All Questions
783 questions
11
votes
1
answer
896
views
Decide if a matrix is transposable
A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...
- 4,729
0
votes
2
answers
102
views
Does Max Flow produce uniform results? [closed]
I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have ...
- 400
1
vote
3
answers
138
views
Literature request: Function that depends on a linear optimization problem [closed]
my question is about functions of the following form:
$$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$
where $\mathbf{x},\mathbf{b}, $ ...
- 1,991
3
votes
1
answer
553
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 ...
- 4,713
1
vote
1
answer
600
views
Closed Form Solution for Optimization Problem over the Space of Rigid Transforms
Is there a closed form solution to this constrained optimization problem:
\begin{equation}
\min_{R \in SO(3),\, \mathbf t \in \mathbb R^3} = \sum_{i = 1}^N \| M_i(R \mathbf p_i + \mathbf t) \|^2,
\...
- 778
0
votes
1
answer
116
views
Matrix equation of the form $C A C^\intercal = D$
Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...
- 323
7
votes
1
answer
386
views
Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?
Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:
The all-one vector $j$ is contained in the conic hull of $col(A)$.
...
CommunityBot
- 1
0
votes
1
answer
145
views
How to solve this optimization problem efficiently? [closed]
Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
...
- 111
1
vote
1
answer
123
views
LASSO problem but with a maximization instead of minimization
I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
- 20.2k
2
votes
1
answer
134
views
Quadratic matrix equation for $X\in \mathbb{C}^{ n\times p}$ with Hermitian parameters
Let $A\in \mathbb{C}^{ n\times n}$ and $B \in \mathbb{C}^{p \times p}$ be Hermitian matrices with $p < n$.
Find matrix $X$ such that $X^*AX=B.$
Solution in the case of positive definite $A$ and $...
- 3,741
-1
votes
1
answer
88
views
sparse data fitting problem [closed]
I am a new learner of optimization, and I am confused by the question below, (how to change a 0-norm constrain into binary and linear constrain ?)
Given a sparse data fitting problem:
$ minimize \...
- 1,517
3
votes
0
answers
244
views
An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,712
4
votes
1
answer
296
views
Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
CommunityBot
- 1
6
votes
1
answer
218
views
Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?
We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
- 12.5k
1
vote
0
answers
74
views
How to minimize n-polytope's bounding box with linear transformation?
I am working on an exact algorithm for integer linear programming for my master's thesis:
$Ax\leq b, x \in \mathbb{Z}^n$
$cx\rightarrow min$
For my idea to work out, I need a guarantee that n-...
- 11
0
votes
1
answer
540
views
Computing spectrum of convex combination of SPD matrices given individual spectral decompositions
Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
- 133
2
votes
1
answer
3k
views
Lanczos algorithm for finding $k$ smallest eigenvector
I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
- 4,729
7
votes
1
answer
356
views
Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?
(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
- 188k
5
votes
1
answer
260
views
Numerical minimization spectral norm under diagonal similarity
This question is a follow up.
Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e.,
$$
s(A) = \inf_{D} \lVert D^{-1} A D\...
- 909
17
votes
3
answers
6k
views
The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
- 13.6k
1
vote
0
answers
85
views
"Barrier functions" in function spaces [closed]
In general the idea of a "barrier function" (as in the context of "interior point methods") can possibly be thought of as defining a function corresponding to an open subset of $\mathbb{R}^n$ such ...
- 2,246
0
votes
1
answer
490
views
Relax a rectangular linear assignment problem
I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
0
votes
1
answer
3k
views
Cholesky decomposition – non-positive definite matrix
In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
- 4,729
0
votes
0
answers
232
views
Smallest eigenvalue of a sparse matrix (updated)
Let $D_{1}$ be $(m-1)n \times mn$ matrix (that is, $(m-1)n$ rows and $mn$ columns) and $D_{2}$ be $m(n-1) \times mn$
defined as
$$\begin{cases}
D_{1}[(m-1)(j-1)+i ; m(j-1)+i] & = -1 , \\
D_{1}[(m-...
- 133
2
votes
1
answer
502
views
Feasibility Mixed integer Linear programming with quadratic constraints?
Consider the mixed integer program
$$Ax\leq b$$
$$By\leq c$$
$$\begin{bmatrix}x&y\end{bmatrix}C\begin{bmatrix}x\\y\end{bmatrix}+D\begin{bmatrix}x\\y\end{bmatrix}\leq d$$ where $x$ are integer ...
- 1,517
11
votes
0
answers
764
views
Fast computation of matrix product $AXA^T$ with fixed $A$?
Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
3
votes
1
answer
266
views
Strong polynomial algorithm for linear programming
What is the current state of finding a strong polynomial algorithm for linear programming? Is there any reference?
-1
votes
2
answers
114
views
On OR condition in Linear Programming with exponentially many constraints [closed]
Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
- 128k
5
votes
2
answers
6k
views
Computing the largest eigenvalue of a very large sparse matrix
I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter $w$, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue ...
0
votes
0
answers
99
views
Is this Graph Iteration Already Known?
When attempting to set up an ILP formulation for a weight-minimal cubic spanning tree (i.e. one with vertex degrees either 1 or 3) I needed connectivity constraint, but misremembered the contents of ...
CommunityBot
- 1
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 761
8
votes
1
answer
1k
views
Square root of a large sparse symmetric positive definite matrix
I am trying to calculate
$$Y = A^{\frac 12} X$$
where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
- 6,891
0
votes
0
answers
46
views
linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
1
vote
1
answer
170
views
Optimization of a continuous function
This is more like an optimization problem but any solution is appreciated.
I have a data set with input specifying power(demand) to be generated for a particular time period(TP).
Input:
Time --- ...
- 61.9k
1
vote
1
answer
652
views
Compute the eigenvectors corresponding to the $k$ smallest eigenvalues w.r.t high dimensional symmetric sparse matrix?
So far as I know:
The power iteration method can only get the eigenvector corresponding to the largest eigenvalue;
The inverse power iteration method requires that the matrix is invertible;
The QR ...
- 20.2k
3
votes
2
answers
437
views
convex polytope integer points
is there a simple proof for the following lemma:
An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
- 1,991
1
vote
0
answers
66
views
On number of solutions by simplex and number of solutions in total in a linear optimization problem?
This is more of a clarification query.
Mizuno http://www2.ims.nus.edu.sg/Programs/012opti/files/talk_mizuno1.pdf says if we give a linear optimization problem
$$\max c'x$$
$$Ax\leq b$$
where $A\in\...
CommunityBot
- 1
2
votes
0
answers
74
views
Possible ordering of coordinates in a linear subspace [closed]
This question was asked on Mathematics Stack Exchange with no answers.
Let $X$ be a linear subspace of $\mathbb{R}^n$. For how many permutations $p$ on $1,...,n$ does there exists $x$ in $X$ with $...
- 2,821
2
votes
1
answer
91
views
Linear program with one quadratic condition convex in domain of interest polynomial time solvable?
$c\leq xy$ is not a convex condition.
However we know $c\leq xy$ is convex in domain $x,y>0$ in $\mathbb R^2$ for any fixed $c\in\mathbb R$.
Is $c\leq x_1y_1+\dots+x_ny_n$ with $0\leq x_1,\dots,...
- 54.2k
1
vote
0
answers
62
views
LP Constraints for Bridgeless Cactus Graphs
When trying to determine the optimal bridgeless spanning cactus graph of a weighted, symmetric graph, I got stuck.
What I do not know how to capture, is
the variable number and sizes of the cycles
...
CommunityBot
- 1
0
votes
0
answers
369
views
Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
1
vote
0
answers
60
views
On the defect of a flow network
This problem in graph theory was actually motivated by some problems in Theory of Fractals.
To formulate the problem I need to recall some definitions related to flow network.
A flow network is a ...
CommunityBot
- 1
1
vote
0
answers
86
views
Infinite system of equations with finitely many constraints
During my research I have stumbled upon the following issue concerning infinite systems of linear equations. I do not have much practice in such settings, so I am asking you whether the following ...
2
votes
1
answer
255
views
Efficient algorithm for solving a convex quadratic program [duplicate]
Let $A \in \mathbb{R}^{n \times m}$ and $b \in \mathbb{R}^n$. Suppose $m \ll n$. How to solve this quadratic program efficiently?
$$\min_{x \in \mathbb{R}^n} \frac{1}{2} x^\top AA^\top x + b^\top x$$
5
votes
0
answers
162
views
Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
2
votes
1
answer
769
views
Is it possible to perform PCA on a multimensional array?
Normally you have a matrix n x p and apply PCA to it. But in the article I'm reading, the author considers that the matrix has points in it. So instead of being n x p, it'd be, say, n x p x 2. He then ...
- 63.9k
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:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
3
votes
1
answer
328
views
Numerical iterative methods for Poisson equation
Given a domain $\Omega \subset \Bbb R^n$ and $\Delta\varphi=f$ where $\varphi:\Bbb R^n \to \Bbb R$ is unknown and $f:\Omega\to \Bbb R$ is a blackbox function (for each $\bf x$ it provides $f({\bf x})$,...
- 224
8
votes
1
answer
2k
views
Finding Toeplitz matrix nearest to a given matrix
For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change.
Specifically I want find the Toeplitz ...
CommunityBot
- 1
1
vote
0
answers
246
views
How to solve a large linear programming problem? [closed]
I have the linear programming problem in $\mathbf x \in\mathbb R^n$
$$\begin{array}{ll} \text{minimize} & \mathbf c^T\mathbf x\\ \text{subject to} & \mathbf A\mathbf x \leq \mathbf b\end{...