All Questions
782 questions
0
votes
1
answer
143
views
$\mathrm{ILP}$-formulation for Minimum Maximal Matching (MMM) Problem
Despite some online searching I couldn't find examples of dedicated Integer Linear Programs ($\mathrm{ILP}$s) for determining smallest matchings, that are not contained in a larger one.
It seems that ...
- 13.2k
1
vote
1
answer
181
views
Linear programming with "nice" matrices
Consider the following linear programming problem
\begin{array}{ll}
\text{minimize} & \mathrm 1^{\top} \mathrm x\\
\text{subject to} & v\le \mathrm A \mathrm x \le u\\
& \mathrm x \geq ...
- 650
1
vote
0
answers
38
views
Efficient solution to linear matrix equations
A general form for a linear matrix equation can be written as
$$AX + XB + \sum C_iXD_i$$
If $C_i$ and $D_i$ are all 0, then this simplifies into a well known and studied matrix equation that has an ...
- 11
2
votes
1
answer
875
views
Interpreting mincost flow dual variables
Consider the task of finding flow of size $b$ with minimum possible cost.
It may be formulated as linear programming in a following way:
$$\boxed{\begin{gather}
\min\limits_{f_{ij} \in \mathbb R} &...
- 1,171
3
votes
0
answers
122
views
Preconditioners for $Ax=y$ that rely on hierarchical statistical modeling
Solving $Ax=y$ exactly can be done as:
fit a linear autoregressive model by treating rows of $A$ as data
apply this model to $A^T y$
Imperfect predictive model corresponds to an approximate inverse ...
- 2,442
1
vote
1
answer
59
views
Does norm of discrepancy decrease monotonously in CGLS/CGNR
I am the author of the package for tomographic reconstruction https://github.com/kulvait/KCT_cbct I have implemented CGLS/CGNR , algorithm which applies conjugate gradients on normal equation
$$
A^\...
- 151
2
votes
0
answers
27
views
Solve sparse system with nested inverse
What is the most efficient way to solve an equation
\begin{align*}
(A\,E^{-1}\,C) x = b, \qquad A\in \mathbb{R}^{m\times n}, \, E \in \mathbb{R}^{n\times n}, \, C\in \mathbb{R}^{n\times m}
\end{align*}...
- 21
0
votes
1
answer
64
views
Round Robin volleyball Tournament [closed]
Consider a set of N teams (N even number) that must make a
Round Robin Tournament. To each pair i; j, i ≠ j, of teams there is associated level
of interest si,j ∈ {1;2;3} of the match between them (1 =...
7
votes
2
answers
244
views
Finding $\theta$ such that at least one eigenvalue of $A(\theta)$ is real
Is there a known method to find a set of $\theta$ such that at least one eigenvalue of $A(\theta)$ is purely real?
Assume $A(\theta)$ is a real square matrix whose elements are linear functions of a ...
- 433
2
votes
2
answers
243
views
Sum over exponentiated bilinear form in finite-field vector space
Let $A$ be a linear map over the finite-field vector space $(\mathbb F_2)^n$, i.e., an $\mathbb F_2$-valued $n\times n$ matrix, not necessarily symmetric. I'm interested in the sum
$$Z(A) = \sum_{X\...
- 3,001
2
votes
1
answer
227
views
Solving linear programming without solving linear programming
Let $v_1, \cdots, v_n$ be vectors in $\mathbb R^k$, and let $M$ be the Gram matrix of them.
It's possible to determine from $M$ and $k$ whether the only vector that has nonnegative inner product with ...
- 9,501
2
votes
0
answers
121
views
Proving some properties of the Landweber–Fridman iterates
$\newcommand\norm[1]{\lVert#1\rVert}$Let $B\in \mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Assume that $x\in \mathbb R^n$ is the solution of $Bx=w$ for some given $w\in \mathbb ...
- 21
2
votes
1
answer
372
views
Who called Farkas' fundamental theorem a lemma?
Farkas proved his famous result (which, nowadays, is fundamental in optimization theory) in 1902 and called it Grundsatz der einfachen Ungleichung which may be translated as fundamental theorem of ...
- 16.4k
17
votes
4
answers
6k
views
Why is fast matrix multiplication impractical?
I am wondering why fast matrix multiplications are impractical, especially for Boolean matrix multiplication.
I read some content saying fast matrix multiplications are impractical because of large ...
- 279
1
vote
0
answers
274
views
Find the eigenvectors from the QR algorithm in the unsymmetric case
It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix $A\in\mathbb{R}^{n\times n}$.
I implemented a version ...
- 11
1
vote
0
answers
198
views
Complexity of singular value decomposition using matrix multiplication oracles
Suppose I have an $n\times m$ real matrix $A$, $n\ll m$ with full row rank $(\mathrm{rank}(A) = n)$. I have an oracle that can compute $Ax$ or $A^T y$ for any $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
- 662
1
vote
1
answer
331
views
Finding a special solution in a solution set over F2
Given a solution set of a linear system of the following form
$$
\{ \begin{bmatrix}
x_{1} \\
\vdots \\
x_{n}
\end{bmatrix} = \vec{v_1} * x_1 + \dots + \vec{...
- 51
0
votes
1
answer
396
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
- 1
0
votes
1
answer
118
views
Proving maximum value of a determinant of $I - B$, where $B$ is nonnegative matrix
I have the following setting:
Let $0 \leq r < 1$ and let $\{z_i\}_{i=1}^k$ be $k$ complex numbers such that $|z_i| \leq r$ for all $i$.
Moreover, $r + \sum_{i=1}^k 2Re(z_i) \geq 0$
I am interested ...
- 59
4
votes
2
answers
315
views
Connecting $2n$ points in $\mathbb R^2$ with line segments s.t. each point belongs to exactly one line segment
I'm trying to do a certain simulation related to the toric code and I'm looking for an algorithm that connects $2n$ points ($n \in \mathbb Z_+$) in $\mathbb R^2$ with line segments with the following ...
- 269
3
votes
2
answers
2k
views
Iterative methods for linear system with non-diagonally dominant matrix
I have a linear system
\begin{align*}
\left[\begin{array}{cccc}
1 & 2 & 1 & -1 \\
3 & 2 & 4 & 4 \\
4 & 4 & 3 & 4 \\
2 & 0 &...
- 65
0
votes
1
answer
319
views
Correct way to conduct equilibrium scaling of linear/integer/MIP program
I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
- 21
1
vote
0
answers
61
views
Linear programming robustness to input perturbations
I'm running a linear program whose parametrization depends on the output of a neural network. I was wondering if there exist results on how robust linear programs are towards perturbations in their ...
- 11
2
votes
1
answer
644
views
How to maximise infinity norm of $x$ with constraint $Ax \le b$ using linear program? [closed]
I want to maximise the infinity norm of $x$, subject to constraint: $Ax \le b$. I think you can use a linear program to solve this, but how do you go about formulating it?
1
vote
1
answer
223
views
Does Wilkinson's shift need to be discontinuous?
Given a symmetric Hessenberg matrix $A = \left[\begin{matrix}\ddots& \vdots & \vdots\\\dotsb & a & b\\\dotsb& b & c\end{matrix}\right]$, the Wilkinson shift $\mu$ employed in ...
- 4,943
0
votes
0
answers
115
views
Explicit equation for border of the Minkowski sum of sets
Assume we have sets of the form
$$
M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}
$$
where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$.
Goal
I am looking for an (explicit) representation ...
- 385
1
vote
0
answers
98
views
Solution of a simple optimization problem
Let $\mathbf{U}_1$ and $\mathbf{U}_2$ be two arbitrary unitary matrices and $\mathbf{D}$ be a diagonal matrix. What is the solution of the following optimization problem?
\begin{align}
\min_{\mathbf{...
- 287
0
votes
0
answers
124
views
The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
5
votes
1
answer
251
views
Smooth, non-analytic functions of non-normal matrices
My apologies if this isn't a well-enough-posed question, I think I'm partly unsure of what exact question to even ask.
There are many different ways in which we can take a function of a matrix.
We ...
- 854
1
vote
0
answers
35
views
How to chose the start vector for the MTZ variables
In the context of LP-formulations for the Traveling Salesman Problem the MTZ constraints prevent subtours via $n$ (i.e. effectively $n-1$) additional variables $$u_1=1\\2\le u_2,\,\dots ,\,u_n\le n\\ ...
- 13.2k
2
votes
1
answer
421
views
Is it possible to obtain orthogonal (but not normalized) vectors after QR factorization?
After QR decomposition of a matrix, $M$, the columns of Q are orthonormal. Is it possible after obtaining Q, we recover unnormalized column vectors from $Q$? For example, the matrix M has the ...
- 879
1
vote
0
answers
200
views
Drawing a 3D object in a 3D environment, and converting to math [closed]
So I have been granted a free time and I want to work on a project but first I had to research.
As we know, lines have infinite points, and with lines, we can create infinite shapes. I want to let ...
- 11
0
votes
0
answers
108
views
How to find a set given its support function
Let $\mathcal{U}$ be a convex and compact set. Its support function is defined as $\delta^*(v|\mathcal{U})=\sup_{u\in \mathcal{U}} v^T u$. Assume that we are given the support function $\delta^*(v|\...
- 53
0
votes
0
answers
40
views
Subtour-gluing constraints for ILP formulation of TSPs
If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-...
- 13.2k
3
votes
0
answers
147
views
Convolution integral and its matrix representation
My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, ...
- 879
0
votes
0
answers
96
views
Why is Gaussian distribution always chosen for smoothed analysis?
I came across the algorithmic perfomance analysis model of smoothed analysis. In all references that I read a Gaussian distribution was used for perturbation (e.g. Spielman and Teng 2004 for the ...
- 29
1
vote
1
answer
82
views
Do we really need degree constraints for ILP formulations of TSP problems
The Dantzig-Fulkerson ILP-formulation of the symmetric TSP is
$$\min\sum\limits_{i=1}^{n-1}\sum\limits_{j=i+1}^n c_{ij}x_{\lbrace i,j\rbrace}\quad\text{s.t.}\\ \sum\limits_{j\ne i,\,j=1}^n x_{\lbrace ...
- 13.2k
1
vote
0
answers
58
views
Second-order envelope theorem for linear programming
Consider parameterized linear programming $V(\theta) = \max_x \langle c(\theta),x\rangle$ s.t. $A(\theta)x\leq b(\theta)$, $x\geq 0$. Let's also assume $c,A,b$ are infinitely differentiable with ...
- 373
1
vote
0
answers
37
views
Sum of all integer binary solutions of a TUM linear system
I have the following problem: $A x = b$ where $A$ is a $m \times n$ total unimodular matrix (TUM) with entries in $\{0,1\}$ and $b$ is a $m$-vector of strictly positive integers. Let $\mathcal X$ be ...
- 11
2
votes
1
answer
91
views
Where has the necessary and sufficient condition for the convergence of the unshifted QR algorithm been stated?
I've obtained a necessary and sufficient condition for the unshifted QR algorithm to converge. The condition for the algorithm to converge for a square matrix $M$ is:
For any two eigenvalues $\lambda$...
- 4,943
2
votes
0
answers
194
views
Maximize the product of Hadamard matrix and a vector
Let $X$ be an $n \times n$ Hadamard matrix (i.e. entries are in $\{-1,1\}$ and rows are orthogonal). For my application, we can assume $n=2^k$.
Given a vector $\bf{w} \in R^n$, I want to find the $X^*$...
- 21
6
votes
2
answers
2k
views
Complexity of rectangular matrix multiplication
I am interested in the complexity of multiplying two matrices $A$ and $B$, i.e. to compute $AB$.
From [Le Gall and Urrotia], I know that:
if $A$ and $B$ are square-matrices of size $n$, then this can ...
- 562
0
votes
0
answers
165
views
Minimum circumscribed ellipsoid of $\mathcal H$-polytope
Given matrix $A \in \mathbb{R}^{m \times n}$ and vector $b \in \mathbb{R}^n$, consider the $\mathcal H$-polytope $P$ defined as follows
$$ P := \left\{ x \in \mathbb{R}^n : Ax \leq b \right\} $$
I ...
- 235
0
votes
0
answers
137
views
Any technique for linearization, or linear approximation?
Consider the following Matrix constraint:
$$
\begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0
$$
where $\Sigma_b$ is a known positive definite ...
0
votes
0
answers
227
views
Decomposition of symmetric block matrix
I came across this question and got really interested about it. There, the OP asks whether is possible to decompose a $2n \times 2n$ block matrix:
$$ \begin{pmatrix}
X & I \\
I & Y
\end{...
5
votes
1
answer
124
views
Is Sun's spectral variation bound for normal matrices optimal?
In On the variation of the spectrum of a normal matrix, Sun proves the following result (Corollary 1.2):
Let $A$ be an $n$-square normal matrix and $B$ an arbitrary $n$-square matrix. Then $$ \min_{\...
- 283
0
votes
1
answer
131
views
How hard is a linear programming with a bounded constraint?
Background: I am reading Greg Kuperberg's answer to the question Deciding membership in a convex hull. I am thinking about the complexity of ''Deciding membership in a convex hull''.
Restate the ...
0
votes
1
answer
266
views
Using QR or SVD to sum up finite number of matrices
Problem
I was wondering if there are any theoretical results that tackle the following problem:
Construct the following matrices $\mathbf{\mathcal{S}_{1}},\mathbf{\mathcal{S}_{2}},\ldots,\mathbf{\...
2
votes
0
answers
75
views
Dense matrix vs sparse matrix, when they have same number of nonzero elements
I came across a new way in the literature to solve PDE problems numerically, which is called 'Patch Reconstruction'. One example paper is: Li, R., Sun, Z., Yang, F., & Yang, Z. (2019). A finite ...
- 21
3
votes
0
answers
282
views
Continuum of Lagrange multipliers, duality gap, and minimax theorem
Suppose I have a linear optimization problem involving random variables on some (infinite) probability space $\Omega$. For example, need to maximize expectation $E[Q]$ of random variable $Q$ subject ...
- 781