All Questions
608 questions
3
votes
1
answer
289
views
On decomposition of polytopes
Given $m$ number of convex polytopes each with $v$ vertices and described by $h$ hyperplane inequalities in $\mathbb R^t$ are there operations on these polytopes that combine then to give an $v^{\...
0
votes
0
answers
232
views
What do square roots as minimums have to do with Harmonic numbers?
In an earlier question where I conjectured (and GH from MO confirmed) that the von Mangoldt function is the limit at s=1 of a certain Dirichlet series:
$$\Lambda(m)=\lim_{s\to 1+}\zeta(s)\sum_{d\mid ...
- 1,183
5
votes
1
answer
424
views
What is the LP gap of vertex cover in planar graphs?
What is the LP gap of vertex cover in planar graphs?
The LP I refer to is min $\sum_{e \in E } c_e x_e \ \ $ subject to $ \ \ x_v + x_u \geq 1 \ \ \ \forall uv \in E $
$ c_e \geq 0 $ are ...
- 26
3
votes
0
answers
178
views
Uniqueness of projection under spectral norm
I am considering
$$
\min_{M\in \mathcal{M}} \|X - M\|:=x \neq 0,
$$
where $X$, $M$ are $m\times n$ matrices, $\|\cdot\|$ is spectral norm and $\mathcal{M}$ is a matrix subspace. I wonder to what ...
- 131
1
vote
0
answers
36
views
Linear programming with a convergent coefficient
The following linear programming problem
$x_n = \arg\min c_n'x \mbox{ subject to } Ax<b$
has a changing coefficient $c_n$. We have that $c_n\rightarrow c_*$. What happens to the solution $x_n$? ...
- 19
2
votes
1
answer
243
views
Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
- 1,826
2
votes
0
answers
29
views
When do projection maps of polyhedra factor?
Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...
- 1,826
3
votes
1
answer
152
views
A question about polytopes related to linear programming
Given linear functions $f_1({\bf x}),\dots,f_n({\bf x})$ on ${\bf R}^m$, let $K = \{(a_1,\dots,a_n) \in {\bf R}^n:$ the $n$ halfspaces $\{{\bf x}: f_i({\bf x}) \leq a_i\} $ have nonempty intersection$\...
- 24.2k
1
vote
0
answers
126
views
Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
- 1,826
1
vote
0
answers
129
views
Is there a method to cut a hypercube into disjoint cubes [closed]
Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$
so we can cut a hypercube into at least $n+1$ disjoint parts.
Is there a method how can one do that?
- 139
1
vote
0
answers
65
views
Covering a simplex efficiently by efficiently describable polytopes?
Take a standard simplex or cube in $\mathbb R^n$.
Is there a way to cover it with $O(poly(\log n))$ convex polytopes each describable by only $O(poly(\log n))$ half-plane inequalities?
If not what ...
- 1,826
1
vote
1
answer
125
views
Projections of particular simplex yielding boundary of a regular polygon?
What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
- 548
0
votes
1
answer
39
views
Gluing simplices through a common point/ realisation of a convex simplicial polytope
Given $m≥d+1$
a positive integer, is it always possible to find m d-dimensional simplices $\Delta_i=\mathrm{Conv}(M,V_{i,1},…,V_{i,d})$ such that
1) they all share the common vertex M
2) the ...
- 14k
1
vote
1
answer
126
views
Quantifier elimination and where is this quantified convex program in the polynomial hierarchy?
I have a quantified convex program of the form that I need to solve
$$\exists(x_{1,1},\dots,x_{1,n})\in\mathbb R^n\quad\forall(x_{2,1},\dots,x_{2,n})\in\mathbb R^n$$
$$\vdots$$
$$\exists(x_{2t-1,1},\...
- 14k
7
votes
2
answers
909
views
Formula for volume of a convex polytope
So I've been searching around the internet for some answers to this, but I currently have a set of linear constraints: $Ax = b, Cx \le d$ for matrices $A \in \mathbb{R}^{n \times m}$, $b\in \mathbb{R}^...
- 984
1
vote
0
answers
24
views
Simple monotonicity property for coordinate descent and linear objective functions
Let $S \subset \mathbb{R}^n$ satisfy $0\leq x_1\leq\dots\leq x_n$ for all $\mathbf{x}\in S$, among other (possibly nonconvex) constraints, and suppose in addition that $\sum_{i=1}^n x_i \geq 1$ for ...
- 4,049
0
votes
0
answers
89
views
Why there isn't lexicographically smallest solution to a bounded linear program?
I am learning computational geometry when I run into this confusion. "A bounded 2D linear program may not have a lexicographically smallest solution", as the book says. I wonder why? I think we can ...
- 1
1
vote
0
answers
37
views
Fast certficate of negativity for objective value of mixed-integer linear program
Let $c \in \mathbb R^n$, $A \in \mathbb R^{m \times n}$, $b \in \mathbb R^m$, and $I \subseteq \{1,2,\ldots,n\}$. Consider the Mixed integer linear program (MILP)
$$
\begin{split}
f^* = &\max \; ...
- 6,853
2
votes
1
answer
270
views
What optimization problems have solutions with few nonzeros?
Consider the following optimization problem, with $n$ variables and $m$ linear constraints:
\begin{align}
\text{maximize} && c_1 x_1 + \cdots + c_n x_n &
\\
\text{subject to} && a_{...
- 14k
13
votes
3
answers
835
views
Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any ...
- 109k
7
votes
1
answer
374
views
Sampling uniformly from the vertices of a polytope
I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...
- 15.4k
1
vote
1
answer
330
views
How good is the LP relaxation?
Consider the optimization problem
\begin{align}
\max_{x\in\mathbb{R}^n}~c^Tx~, \text{ s.t. } Ax=b,~x_i\in\{0,1\}~\forall i
\end{align}where $c,b\in\mathbb{R}_{+}^n$ and $A\in\mathbb{R}_{+}^{n\times n}$...
0
votes
2
answers
102
views
Does Max Flow produce uniform results? [closed]
I am interested in using Max Flow algorithm. I want to simulate transfer of quantity. Anyway, I am unsure of some thing.
Does Max Flow algorithm produce uniformly distributed max flow?
I have ...
- 400
1
vote
3
answers
138
views
Literature request: Function that depends on a linear optimization problem [closed]
my question is about functions of the following form:
$$ f(t) = \max_{\mathbf{x}}~ \mathbf{c^T x} ~ {\rm s.t. \mathbf{Ax} +t \cdot \mathbf{a} \leq \mathbf{b}}, $$
where $\mathbf{x},\mathbf{b}, $ ...
- 1,991
3
votes
1
answer
553
views
Calculate the discrete set of points B which are in the convex hull of the set of points A
This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...
- 4,713
6
votes
0
answers
92
views
Tiling with Horn's polytopes
Let $n\ge2$ be an integer. Consider the hyperplane $H_n$ of ${\mathbb R}^n$ defined by the equation $x_1+\cdots+x_n=0$ and then the sector $P_n\subset H_n$ defined by the inequalities $x_1\le\cdots\le ...
- 52.3k
7
votes
1
answer
386
views
Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?
Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:
The all-one vector $j$ is contained in the conic hull of $col(A)$.
...
CommunityBot
- 1
0
votes
1
answer
145
views
How to solve this optimization problem efficiently? [closed]
Let, $D\in\mathbb{C}^{1\times M}$ is a row vector with $M$ elements
$V\in\mathbb{C}^{3^M\times M}$ is a given matrix
$T$ is a scalar (real and $>1$)
$\textbf{The problem at hand is as follows:}$
...
- 111
1
vote
1
answer
123
views
LASSO problem but with a maximization instead of minimization
I have the following optimization problem (like the LASSO problem but with maximization instead of minimization):
$\mathbf{maximize}_{\boldsymbol{\alpha}} \|\mathbf{x} -\mathbf{A}\boldsymbol{\alpha}\|...
- 20.2k
-1
votes
1
answer
88
views
sparse data fitting problem [closed]
I am a new learner of optimization, and I am confused by the question below, (how to change a 0-norm constrain into binary and linear constrain ?)
Given a sparse data fitting problem:
$ minimize \...
- 1,517
1
vote
0
answers
74
views
How to minimize n-polytope's bounding box with linear transformation?
I am working on an exact algorithm for integer linear programming for my master's thesis:
$Ax\leq b, x \in \mathbb{Z}^n$
$cx\rightarrow min$
For my idea to work out, I need a guarantee that n-...
- 11
2
votes
0
answers
41
views
Efficient $H$ representation of matrices with distinct cyclic shift permuted entries
Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where
$v_1,\dots,v_n$ when written as columns of ...
- 13.9k
17
votes
3
answers
6k
views
The cone of positive semidefinite matrices is self-dual? (reference needed)
I'm seeking a reference for the following fact.
The cone of positive semidefinite matrices is self-dual (a.k.a. self-polar).
This result is relatively easy to prove, has been known for a long time,...
- 13.6k
1
vote
0
answers
85
views
"Barrier functions" in function spaces [closed]
In general the idea of a "barrier function" (as in the context of "interior point methods") can possibly be thought of as defining a function corresponding to an open subset of $\mathbb{R}^n$ such ...
- 2,246
0
votes
1
answer
490
views
Relax a rectangular linear assignment problem
I wonder if there is any literature on the following problem
$$\begin{array}{ll} \underset{X \in \mathbb R^{m\times n}}{\text{minimize}} & \displaystyle\sum_{i,j} C_{i,j} X_{i,j}\\ \text{subject ...
2
votes
1
answer
502
views
Feasibility Mixed integer Linear programming with quadratic constraints?
Consider the mixed integer program
$$Ax\leq b$$
$$By\leq c$$
$$\begin{bmatrix}x&y\end{bmatrix}C\begin{bmatrix}x\\y\end{bmatrix}+D\begin{bmatrix}x\\y\end{bmatrix}\leq d$$ where $x$ are integer ...
- 1,517
3
votes
1
answer
266
views
Strong polynomial algorithm for linear programming
What is the current state of finding a strong polynomial algorithm for linear programming? Is there any reference?
-1
votes
2
answers
114
views
On OR condition in Linear Programming with exponentially many constraints [closed]
Suppose we have two linear programs $Ax\leq b$ and $Bx\leq c$ is there a way to combine them into one program of possibly a larger dimension $Cy\leq d$ such that projection of vectors $y$ into a ...
- 128k
1
vote
0
answers
47
views
What separates a cyclic polytope from a projective polytope?
I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries.
The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
- 151
0
votes
0
answers
99
views
Is this Graph Iteration Already Known?
When attempting to set up an ILP formulation for a weight-minimal cubic spanning tree (i.e. one with vertex degrees either 1 or 3) I needed connectivity constraint, but misremembered the contents of ...
CommunityBot
- 1
19
votes
4
answers
1k
views
Applications of linear programming duality in combinatorics
So, I know that one can apply the strong LP duality theorem to specific instances of maximum flow problems to recover some nontrivial theorems in combinatorics, such as Hall's theorem, Koenig's ...
- 761
0
votes
0
answers
46
views
linear inequalities and reference request
I have proved and am using the following simple lemma in my current research problem:
Let $\{a_1,...,a_m\}$ and $\{b_1,b_2,...,b_n\}$ be set of positive numbers such that $\sum_{i=1}^m a_i < \sum_{...
- 1,269
1
vote
1
answer
170
views
Optimization of a continuous function
This is more like an optimization problem but any solution is appreciated.
I have a data set with input specifying power(demand) to be generated for a particular time period(TP).
Input:
Time --- ...
- 61.9k
3
votes
2
answers
437
views
convex polytope integer points
is there a simple proof for the following lemma:
An unbounded convex polytope (defined by linear constraints) has either zero integer points or infinite many integer points.
- 1,991
1
vote
0
answers
66
views
On number of solutions by simplex and number of solutions in total in a linear optimization problem?
This is more of a clarification query.
Mizuno http://www2.ims.nus.edu.sg/Programs/012opti/files/talk_mizuno1.pdf says if we give a linear optimization problem
$$\max c'x$$
$$Ax\leq b$$
where $A\in\...
CommunityBot
- 1
2
votes
0
answers
76
views
Possible ordering of coordinates in a linear subspace [closed]
This question was asked on Mathematics Stack Exchange with no answers.
Let $X$ be a linear subspace of $\mathbb{R}^n$. For how many permutations $p$ on $1,...,n$ does there exists $x$ in $X$ with $...
- 2,821
2
votes
1
answer
91
views
Linear program with one quadratic condition convex in domain of interest polynomial time solvable?
$c\leq xy$ is not a convex condition.
However we know $c\leq xy$ is convex in domain $x,y>0$ in $\mathbb R^2$ for any fixed $c\in\mathbb R$.
Is $c\leq x_1y_1+\dots+x_ny_n$ with $0\leq x_1,\dots,...
- 54.2k
1
vote
0
answers
62
views
LP Constraints for Bridgeless Cactus Graphs
When trying to determine the optimal bridgeless spanning cactus graph of a weighted, symmetric graph, I got stuck.
What I do not know how to capture, is
the variable number and sizes of the cycles
...
CommunityBot
- 1
0
votes
0
answers
369
views
Finding a point in the relative interior of the convex hull of a set of integer-valued vectors
Let $X \subset \mathbb{Z}^n$ be the set of integer-valued vectors satisfying a system of linear constraints. We can suppose that $X$ is the set of integral points in a given polyhydral set $Y \subset \...
- 136
1
vote
0
answers
60
views
On the defect of a flow network
This problem in graph theory was actually motivated by some problems in Theory of Fractals.
To formulate the problem I need to recall some definitions related to flow network.
A flow network is a ...
CommunityBot
- 1