11
votes
3answers
473 views

finding the most-isolated point in a high-dimensional cube

I have a set of points {$x_1,\ldots,x_n$} located in the d-dimensional unit cube $[0,1]^d$. $n$ is about 1000 and $d$ is about 25. I'd like to find $\max_{\omega\in [0,1]^d}\min_{i=1,\ldots,n} ...
10
votes
1answer
520 views

Complexity of a weirdo two-dimensional sorting problem

Please forgive me if this is easy for some reason. Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$. I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
6
votes
2answers
814 views

Conic hulls and cones

Suppose I have a number of vectors in $\mathbb{R}^n.$ The first question is: what is the most efficient algorithm to compute their "conic hull" (the minimal convex cone which contains them)? The next ...
4
votes
1answer
2k views

Homogeneous system of polynomial equations

Hi all, Previously I asked a question that currently has no satisfactory answer Least sum squares given constraints on subcomponents It comes from an engineering problem. I was thinking to formulate ...
2
votes
2answers
307 views

formulate edge length problem as convex optimization problem

I want to us convex optimization to describe a problem in computational geometry. Let $E = (E_1, E_2,\ldots , E_m) $ be a sequence of line segments in the plane, where $E_1$ and $E_m$ may be points ...
5
votes
1answer
595 views

Minimizing variance of distances between points when mean distance is fixed

In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d ...