All Questions
783 questions
3
votes
1
answer
4k
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 $C>0$ ...
- 155
1
vote
0
answers
126
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 $M_k$...
- 4,829
8
votes
1
answer
1k
views
Norm of inverse confluent Vandermonde matrix
Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as
$$V=
\begin{bmatrix}
v_{1,0}&v_{2,0}&\dots&...
CommunityBot
- 1
2
votes
0
answers
184
views
Checking for error in conjugate gradient algorithm
What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
- 489
4
votes
2
answers
1k
views
Set Cover:Greedy vs LP
Hi
Both, the greedy and the LP approach for Set Cover give a O(log n) approximation. Is there some inherent difference on the two approximation approaches?
thanks
- 183
1
vote
1
answer
639
views
Approximate Set Cover Problem by Rounding
Here is the simple algorithm for approximating set cover problem using rounding:
Algorithm 14.1 (Set cover via LP-rounding)
Find an optimal solution to the LP-relaxation.
Pick all sets $S$ for ...
CommunityBot
- 1
0
votes
1
answer
165
views
Nonlinear matrix equation 2
Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$
$Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \...
CommunityBot
- 1
3
votes
1
answer
3k
views
Nonlinear matrix equation
Solve the following nonlinear equations for $v$ and $w$
$Avv^TAw=\lambda_1v+\lambda_2w$
$Aww^TAv=\lambda_1w+\lambda_2v$
$v^Tw=w^Tv=0$
$v^Tv=w^Tw=1$
where $\lambda_1, \lambda_2, \lambda_3$ are ...
- 41
0
votes
1
answer
2k
views
Finding linearly independent columns of a large sparse rectangular matrix
I have a problem that necessitates solving a large non-negative least-squares
problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols)
and nearly binary. However, A is not ...
- 30.1k
1
vote
2
answers
444
views
Levenberg-Marquadt near the minima for non-zero-residual problems
I'm using the LM algorithm to do gradient descent in a model fitting context. I'm minimizing:
$$
c(x) = \sum ( f_i(x) - y_i )^2
$$
I'm noticing that after a few steps when I'm close to the minima, I ...
- 561
0
votes
0
answers
103
views
Gauss-Newton for quotient functions
I'm optimizing a function of the form
$$
\sum \frac{ \|\mathbf{f_i}(x)\|^2 }{ g_i(x)^2 + h_i(x)^2 }
$$
where $x$ is a real vector, $\mathbf{f}(x)$ is a real vector, and $g(x)$ is a scalar. My first ...
- 561
1
vote
1
answer
769
views
Results for minimizing the norm w.r.t a unitary matrix
Suppose $x \in \mathbb{R}^n$, $B,U \in \mathbb{R}^n\times\mathbb{R}^n$ and $U$ a unitary matrix. Define $g_{U}(x) = || BUx||$ where $||.||$ is some norm or norm-ish function on $\mathbb{R}^n$ (not ...
- 3,059
0
votes
0
answers
194
views
A linear program related question
Dear all, recently, I encountered the following problem. It is closely related to the order of growth for UMD constants of all $n$-dimensional Banach lattice.
Let $\alpha^k \in (\alpha_1^k, \alpha_2^...
- 769
0
votes
0
answers
79
views
Computing maximum point for minimal function of a family of linear functions
Let $x \in S^n $ where $S^n = ${$ [x_1,x_2,...,x_{n+1}]\in \mathbb{R}^{n+1} \mid x \ge 0 , \sum x_i = 1 $} and let $f_i : I^n \to \mathbb{R}$ be a finite $m$-sized family of LINEAR functions such that:...
- 11
0
votes
1
answer
130
views
Cascading minimization problems
Hi all. Suppose I have a linear programming problem on the vector variable $x$ that has many solutions and let $U$ be the set of these solutions. Suppose I have a second LP problem on $y \in U$. ...
- 3,934
3
votes
2
answers
10k
views
linear programming with OR restrictions
Hi all. I have a linear program with the restriction that every variable can be zero or greater than or equal to a positive constant. That is:
minimize: $w^Tx$
subject to: $Ax=b$, $Cx \le d$ and for ...
- 57
0
votes
2
answers
891
views
Find both maximum and minimum values in linear programming problem
Hi all. I have a linear programming problem where I need to find both maximum and minimum values of the objective function. The optimal points are not relevant.
Is there an efficient way to do so?
- 96.4k
1
vote
2
answers
242
views
what method can I employ to solve this optimization problem which involves \min?
The optimization problem is:
maximize $$\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}\...
- 101
0
votes
1
answer
205
views
SDP Algorithms/ maximally complementary solutions
Hello,
I was wondering if there are algorithms for (linear) Semidefinite Programs (SDP) out there, that converge towards a maximally complementary solution, even if strict complementary does not hold.
...
CommunityBot
- 1
9
votes
2
answers
1k
views
polynomials with minimal $L_\infty$ norm on multiple disjoint intervals
It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
- 91.8k
1
vote
2
answers
134
views
LP/QP with not-so-constant linear constaints
I have an otherwise standard LP or PSD QP problem as below:
$\min\limits_x {c}' x$ subject to $Ax\leq b$
or
$\min\limits_x \frac{1}{2}{x}' Qx + {c}' x$ subject to $Ax\leq b$
the only exception ...
- 54.2k
0
votes
0
answers
783
views
LP relaxation for ILP\IP (integer linear programming)
I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
3
votes
1
answer
533
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 ...
- 276
0
votes
1
answer
193
views
Ease of calculation of norm
I have SPD matrix A and two vectors z and b.
Is there exist a norm where I can calculate $||A^{1/2}b-z||$ without having to calculate $A^{1/2}b$ explicitly ?
- 3,934
2
votes
1
answer
6k
views
sum of maxima vs the maximum of the sum
Consider the following integer program
$$
\begin{align}
\max &\sum\nolimits_{i}\sum\nolimits_{j} U_i(j)\cdot x_{i,j}\\
\text{subject to}& \sum_{i}x_{i,j}\cdot f\left(i,j\right)\leqslant c_j,&...
- 1,253
5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 12.7k
4
votes
2
answers
4k
views
Dual Norm For Sum of 2-Norms
What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$
$\|\mathbf{x}\| = ...
- 9,486
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 ...
CommunityBot
- 1
2
votes
3
answers
2k
views
Efficient Algorithm For Projection Onto A Convex Set
Given $\mathbf{x} \in \mathbb{R}^n$ and $\tau$ a scalar, I would like to solve the following Euclidean projection problem:
$\underset{\mathbf{p}}{\mathrm{argmin}} \; \|\mathbf{p}-\mathbf{x}\|_2
\;\;
\...
- 3,934
2
votes
0
answers
230
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 ...
- 221
0
votes
0
answers
136
views
Finding peaks and determining noise
Hello ,
Im having one matrix which is product of two FFT transforms of one fits image ( astronomical image ). In that matrix you could find 3 peaks. One largest in center, and two around central ...
- 101
0
votes
1
answer
226
views
Continuity of Lexicographic Minimum Solution of a parametrized LP problem
Given a parametrized LP problem
find x, that minimizes F*x
such that Ax <=Bt+D
where t is a parameter.
And suppose C(t) is a set of all optimal solutions of LP with parameter t.
Let x_L(t) be ...
CommunityBot
- 1
3
votes
1
answer
347
views
Grading a non-graded poset as squeezed as possible
Here is a curiosity question (motivated by the recent revamp of ranked-poset routines in Sage).
Let $P$ be a finite poset. We look for a family $\left(a_p\right)_{p\in P}$ of real numbers summing up ...
- 33.8k
3
votes
2
answers
3k
views
how to model a linear program with step-like cost function in the objective
I have a large linear program with the following details.
d1 to di are the variables, where di is an integer. The constraints are a series of inequalities of the form
d1 < d3 < d7 < d23 (...
CommunityBot
- 1
3
votes
2
answers
195
views
Inflate a simplex, change rows to make the rank n
I have a simplex, n + 1 points in $\mathbb{R}^n$,
which may have rank $r < n$.
Is there a cheap way of "inflating" it to rank $n$,
changing a few, all but $r$, of the points ?
The points are ...
- 15.5k
10
votes
1
answer
411
views
Network flows with capacities on pairs of edges
Take a standard network flow problem: a directed graph with nonnegative capacities on each edge, a source $s$, a sink $t$. We all know how to find the maximum flow from $s$ to $t$.
Now add edge-pair ...
- 37.7k
2
votes
1
answer
815
views
A question for solutions of perturbed linear systems
Consider a linear system
$$Ax=b\qquad (*)$$
and a sequence of perturbed linear systems $$(A+\delta A_n)x=b+\delta b_n. \qquad (n)$$
Suppose that all the linear systems are consistent (i.e., ...
CommunityBot
- 1
2
votes
1
answer
848
views
Algorithm for satisfiability of inequalities.
I am looking for an algorithm for checking the satisfiability (with natural values) of a set of inequalities made of variables and natural numbers, for example: $u < v, u \leq z, 3 \leq v$.
In ...
- 37.7k
3
votes
0
answers
130
views
Computing the norm of the columns of an implicitly defined matrix
I have an $n \times n$ matrix $M = \Sigma W$ where $\Sigma$ is diagonal and $W$ orthogonal. $W$ is implicitly defined, i.e. I can only perform matrix-vector products (but I also have access to $W^T$).
...
7
votes
1
answer
360
views
Does the Hirsch conjecture hold for $n < 2d$?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
- 1,502
2
votes
1
answer
137
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 ...
CommunityBot
- 1
10
votes
2
answers
3k
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.
- 2,239
3
votes
0
answers
220
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 ...
- 131
3
votes
0
answers
1k
views
How many iterations are required for the Lanczos algorithm to converge?
I am trying to find the n smallest eigenvalues and eigenvectors of a NxN SPD matrix using Lanczos method. What is the number of iterations usually required? I mean, does it scale as $O(N)$ or $O(\sqrt{...
- 31
7
votes
0
answers
120
views
How do I find elements of an algebra which generate an algebra contained in a fixed subspace?
Suppose $V$ is a linear subspace of a finite dimensional $C^*$-algebra $A$. (Feel free to assume $A$ is a multi-matrix algebra over $\mathbb C$.)
I would like to find $x \in V$ such that $\mathbb C [...
- 7,800
2
votes
2
answers
2k
views
Solving a system of equations/inequalities that have trigonometric functions on the left-hand side
Is there any known (symbolic) method that solves a system of equations/inequalities that have trigonometric functions on the left-hand side of the system?
Ex) Find $x,y,\theta \in \mathbb{R}$ that ...
- 8,688
0
votes
1
answer
695
views
Minimum distance between two data sets
Suppose we have two sets of data, $X$ and $Y$, each of which contains
$10$ positive numbers. Now let us order the data sets $X=\left\{ x_{1},\cdots,x_{10}\right\}$,
$x_{1}\ge\cdots\ge x_{10}>0$ and ...
- 28.6k
4
votes
1
answer
2k
views
solving multiple linear programming problems with the same set of constraints
Hi,
I need to solve a set of linear programs of the form:
Problem $i$: $\quad \max c_i \cdot x$ s.t. $ A x \leq b$.
The $c_i$'s are different vectors so each problem has a different objective ...
CommunityBot
- 1
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
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), ...
- 54.2k