All Questions
985 questions
3
votes
0
answers
169
views
Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)
Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of $...
- 41
1
vote
2
answers
660
views
constructing a curve dividing two sets of points
Lets assume I have two sets of points, each characterized as being "A" or "B", respectively, that are in a Euclidean plane. Theoretically these two sets are samplings from a space that has ...
- 11
1
vote
2
answers
444
views
Levenberg-Marquadt near the minima for non-zero-residual problems
I'm using the LM algorithm to do gradient descent in a model fitting context. I'm minimizing:
$$
c(x) = \sum ( f_i(x) - y_i )^2
$$
I'm noticing that after a few steps when I'm close to the minima, I ...
- 561
2
votes
0
answers
148
views
Derivation of gradient of SSE in Geodesic Regression
On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
- 5,405
1
vote
0
answers
64
views
Maximize discrete harmonic function at given point
Let $n>0$, and let $S_n$ denote the discrete square
$S_n=[|-n,n|]^2$ (so $S_n$ has $(2n+1)^2$ elements). Let $K_n$ denote the set of four corner points $\lbrace (\pm n,\pm n)\rbrace$, and $C_n=S_n\...
- 621
2
votes
0
answers
149
views
How to solve the following generalized quadratic programming problem [closed]
I want to solve a generalized form of a quadratic programming problem
$$\min_x \left(\sqrt{x^TPx}+\sqrt{x^TQx}\right)^2+c^Tx$$,
$$\textrm{ s.t. } Ax\le b.$$ Here, $P$ and $Q$ are both positive ...
- 21
0
votes
1
answer
82
views
Introducton books for $\frak{E}_p(I)$
Are there any good books different from abstract harmonic analysis by hewitt to study $\frak{E}_p(I)$. where $\frak{E}_p(I)$ is: Let $I$ be an arbitrary index set. For each $i\in I$ let $H_i$ ...
- 209
5
votes
2
answers
1k
views
Applications of minmax theorem(s)
Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X \...
0
votes
1
answer
123
views
Practical Algorithm for Comparing the Discrepancy of Point Sets (on Unit Hyper Spheres)
I have devised a simple geometric algorithm for generating a sequence of points on unit hyper spheres; that algorithm depends on a single real parameter, which I would like to optimize in order to get ...
- 13.2k
4
votes
1
answer
288
views
Equivalent method for maximum likelihood estimation of covariance parameters
My goal is to estimate the parameters of a covariance matrix $\Omega$, by maximizing the following log-likelihood function:
$$\log L(\vec\tau, \rho, \sigma \mid W, X) = -m\ln(\left | \Omega \right |) ...
- 141
1
vote
1
answer
2k
views
Can one efficiently optimize over the inverse of matrix?
Hello,
I have the following problem:
Find a non-negative matrix $L$ (i.e. $L_{i,j} \geq 0$ for all $i,j$), $L \neq I$ so that $A(I-L)^{-1}y \geq 0$ (the inequality must hold for each component), ...
- 53
2
votes
0
answers
126
views
Intersecting balls with convex regions and a bisector thereof
This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...
- 113
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
1
vote
0
answers
55
views
Separation on discrete set
Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$.
Define linear functions $f(x)= a_1x_1+ \...
- 61
0
votes
0
answers
63
views
Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere
Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...
user76479
1
vote
1
answer
769
views
Results for minimizing the norm w.r.t a unitary matrix
Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not ...
- 322
1
vote
0
answers
526
views
minimizing the sum of euclidean norms
minimizing the sum of euclidean norms with box constraints
I am a graduate student in computer science, making a thesis on uncertainty geometry. During my thesis I came across the following ...
- 13
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
2
votes
1
answer
126
views
Is it possible to represent non-linear ranking type constraints as equivalent linear constraints?
I have formulated a linear program with binary indicator variables $z_i(a)$ which is equal to $1$ if the $i^{th}$ document is of rank $a$ and $0$ otherwise.
The other variables in the linear program,...
- 121
2
votes
0
answers
53
views
Facet counting argument for polytopes
Consider a pair of piecewise-linear cobordant $n$ dimensional polyhedra $P_1, P_2$ sitting in $\mathbb{R}^{n+2}$ (with some fixed orientation).
Let $O$ be an $n+1$ dimensional piecewise-linear ...
- 2,689
1
vote
2
answers
2k
views
Regular vs. Irregular Vertices in a Mesh
Hi everybody,
Reading about Geometry Processing, I have realized that people in this area are very interested in regular vertices(degree=6) rather than irregular ones.
Can anybody give me reasons ...
- 111
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
2
votes
0
answers
5k
views
A system of linear equations with linear constraints
Mathematical problem.
Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\...
0
votes
2
answers
118
views
Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
- 2,239
1
vote
0
answers
1k
views
How to solve simple bilinear equations under extra linear constraints
Hello,
This is the full version of a question I asked earlier. I am trying to understand whether finding a solution to the following bilinear system is computationally hard or easy:
$\lambda_i^T u_{...
- 53
2
votes
1
answer
104
views
Standard names and methods for this type of fitting minimization
In material science research, we have come across the following type of problem.
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
...
- 867
2
votes
1
answer
1k
views
Linear Programming Cost Function [closed]
I need to add the following to my LP problem:
If the amount of workers hired in period $t$ ($H_t$) is higher than 25, the hiring cost is only 1 instead of 1.2.
Example: if 30 workers are hired in ...
0
votes
2
answers
244
views
Rewrite optimization objective
Hi,
I wanted to ask, under which conditions can one rewrite the optimization objective
$\min_x f(x)\;\;\;s.t.\;\;\;g(x) \leq s$
as
$\min_x g(x)\;\;\;s.t.\;\;\;f(x) \leq t$
I have particular ...
- 409
6
votes
0
answers
317
views
Variant of orthogonal Procrustes problem
The orthogonal Procrustes problem seeks a matrix $M$ that minimizes $||AM-B||_F$ subject to $M^TM=I$, where $M$ is $d\times d$ and both $A$ and $B$ are $n\times d$. Geometrically, $M$ rotates a set of ...
- 61
1
vote
1
answer
246
views
Eigenvalue problem with quadratic constraints
$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$
where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = (x_{1},x_{2},.....
- 45
0
votes
2
answers
2k
views
Detect if directed cycle is clockwise or counterclockwise in 3D [closed]
I need to check if cycle given by $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3)$ is clockwise or counterclockwise. I have found this answer: Detecting whether directed cycle is clockwise or ...
- 101
1
vote
1
answer
296
views
Deducing Linear Inequalities
Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...
- 185
2
votes
2
answers
640
views
Sorting a binary matrix diagonal in polynomial time while preserving rows
Is there a polynomial time solution to sort an arbitrary binary square matrix in polynomial time by rows so that the diagonal contains a 1 if any row contains a 1 in that column?
For example given ...
- 121
1
vote
1
answer
573
views
'Reference' request: Program to work with cyclic quotient singularities.
I'm looking for program code to deal with cyclic quotient singularities on normal surfaces. In particular, at the moment I need that given a singularity $p$ like $p=\frac{1}{n}(1,a)$ the code computes:...
- 2,215
3
votes
1
answer
141
views
Number of lattice polytopes contained in a given lattice polytope?
Given a (convex) lattice polytope, suppose we want to list or count all (convex) lattice polytopes (of the same dimension) contained in it. Are there efficient ways to do this?
- 495
2
votes
1
answer
227
views
Arrangements of hyperplanes
Fix $n>0$ and $X\subseteq\mathbb{R}^n$. A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form
$$
f(\bar{x})=a_1x_1+\ldots+a_nx_n+b
$$
for some $a_i,b\in\mathbb{R}$.
Suppose we ...
- 3,274
2
votes
1
answer
115
views
mean length of the non-crossing graphs on n points
My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...
- 1,352
0
votes
1
answer
116
views
Paper on unit disk graphs
I was wondering if anybody knows of a 'link' to the paper by Marathe 1995 et al on analysis of the greedy algorithm for finding a Max independent set in Unit Disk Graphs?
- 903
2
votes
1
answer
130
views
Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?
In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
- 121
1
vote
0
answers
85
views
Smallest sum of original column entries in 2d matrix [closed]
I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this:
...
- 111
1
vote
2
answers
306
views
Name of operations on two vectors
Suppose we have two vectors $x\in \mathbb{R}^n$ and $y\in \mathbb{R}^m$.
I could define the mapping
$$
T: \mathbb{R}^n\times \mathbb{R}^m \rightarrow \mathbb{R}^{n\times m}
$$
as follows
$$
T(x,y) = ( ...
1
vote
2
answers
172
views
Linear Programm with matrix [closed]
Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
- 11
1
vote
2
answers
242
views
what method can I employ to solve this optimization problem which involves \min?
The optimization problem is:
maximize $$\min(\sum\limits_{i=1}^N \log\left(a_{1,i}+\frac{b_{1,i}}{c_{1,i}+d_{1,i}x_i}\right),\sum\limits_{i=1}^N \log\left(a_{2,i}+\frac{b_{2,i}}{c_{2,i}+d_{2,i}x_i}\...
- 764
1
vote
1
answer
240
views
Conditions for differentiability of minima and minimizers of linear functionals?
Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$.
For every continuous linear functional $F$ on $B$, define
$V(F)=min_{c\epsilon C} F(c)$ and
$S(F)= { \lbrace c \epsilon C :...
- 13
1
vote
0
answers
517
views
Solving 3D equation system (inverse-projecting a triangle)
Please, how is the equation system below named exactly (to search further literature)?
Does it have an analytical solution? If it doesn't, then what could be the fastest numerical method for it (...
- 119
6
votes
0
answers
97
views
Finding the optimal mixture of two convex functions
I am trying to find an efficient way to solve the problem $$\min_{p,x_1,x_2} p\cdot f(x_1)+ (1-p) \cdot f(x_2)~~~~~ s.t.\\p\cdot g_1(x_1) + (1-p)\cdot g_2(x_2)\leq 1 \\ 0\leq p \leq 1$$ where $x_1,x_2\...
0
votes
0
answers
585
views
Determining the simplices in freudenthal triangulation
HI,
I have a doubt on Freudenthal Triangulation. I want to partition a simplex into finer simplices.
The FT gives me the vertices of the simplices which partition my original simplex into finer ...
- 173
1
vote
0
answers
20
views
Calculating Cost-Optimal 1-Factors in Digraphs
I need to find a cost-optimal 1-factor in a positively weighted, directed, regular graph $G(V,A)$ without antiparallel arcs, i.e. given $$\text{deg}_{\text{in}}(u)=\text{deg}_{\text{in}}(v)=\text{deg}...
- 13.2k
4
votes
1
answer
2k
views
How to find which subset of bitfields xor to another bitfield?
I have a somewhat coding-oriented problem. I have a bunch of bitfields and would like to calculate what subset of them to xor together to achieve a certain other bitfield, or if there isn't a way to ...
- 41
1
vote
1
answer
639
views
Approximate Set Cover Problem by Rounding
Here is the simple algorithm for approximating set cover problem using rounding:
Algorithm 14.1 (Set cover via LP-rounding)
Find an optimal solution to the LP-relaxation.
Pick all sets $S$ for ...
