All Questions
1,022 questions
0
votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
3
votes
2
answers
2k
views
ILP for minimum edge coloring problem
We know that for a graph $G=(V,E)$, minimum edge coloring is a coloring of
$E$, i.e., a partition of $E$ into disjoint sets $E_1, E_2, \dots, E_k$ such
that, for $1 \leq i \leq k$, no two edges in $...
- 45
5
votes
2
answers
2k
views
Multiplicative gradient descent?
The normal gradient descent is additive: $w_{t+1}=w_t-\lambda_t\nabla f(w_t)$, but is there a multiplicative gradient descent that looks something like $w_{t+1}=w_t[-\lambda_t\nabla f(w_t)]$?
I know ...
- 211
6
votes
1
answer
1k
views
Speed up Linear programming
I have a linear programming problem like this:
minimize $c^t X$
under the constraint that $AX \ge b$.
I will need to solve this linear programming problem online many times. I need it to be as fast ...
- 83
2
votes
1
answer
143
views
Find base of kernel with as many 0 as possible
I have a 400x132 rectangular matrix with only 0 and 1.
I am looking for the linear combinations of the columns of the matrix that sum to 0.
For example C1 + C2 - C3 = 0.
I want to find the linear ...
- 83
2
votes
2
answers
163
views
Maximum possible number of similar three-colored triangles
I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...
- 628
2
votes
2
answers
219
views
Boundedness of ratio of linear functions
Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \...
- 23
-3
votes
2
answers
546
views
Hexagon Formed by connecting Trisections of triangle sides [closed]
Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
2
votes
1
answer
423
views
Tools for Removing Radicals from Equations
I am currently doing some investigations on Sylvester's 4 Point Problem Probability of 4 Points being in Convex Configuration
and repeatedly face the problem of solving equations between sums of ...
- 13.2k
0
votes
1
answer
546
views
Solution of infinite dimension linear system
Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$.
For fix n,
we can construct n dimension linear equation ...
- 111
3
votes
2
answers
437
views
convex polytope integer points
is there a simple proof for the following lemma:
An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
- 39
1
vote
0
answers
280
views
A conjecture about cross sections of a pyramid [closed]
Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.
This is a conjecture I came across while trying to solve this problem.
I was ...
- 1,090
1
vote
0
answers
140
views
Reduce a Combinatorial problem
It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...
- 11
0
votes
2
answers
708
views
Approximate solution to large mixed integer programming problem
What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...
user24451
3
votes
1
answer
966
views
Continuity of minimizers to distance function from point to convex set
Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$):
$\min_{x\in U}\|x-y\|$.
I believe the minimizer $x_{U}^{*}$ is ...
- 81
1
vote
1
answer
624
views
a closed form lower bound solution for linear programming
Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it?
to clearly express the problem assume that
$$
z(\mathbf{a,B,c})=\mathop {\inf} ...
- 275
3
votes
1
answer
2k
views
Conditions for a parametric curve to avoid self-intersection?
Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the ...
- 151k
3
votes
1
answer
340
views
Name search for special Linear Integer Program
I am looking for a name for the following question in literature!
(and if you know it, then it would be great)
I couldn't find it and due to wide audience here, presumably you know more. Thank you
$...
- 31
2
votes
1
answer
191
views
Optimization problem whose cardinality never exceeds 7 for some reason
I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...
0
votes
1
answer
452
views
Large scale least squares of non symmetric and non square problems
Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric ...
- 125
2
votes
0
answers
286
views
Probability of 4 Points being in Convex Configuration
Background of my question is, that I would like to implement a parallel preprocessing for a constructing the convex hull of very huge number of points in the euclidean plane;
the idea is to process 4-...
- 13.2k
1
vote
1
answer
70
views
Heuristic for choosing n-vectors from n-sets
my given problem is:
choose n-vectors from n-sets (one vector from each set) so that the biggest element in the sum of the chosen vectors is minimal. Unfortunately the problem is NP-hard. So I'm ...
- 11
3
votes
1
answer
554
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
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 ...
- 143
6
votes
1
answer
546
views
Reference request: a differential equation in elementary geometry
15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange.
My question is where the following differential equation arises naturally and where it ...
2
votes
0
answers
91
views
Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings
In one paper from 1980 I found a note that there are no known algorithms for solving
homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring.
(The non-...
1
vote
2
answers
172
views
Linear Programm with matrix [closed]
Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
- 11
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
4
votes
2
answers
387
views
Points contained in a disk [closed]
I have a question, but not sure how to prove this.
We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points.
Conjecture: There ...
- 65
4
votes
1
answer
750
views
submatrix of a given size with maximum frobenius norm
Let $I\subset \{1,2,\ldots,n\}$, and let $|I|$ denote its cardinality. Now given a Hermitian matrix $\mathbf{A}\in\mathbf{C}^{n\times n}$. I am interested in finding the subset $I$ that maximizes the ...
- 859
2
votes
0
answers
120
views
integrality of a linear program -- binary equality constaints
Consider the following linear program:
$\left\{
\begin{array}{l}
\underset{x}{max} \;\;c^Tx\\
[I, \;B]x = \mathbf{1}\\
x\geq 0
\end{array}
\right.$
where $c$ is a vector ...
- 127
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 ...
- 49
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 363
4
votes
2
answers
212
views
combinatorial and linear duality
Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The combinatorial dual $\...
- 987
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
7
votes
2
answers
484
views
How to find a tetrahedron that covers four points?
I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular:
Given four distinct real numbers $a_1$, $a_2$, $a_3$, $a_4$, is there a ...
- 221
1
vote
0
answers
75
views
Are there any known bounds on the value of solutions of linear integer programming?
Given a linear objective function and a system of linear constraints; are there any known bounds on the values of (positive) integral solutions in terms of the coefficient matrix of the constraints?
...
- 111
1
vote
1
answer
4k
views
Maximizing linear objective function with absolute values
This has be asked on other forums, though couldn't
find authoritative answer.
I have a linear program over the reals and don't
want to introduce integer or binary variables.
The objective function ...
- 25.4k
2
votes
2
answers
842
views
Finding the maximum of a multivariate polynomial of degree one
I need to find the global maximum of the function
\begin{align}
f\left(x\right) & = p_1 \max\left(\sum a_{1i} x_{1i}, \sum b_{1i} x_{1i}\right) - \sum c_{1i} x_{1i} \\
&+\ldots \\
&+ p_n ...
user24451
4
votes
1
answer
611
views
What is the correct preposition? (And is there one?)
I just stumbled upon a linguistic problem I wasn't able to resolve via web search. Suppose we're given some geometric set $A$ and subset $B\subset A$. Isn't there a compact way of saying that there ...
- 3,513
3
votes
1
answer
804
views
Approximation of a convex body by a contained polytope
This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes.
For a given $\delta$, let $n_\delta$ be the number of faces ...
- 591
3
votes
0
answers
48
views
Number of not self-intersecting closed paths spanning $n$ iid uniform points
Let $X_1,X_2,\dots,X_n$ be independent uniform variables in the square. What is the number of piece-wise linear paths which vertices are all the $X_i$ and that do not self-intersect? In other words, ...
- 1,352
3
votes
2
answers
264
views
Successive Inner or Outer Approximation of Simple Polygons with Hierarchies of Implicit Functions
The problem I want to solve, is to quickly decide, whether a point $p=(x^*,y^*)$ is inside or outside of a polygon $P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with $n$ potentially very large.
...
- 13.2k
1
vote
0
answers
493
views
Complexity of Nested Linear Optimization
My question is motivated by the fact, that among other ways, it is possible to restrict a variable to two discrete values, e.g. the prototypical $0$ and $1$, via an optimization constraint:
$$\max(\...
- 13.2k
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{...
- 591
0
votes
1
answer
100
views
generalization from linear programming solution [closed]
I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that ...
- 109
0
votes
4
answers
1k
views
Characterization of angles trisectable with straightedge and compass
Lindemann's proof of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative.
In the following, ...
- 13.2k
3
votes
1
answer
97
views
Number of small projections
Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of $\...
- 133
10
votes
3
answers
2k
views
The intersection of $n$ cylinders in $3$-dimensional space
A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
- 11.4k
7
votes
1
answer
819
views
Has this generalization of a determinant (assigning multiplicities to the rows) been studied?
I'm working on some questions in tropical geometry, and my problem led me to create the following generalization of a determinant:
Let $A$ be an $m \times n$ matrix with $m \le n$, and positive ...
- 1,509