All Questions
740 questions
6
votes
1
answer
1k
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 ...
CommunityBot
- 1
5
votes
0
answers
167
views
A specific case of the $p$-center problem
Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
- 125
2
votes
1
answer
130
views
Fastest 'Oracle' Algorithm for satisfying a single linear constraint on a convex set?
In this paper by Arora, Hazan, and Kale (http://www.cs.princeton.edu/~arora/pubs/MWsurvey.pdf) a method is given for using the Multiplicative Weights Update algorithm to quickly solve Linear Programs ...
- 1,282
5
votes
1
answer
384
views
Examples of Polyhedra with Large Shadows
Let $P \subseteq \mathbb{R}^n$ be a polyhedron described by $\mathcal{O}(n^{c_1})$ inequalities, where $c_1$ is a constant. Moreover, let $M\colon P \to \mathbb{R}^2$ be a linear mapping. I'm looking ...
- 321
1
vote
1
answer
175
views
accelerate convex optimization by proximal projection
I am using level method to solve non-smooth convex programming problem (where the objective function is given by an oracle from another program ):
http://www2.isye.gatech.edu/~nemirovs/Lect_EMCO.pdf
...
- 12.7k
6
votes
3
answers
11k
views
Maximum flow with negative capacities?
I'm trying to compute an (s-t) maximum flow through a network which includes a number of arc pairs ((u,v), (v,u)) that have equal, negative capacities (weights). I'm not aware of any efficient ...
- 11
1
vote
1
answer
4k
views
Maximizing linear objective function with absolute values
This has be asked on other forums, though couldn't
find authoritative answer.
I have a linear program over the reals and don't
want to introduce integer or binary variables.
The objective function ...
- 1
0
votes
1
answer
2k
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. ...
CommunityBot
- 1
5
votes
1
answer
2k
views
Algorithm to minimally connect line segments in Euclidean plane
Suppose you have finitely many line segments in the Euclidean plane. How do you "connect them to form one chain of line segments of minimal length?"
More formally and generally, what I'm looking for ...
- 3,979
7
votes
1
answer
224
views
A polytope with a bound on the sum of any $k$ variables
Let $2\le k\le n-1$ and define the polytope
$$P_k(n) = \lbrace (x_1,\ldots,x_n) \in \mathbb{R}^n :
-1\le x_{i_1}+\cdots +x_{i_k} \le 1 \text{ for all } 1\le i_1\lt\cdots\lt i_k\le n\rbrace.$$
There ...
- 13.6k
2
votes
1
answer
276
views
An optimization problem in complex space
Consider the following optimization problem
$$
\min \| \textbf{Ax-B}\|
$$
$$
s.t.|x_i|=1,i=1,...,n
$$
where $\textbf{x}\in \mathbb{C}^{n}$ is the optimization varaible, $x_i$ is the $i$-th ...
- 14k
1
vote
2
answers
256
views
how to estimate a polyhedron(convex hull) classifier from data sample
Given a set of points $X\in\Re^D$, they have labels $Y\in${$-1,+1$}. I would like to separate the data labeled +1 and the data labeled -1 by a polyhedron.
$min_w \sum_i \xi_i + \frac{1}{2}\|w\|_2^2$
...
CommunityBot
- 1
0
votes
1
answer
389
views
Tetris in 3D with 5 units [closed]
Background: There are 7 "bricks" used in the game of Tetris. These are the 7 unique combinations of 4 unit squares in which every square shares at least one edge with another square. ("unique" in this ...
CommunityBot
- 1
2
votes
1
answer
104
views
Standard names and methods for this type of fitting minimization
In material science research, we have come across the following type of problem.
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
...
- 6,210
2
votes
1
answer
301
views
books on very large scale linear optimization
Recently in my material science research, I have encountered problems of very large scale linear optimization. I read the introductory book "Introduction to Linear Optimization (Athena Scientific ...
- 1,181
3
votes
1
answer
2k
views
Finding a minimum bounding sphere for a frustum
I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible.
I can choose the centre of the sphere to be right in ...
CommunityBot
- 1
0
votes
2
answers
118
views
Inner Product of Given Sum Positive Sequence
Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in A,\,\sup\...
- 2,239
1
vote
1
answer
70
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 ...
- 1,282
19
votes
2
answers
1k
views
Four Dimensional Origami Axioms
What are the axioms of four dimensional Origami.
If standard Origami is considered three dimensional, it has points, lines, surfaces and folds to create a three dimensional form from the folded ...
CommunityBot
- 1
1
vote
1
answer
6k
views
Convert linear programming problem into its standard form [closed]
all,
I met a question that, the cost function of the linear programming problem is a function with absolute value. Here is the problem:
min 3x1+|6x2+3|
st.
|x1+4|+|2x2|<=3
How can I deal with it?...
- 54.2k
1
vote
2
answers
150
views
investigating positivity/negativity of a function [closed]
I'm investigating if and how the positivity or negativity of a multivariable function can be proved. Consider $y_{1},y_{2},y_{3}\in\mathbb{R}$ and the following function
$$f\left(y_{1},y_{2},y_{3}\...
- 79.9k
1
vote
1
answer
1k
views
Linear programming with exponentially many constraints and variables [closed]
From class, I learnt that problems like traveller salesman have a Linear programming representation with exponentially many constraints. Using method of separation, this problem is solved rather ...
- 1,282
3
votes
2
answers
5k
views
Linear program to maximize the minimum absolute value of linear functions ?
I'd like to compute
$\max_{x,t} t$ such that $\forall i$, $t < a_i + |x - b_i|$.
where $a_i,\ldots, a_n$ and $b_1,\ldots,b_n$ are fixed and $x \in [0,1]$.
Can this be solved with a linear ...
- 11
18
votes
1
answer
678
views
Higher dimensional generalization of: Any quadrilateral tiles the plane?
Any (non-self-intersecting) quadrilateral tiles the plane.
(MathWorld image.)
Q. What is the strongest known generalization of this statement to higher dimensions?
I.e., $\mathbb{R}^d$ ...
- 17.4k
0
votes
1
answer
270
views
Generalized assignment problem with no integrality gap
Suppose I am solving the generalized assignment problem, so that I
am given matrices $U$ and $W$ and a vector $c$ (all three of which
have, say, positive entries), and I want to solve
$$\text{...
- 54.2k
5
votes
1
answer
132
views
How to approximate volumes of m-dimensional manifolds with m-dimensional polyhedra
The areas of a sequence of polyhedra approaching a surface need not approach the area of the surface, but there are theorems guaranteeing that this be so. (T. Rado, On the Problem of Plateau, Chapter ...
- 96.4k
4
votes
1
answer
694
views
$\mathcal{H}$-polyhedron under a linear map
Let $P = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$ be a (bounded) polyhedron for $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$, $n,m > 0$.
Moreover, let $M \colon \mathbb{R}^n \to \...
CommunityBot
- 1
2
votes
1
answer
143
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 ...
- 1,513
4
votes
2
answers
250
views
Build a topological polytope with a specified CW-structure
I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...
- 6,282
4
votes
3
answers
560
views
Can anyone suggest a text on polyhedral theory?
Can anyone suggest a text on polyhedral theory? Particularly on increasing the number of faces under projections. 0,1 polytopes
- 1,312
5
votes
2
answers
334
views
Put P inside Q! polygons/polyhedra
We have two Polygons/Polyhedra P and Q.
Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not?
First think about the case of which P ...
- 151k
2
votes
2
answers
219
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,\; \...
- 1,991
0
votes
1
answer
546
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 ...
- 720
1
vote
1
answer
624
views
a closed form lower bound solution for linear programming
Given a linear objective function and a system of linear constraints, is there any known closed form lower bounds for it?
to clearly express the problem assume that
$$
z(\mathbf{a,B,c})=\mathop {\inf} ...
- 1,991
2
votes
1
answer
191
views
Optimization problem whose cardinality never exceeds 7 for some reason
I am working on a problem in which I have a collection of $n$ points, $x_1,\dots,x_n$, in the plane, as well as a positive definite matrix $\Sigma$ and another point $\mu$ in the plane. I am trying ...
CommunityBot
- 1
10
votes
3
answers
2k
views
On maximal regular polyhedra inscribed in a regular polyhedron
Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...
CommunityBot
- 1
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 5,971
1
vote
0
answers
140
views
Reduce a Combinatorial problem
It is given n sets with k vectors. (k is element-wise positive or zero)
Choose one vector of each set so that the biggest element of the sum of the chosen vectors is minimal.
What i also know but is ...
- 11
0
votes
2
answers
708
views
Approximate solution to large mixed integer programming problem
What are the available approaches to find an approximate solution to a large mixed integer programming problem?
I ran my problem in the Gurobi MIP solver.
It can find a feasible solution in ...
- 15.8k
0
votes
1
answer
504
views
$\ell_o$ Minimization (Minimizing the support of a vector)
I have been looking into the problem
$\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...
- 11
1
vote
2
answers
1k
views
Nonstandard Hessian approximations in Gauss-Newton
The Gauss-Newton algorithm optimizes functions
$$
E(x) = \sum f(x)^2
$$
by approximating f as (locally) linear, in which case the Hessian of $E$ is approximated as
$$
H = 2 \sum {J_f}^T J_f
$$
Now ...
CommunityBot
- 1
0
votes
1
answer
452
views
Large scale least squares of non symmetric and non square problems
Given a system like $b=Ax$ with an non symmetric and non square $A$ I would like to solve it having many elements in $x$ (lets say $10^7$).
There is a large amount of algorithms for symmetric ...
- 3,934
13
votes
0
answers
406
views
Surface area of convex hull [duplicate]
Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?
- 6,210
2
votes
0
answers
91
views
Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings
In one paper from 1980 I found a note that there are no known algorithms for solving
homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring.
(The non-...
3
votes
2
answers
64
views
Enumerating Tri-vertex transitive polyhedra n > 3 faces
How many unique vertex transitive polyhedra exist where each vertex has 3 incident edges
for polyhedra with n (= # faces) > 3 ?
- 37.7k
1
vote
2
answers
172
views
Linear Programm with matrix [closed]
Is there a name for problems like this
min norm(Cx)
Ax = b
where C is a matrix and norm is the maximum norm.
This is kind of like a linear Programm. Could this be rewritten as linear programm? Or Any ...
- 980
0
votes
0
answers
917
views
Inverse problem with a rank-1 update
I hope you can help me out with this. I have to find the solution x to an inverse system
$$
x=A^{-1}b
$$
This inverse problem is basically a least square problem with a rank-1 update.
$$
x=[uv^{T}...
13
votes
2
answers
680
views
What was the Question that led Euler to his Investigations on Polyhedra?
The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and that story is a must in every introduction to graph theory.
...
- 151k
4
votes
1
answer
750
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 ...
- 28.6k
2
votes
4
answers
2k
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 lines:...
- 980