# Tagged Questions

**0**

votes

**0**answers

59 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 ...

**0**

votes

**0**answers

26 views

### Find relationships between events

I have a set of Events $(E_i)_i$ which have a probability $(P_i)_i$.
I am able to write each event as a sum of distinct events that form a partition of the space.
My goal is to find all the ...

**2**

votes

**1**answer

67 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 ...

**2**

votes

**2**answers

101 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,\; ...

**0**

votes

**1**answer

76 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 ...

**1**

vote

**0**answers

119 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 ...

**1**

vote

**1**answer

47 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 ...

**2**

votes

**0**answers

53 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 ...

**1**

vote

**2**answers

48 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 ...

**0**

votes

**0**answers

73 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.
$$
...

**3**

votes

**1**answer

110 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 ...

**6**

votes

**1**answer

242 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 ...

**4**

votes

**0**answers

108 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 ...

**1**

vote

**0**answers

45 views

### Nontrivial Matrix-estimate

I try to proof the following estimate:
\begin{align}
h' W^{-1} H W^{-1} h \geq c h' H h \qquad c>0, \qquad\qquad (1)
\end{align}
where $h\in\mathbb{R}^{K-1}$ and ...

**1**

vote

**1**answer

132 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 = ...

**0**

votes

**1**answer

402 views

### eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...

**1**

vote

**2**answers

172 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

**1**answer

105 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< ...

**1**

vote

**0**answers

75 views

### Arrangements of graphs of linear functions

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Assume $X$ is unbounded.
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 ...

**2**

votes

**1**answer

153 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 ...

**6**

votes

**1**answer

482 views

### Inverse of a totally unimodular matrix

A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $âˆ’1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A ...

**0**

votes

**1**answer

321 views

### Find edge weights that fit given node weights

Let $G = (V,E)$ be a connected simple graph (unweighted, undirected, no selfloops) on $n$ nodes.
Let $\mathbf{d} := (d_1, d_2, ..., d_n) \in \mathbb{R}_{>0}^n$ be a vector of arbitrary given node ...

**1**

vote

**1**answer

720 views

### Schur complement and negative definite matrices

Hello,
My question regards to the Schur complement lemma. Consider the matrix $M=\left( \begin{array}{cc}
A & B\\\
B^T & C \end{array}\right)
$.
According to the lemma $M\geq0$ iff $C>0$ ...

**1**

vote

**0**answers

99 views

### Matrix Minimax problem

I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The ...

**0**

votes

**1**answer

488 views

### Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not ...

**3**

votes

**2**answers

253 views

### Bounding the minimal maximum norm of a solution of a linear system.

I would be grateful for pointing me out a reference to some general bound on the $\ell_{\infty}$ norm of a solution of a linear system. To be specific, suppose that we have an underdetermined linear ...

**0**

votes

**1**answer

256 views

### Solving a system of linear inequalities

Consider the following system of inequalities:
$Ax=b$;
$x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without ...

**1**

vote

**0**answers

147 views

### Consistency of a system of linear equations

I have a system of linear equations in form of $AX=b$ where $A_{m\times n}$, $X_{n\times 1}$ and $b_{m\times 1}$. Coefficient matrix $A$ is quite sparse. However, using a practical LP solver like ...

**1**

vote

**0**answers

230 views

### Totally Uni-modular Matrices

A matrix is totally uni-modular if the determinant of any (square) sub-matrix is {+1, 0, -1}. My question is, "Is there a way to transform(linear or non) a general matrix into a totally uni-modular ...

**4**

votes

**1**answer

215 views

### Checking if one polytope is contained in another

Hi,
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 ...

**2**

votes

**1**answer

106 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 ...

**5**

votes

**2**answers

848 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.

**2**

votes

**0**answers

170 views

### Could SVD be used to optimize the partial inner-products?

Suppose a set $N$ of $n$ distinct points in $m-$dimensional space is given in $X\in\mathbb{R}^{n\times m}$. Also, suppose a subset $L\subset N$, $|L|=l<m<n$, with
$m-$dimensional coordinates in ...

**4**

votes

**1**answer

788 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 ...

**1**

vote

**1**answer

435 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), ...

**1**

vote

**1**answer

377 views

### Need help to find an efficient algorithm for the following problem!

Consider $x$ an n-dimensional vector with $x_i$ is integer in the range $[0 \dots k], k\in N$.
Given $A_{n\times n}$ is the covariance matrix of $x$.
$u$ is a given n-dimensional vector of real ...

**4**

votes

**2**answers

485 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 ...

**2**

votes

**0**answers

225 views

### Linear complementarity problem: principal pivoting algorithm

I'm trying to implement the "Dantzig; van de Panne and Whinston" principal pivoting algorithm for solving symmetric positive semi-definite LCPs from "The Linear Complementarity Problem" book ...

**3**

votes

**1**answer

317 views

### Mathematical Programming with other Algebras than Linear

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.
What analogies are there for ...

**1**

vote

**1**answer

2k 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
...

**2**

votes

**2**answers

203 views

### Maximization of a matrix product by iterative methods

This might not be very difficult, but I think I may have gotten a little confused.
Suppose we are given a matrix A, and would like to find the vector x of modulus 1 which maximises the product xt A x ...

**2**

votes

**4**answers

358 views

### Efficient algorithm for finding the minima of a piecewise linear function

Consider real numbers $a_i$ and $b_i$ for $i=1\dots n$ and define a function by
$f(x) = \max_i ( a_i + b_i x )$
We desire to find $\min_x f(x)$. Obviously this occurs at an intersection of two ...

**2**

votes

**0**answers

238 views

### When is polytope compatible with network flow?

A linear program is the problem of optimizing an linear objective function within some polytope $A$ over $\mathbf R^n$. My question is motivated by the question of when a linear programming problem ...

**3**

votes

**4**answers

530 views

### efficient way to compute the inversion of the following matrix

Hi, there
I have looked it up in the current textbook. The conventional numerical method to compute the inversion of an $n \times n$ matrix requires $O(n^3)$. However, for the following special ...

**1**

vote

**2**answers

820 views

### Inequality-constrained linear-regression, what is the covariance of the estimator?

If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...

**0**

votes

**1**answer

767 views

### For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?

Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...

**6**

votes

**2**answers

3k views

### Solving a system of linear inequalities — what is the dimension of the solution set?

It is well known how to solve a system of linear equations $A{\bf x} = {\bf b}$, but how do we solve a system of linear inequalities $A{\bf x} \leq {\bf b}$?
For the applications I have in mind the ...

**3**

votes

**2**answers

3k 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 ...

**0**

votes

**0**answers

2k 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 ...

**2**

votes

**0**answers

504 views

### Is it possible to use linear programming to solve this problem?

I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group.
Could someone comment on whether this is ...