All Questions
782 questions
6
votes
1
answer
830
views
Dominant eigenvector of a real symmetric tridiagonal matrix
What is the most efficient way to calculate the dominant eigenvector of a real symmetric tridiagonal matrix? What's the corresponding time complexity bound?
Could someone give me a reference for ...
- 61
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 ...
- 21
0
votes
2
answers
1k
views
Degenerate case of linear programming duality?
Let's say we have a maximization linear program that looks like this: maximize $\vec{c}\vec{x}$, subject to $\matrix{A}\vec{x} \leq 0$, $\vec{x} \geq 0$. If we take the dual, we have "minimize $0\vec{...
- 2,009
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
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 ...
1
vote
4
answers
978
views
Maximum average value within a rectangular bounding box
The goal is to expedite detection using the sliding window approach. In other words, an object classifier is known and I need to find where the possible locations of this object are in an image. This ...
- 111
1
vote
2
answers
1k
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 $(A' A)^{-1}$
...
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
9k
views
what is the difference between the revised simplex method andthe full tableu?
No to sound naive but they look like they include the same steps to me, one's just the algorithmical representation of the other. Thanks in advance.
- 111
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 ...
- 233
2
votes
0
answers
281
views
Recovering a piecewise affine function
Lets say I have an piecewise affine convex function $f(x_1,x_2)$, on which the following operations are possible:
Computing $f(x_1,x_2)$.
Computing a subgradient to $f$ at $(x_1,x_2)$
Computing all ...
- 567
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
0
votes
1
answer
1k
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 ...
- 135
38
votes
10
answers
18k
views
Fast matrix multiplication
Suppose we have two $n$ by $n$ matrices over particular ring. We want to multiply them as fast as possible. According to wikipedia there is an algorithm of Coppersmith and Winograd that can do it in $...
- 1,791
5
votes
2
answers
3k
views
Continuous Linear Programming: Estimating a Solution
I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the ...
- 373
4
votes
1
answer
275
views
Symmetry of the integer gap
Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...
- 183
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 \...
1
vote
3
answers
2k
views
How to solve Linear Programming problem with tighter Integer Programming constraints
I want to learn a bit about Linear Programming.
After some research, I decided to solve the Cutting Stock problem as an example to learn. After doing some more research, I feel like I finally ...
9
votes
1
answer
6k
views
Proving that a binary matrix is totally unimodular
I'm working on a set of problems for which I can formulate binary integer programs. When I solve the linear relaxations of these problems, I always get integer solutions. I would like to prove that ...
- 213
10
votes
3
answers
6k
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 ...
- 7,857
6
votes
2
answers
8k
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 $...
4
votes
1
answer
866
views
When is a triangular matrix totally unimodular?
I have a {0,1}, invertible, triangular matrix, that I would like to show is totally unimodular. Are there any known results on the total unimodularity of classes of triangular matrices?
- 1,182
5
votes
1
answer
271
views
Feasibility of linear programs
It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?
- 201
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 $\...
4
votes
0
answers
790
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 ...
- 41
15
votes
9
answers
9k
views
Exponential of large matrices
I want to make a diffusion kernel, which involves $e^{\beta A}$, where A is a large matrix (25k by 25k). It is an adjacency matrix, so it's symmetric and very sparse.
Does anyone have a ...
- 151
6
votes
3
answers
2k
views
A simple infinite dimensional optimization problem
I'd be grateful for a reference for the following result, which I believe to be true, and
should be well-known.
Let the continuous functions $f_0,f_1,\cdots,f_n: [0,1]\rightarrow [0,\infty)$ be ...
- 520
29
votes
6
answers
8k
views
How to find a closest integer point to the intersection of two lines?
Here's a question that originates from StackOverflow.
Given are two lines on a plane, specified by equations ($a x + b y = c$) with integer coefficients. The lines aren't parallel and they don't ...
- 391
1
vote
0
answers
1k
views
Covariance matrix formula interpretation - what am I missing?
I'm reading a paper that outlines the calculation of a covariance matrix like the following:
$C=\displaystyle\sum^{N_b}_{i=1}\vec{x}_i\vec{x}_i^T$
What is the order of this matrix? My interpretation ...
- 111
0
votes
2
answers
4k
views
Linear programming piecewise linear objective
I am fairly new at linear programming/optimization and am currently working on implementing a linear program that is stated like this:
max $\sum_{i=1}^{k}{p(\vec \alpha \cdot \vec c_i)}$
$s.t. $
$|\...
- 3
10
votes
1
answer
2k
views
Sum of difference moduli vs. sum of modulus differences
This is a failed attempt of mine at creating a contest problem; the failure is in the fact that I wasn't able to solve it myself.
Let $x_1$, $x_2$, ..., $x_n$ be $n$ reals. For any integer $k$, ...
- 33.8k
18
votes
3
answers
3k
views
Deciding membership in a convex hull
Given points $u, v_1, \dots,v_n \in \mathbb{R}^m$, decide if $u$ is contained in the convex hull of $v_1, \dots, v_n$.
This can be done efficiently by linear programming (time polynomial in $n,m$) in ...
- 667