,\,\...
- 13.2k
0
votes
1
answer
86
views
Lattice-point-free body diameter
The following interesting problem was asked at Aops and I wonder if it was based on some research paper:
Let $K$ be a convex body in $\mathbb R^2$, such that the diameter of $K$ is less than $\sqrt2$....
- 3,153
0
votes
1
answer
110
views
Detecting non-negativity of a single constraint by polyhedral constraints - $I$
We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...
- 13.9k
0
votes
1
answer
76
views
A question on graph partitioning
Given a connected un-directed simple graph $G=(V,E)$, is there a polynomial time algorithm to find the smallest subset $S$ of $V$ such that each node in $V \setminus S$ has at least 50% of its ...
- 1,216
0
votes
1
answer
92
views
Can we prove that simple polygons can always be split in half (vertex-wise) by diagonals?
Can we prove that for any simple polygon with more than 3 vertices there always exists a diagonal which:
is inside the polygon
doesn't intersect with any edges
splits the polygon in two polygons in ...
