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12 votes
1 answer
5k views

Closest 3D rotation matrix in the Frobenius norm sense

Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm: \begin{equation} \|R-M\|_F \end{equation} Is there a closed form solution for $R$, or is it ...
Alex Flint's user avatar
11 votes
3 answers
6k views

Random Sampling a linearly constrained region in n-dimensions...

Hi, So here is my problem: Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$ $x_n \le c_n$ and $\sum_{n=1}^N x_n = 1$ find an ...
user1's user avatar
  • 113
6 votes
6 answers
3k views

Circumference of Convex Shapes

Here is a puzzle I found in Mitteilungen der DMV (roughly, "Letters of the German Society of Mathematicians"), issue 19/2011. It was posed by Alfred Schreiber in "Wie man Hasen fangt" (How to catch ...
Matthias Goergens's user avatar
3 votes
1 answer
327 views

LP Constraints for Connected Subgraphs of Fixed Size

Question: how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$? ...
Manfred Weis's user avatar
  • 13.2k
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 ...
user42804's user avatar
  • 1,121
19 votes
2 answers
2k views

Is the tensor product of polyhedra a polyhedron?

Conventions: A polytope in a finite-dimensional $\mathbb R$-vector space $V$ is defined to be a convex hull of finitely many points in $V$. A polyhedron in a finite-dimensional $\mathbb R$-vector ...
darij grinberg's user avatar
17 votes
3 answers
2k views

The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$

Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum: $$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$ ...
TOM's user avatar
  • 2,288
5 votes
3 answers
1k views

Algorithm for the intersection of a vector subspace with a cone of non-negative vectors

Hi, I would like to know whether there is some more effective way of how to compute an intersection of a vector subspace of $\mathbb{R}^{n}$ with a cone of vectors with non-negative entries than the ...
Miroslav Korbelar's user avatar
4 votes
1 answer
8k views

Detection of Redundant Constraints

Suppose I pose the following query to a constraint logic programming system: ?- Y <= 6 - X, Y <= (- 4) + 4 * X, Y <= 4 + X / 3. Are there systems that would recognize the last inequality as ...
user avatar
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 ...
Bing's user avatar
  • 49
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} \...
matematicaActiva's user avatar
29 votes
6 answers
8k views

How to find a closest integer point to the intersection of two lines?

Here's a question that originates from StackOverflow. Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
P Shved's user avatar
  • 391
19 votes
4 answers
1k views

Applications of linear programming duality in combinatorics

So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
amakelov's user avatar
  • 997
18 votes
3 answers
3k views

Deciding membership in a convex hull

Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$. This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
Mitch's user avatar
  • 667
16 votes
3 answers
1k views

Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets. Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$. Can $\text{Proj}(P)$ have more than $f$ facets? ...
Pedro Ruiz's user avatar
10 votes
2 answers
3k views

How do you tell if a system of linear inequalities has a solution?

A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
user21816's user avatar
  • 693
10 votes
1 answer
411 views

Network flows with capacities on pairs of edges

Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$. Now add edge-pair ...
Brendan McKay's user avatar
9 votes
1 answer
2k views

Uniform sampling from general simplex with a twist

This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange. Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
Juanito's user avatar
  • 221
9 votes
1 answer
295 views

Definition of packing property

Definition 1: A clutter $C$ is said to have the packing property if $C$ and all of its minors satisfy the König property. where, vertex cover of $C$ is a set of vertices that have non-empty ...
user177523's user avatar
7 votes
2 answers
1k views

Is a given point in the interior of the convex hull of a given finite collection of points?

Suppose I have the convex hull $P$ of a finite collection of points in $\mathbb{R}^d,$ and I want to see whether a point $p$ is contained in $P.$ This is a standard (some would say the standard linear ...
Igor Rivin's user avatar
  • 96.4k
6 votes
1 answer
609 views

Attempt at applying linear programming to the partial sums of the Möbius inverse of the Harmonic numbers

Let $a(n)$ be the Dirichlet inverse of the Euler totient function: $$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$ and let the matrix $T(n,k)$ be: $$T(n,k)=a(\gcd(n,k)) \tag{2}$$ It has been ...
Mats Granvik's user avatar
  • 1,183
6 votes
2 answers
8k views

Existence/Uniqueness of Nonnegative Solutions of Linear Systems of Equations

Suppose we have an $m$x$n$ matrix $A$, with $m\lt n$, and an $m$x$1$ vector $b$. Are there existence and uniqueness conditions characterizing nonnegative solutions of the system of linear equations $...
Adam Clarridge's user avatar
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 ...
bandini's user avatar
  • 491
4 votes
1 answer
891 views

Basic result in semi-infinite linear programming

Consider a standard linear program of the form $$\textrm{minimize}_x~~~~ c^Tx~~~~ s.t. \\ Ax = b \\ x \geq 0$$ with $x\in \mathbb{R}^n$ and $A \in \mathbb{R}^{m \times n}$. It is well known that, if ...
Tom Solberg's user avatar
  • 4,049
4 votes
1 answer
204 views

Reference: Packing under translation is in NP

I am looking for a reference for a result that I am aware of. Let me describe the result. Given a polygon $C$ and polygons $p_1,\ldots,p_n$, it can be decided in NP time, if $p_1,\ldots,p_n$ can be ...
Till's user avatar
  • 479
4 votes
2 answers
2k views

Simplified knapsack problem

There is a problem that I can not solve. Given a set of items (each item has some integer weight) we have to fill bag with some number of copies of these items, with the only restriction that the ...
Leo-guest's user avatar
3 votes
2 answers
791 views

complexity of finding optimal matchings of given fixed size

It is known, that maximal matchings (i.e. matchings with the maximal number of edges) and optimal matchings (i.e. matchings for which the sum of edge weights is optimal) can be calculated in ...
Manfred Weis's user avatar
  • 13.2k
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 ...
bandini's user avatar
  • 491
2 votes
1 answer
2k views

Finding a point farthest away from $k$ points in a polygon

There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized. ...
marc's user avatar
  • 57
2 votes
1 answer
137 views

Design constraint systems over the reals

This question is inspired by the discussion at this problem. Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...
Aaron Meyerowitz's user avatar
2 votes
2 answers
402 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?
ZhongHua Yan's user avatar