All Questions
608 questions
1
vote
0
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86
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Infinite system of equations with finitely many constraints
During my research I have stumbled upon the following issue concerning infinite systems of linear equations. I do not have much practice in such settings, so I am asking you whether the following ...
15
votes
2
answers
3k
views
Given the vertices of a convex polytope, how can we construct its half-space representation?
Let us say I have the vertices of a polytope $V = \{v_1,\dots,v_k\} \subset \mathbb R^n$. Is it possible to write $V$ as intersection of half-spaces using the information from the vertices, i.e., can ...
6
votes
1
answer
2k
views
Approximation of convex hull in high dimension
What are efficient methods (polytime) to compute an approximation of the convex hull in high dimension (say, $30000$) for a given set of points?
Edit:
I am looking for an algorithm for getting the ...
5
votes
0
answers
162
views
Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
1
vote
0
answers
246
views
How to solve a large linear programming problem? [closed]
I have the linear programming problem in $\mathbf x \in\mathbb R^n$
$$\begin{array}{ll} \text{minimize} & \mathbf c^T\mathbf x\\ \text{subject to} & \mathbf A\mathbf x \leq \mathbf b\end{...
1
vote
1
answer
126
views
a linear programming problem
Recently I have a conjecture on decomposing a linear program into smaller ones. I have tested it in Mathmatica by a lot of examples. However, I cannot prove it. I will appreciate if someone can give ...
CommunityBot
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2
votes
1
answer
1k
views
What does the basis of the null space of the constraint matrix of a flow problem look like?
Consider a directed graph $G=(V,\mathbb{A})$ and a set of flow constraints of the following form:
$$ \sum_{(u,v)\in\mathbb{A}}x_{u,v} - \sum_{(v,u)\in \mathbb{A}}x_{v,u} = 0 \forall v \in V$$
...
- 6,051
5
votes
1
answer
394
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Explicit Formula of Delsarte's Linear Programming Upper Bound for $A_q(n,3)$
The problem of giving an explicit formula for $A_q(n,d)$ is sometimes referred to as "the main problem in coding theory." The value of $A_q(n,d)$ is given by the maximum number of codewords in a q-ary ...
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1
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0
answers
44
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In convex optimization we know that the optimum solution is on which hyper plane
We have a standard linear program, I mean a set of inequalities $c_i^Tx\leq b_i$ where $i\in \{1,\ldots ,k\}$ and we want to find $max\{c^Ty| y\in \{\cap \{x|c_i^Tx\leq b_i\}\}$. I put some condition ...
- 19
-1
votes
1
answer
137
views
Does a half plane contain intersection of some other half planes? [closed]
I'm doing research in Optimization and I have found this obstacle in the way.
If we have set of half planes like $c_ix\leq b_i$ where $i\in \{1,\ldots ,k\}$ there is an algorithm(it would be better ...
- 361
35
votes
4
answers
5k
views
Why are optimization problems often called "programs"?
Why are optimization problems often called programs?
linear programming
geometric programming
convex programming
Integer programming
...
2
votes
0
answers
283
views
Derivative with multiple summation operators
I have a defined utility function as Eq.(1), and I am seeking the minimized utility subjects to some constraints. The notation used is as following:
\linebreak
$V$ is the set of nodes, $v_i\in V$; $O$...
- 21
0
votes
1
answer
212
views
Is an exact violated inequality constraint met as equal constraint in optimal solution?
We have a solution which does not satisfied exactly one inequality constraint in linear program. The corresponding dual solution is also feasible. Is it correct this constraint is in equal form in the ...
- 54.2k
9
votes
1
answer
295
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Definition of packing property
Definition 1:
A clutter $C$ is said to have the packing property if $C$ and all of its minors satisfy the König property.
where,
vertex cover of $C$ is a set of vertices that have non-empty ...
- 30.5k
4
votes
0
answers
203
views
$\ell^1$-norm minimization duality
I am looking for an explicit description and discussion of the dual of the $\ell^1$-norm minimization problem $\lVert A x\rVert_1\to\min$, where $A$ is a matrix, and $x$ belongs to the $n$-simplex $\...
- 4,746
1
vote
0
answers
261
views
Prove that the following set of triples forms a convex polytope
Take $a,\,b,\,c,\,d \in \mathbb R_+$ such that $a+b+c+d=1$. Define:
\begin{equation}
x_1 = \min(a+b,\,c+d)\,,\qquad x_2 = \min(a+c,\,b+d)\,,\qquad x_3 = \min(a+d,\,b+c)\;.
\end{equation}
I would like ...
- 6,406
4
votes
1
answer
2k
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What does it mean polynomials share Newton polytope?
I have trouble understanding the connection between polynomials and Newton polytopes. I will try to make a short introduction to my problem and hope you will catch on. In the end I will ask questions.
...
- 984
9
votes
1
answer
2k
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Uniform sampling from general simplex with a twist
This is part of a question I had asked elsewhere, and then some of the links redirected me to CS stack exchange.
Given $0\leq a_1\leq\dots\leq a_D\leq1$ (all strictly positive), I want to draw points ...
CommunityBot
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3
votes
0
answers
3k
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0,1 solution to system of linear integer equations
I have the following problem:
$A x = b$
where $A, b$ - $m \times n$-matrix and $m$-vector of nonnegative integers (respectively).
$x \in \{0,1\}^n $ - vector of binary variables, which need to be ...
CommunityBot
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2
votes
0
answers
43
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Partitioning $n$-space based on linear combinations
I'm trying to figure out the approximate number of areas the positive $n$-space will be divided into if we partition it as follows: we have $k$ linear functions $F_1$, $F_2$, ..., $F_k$ on $n$ ...
- 21
2
votes
1
answer
2k
views
Finding a point farthest away from $k$ points in a polygon
There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized.
...
- 1,228
25
votes
2
answers
2k
views
An Interesting Optimization Problem
You are given n non-negative integers $a_1, a_2 ,, a_n$. In a single operation, you take any two integers out of these integers and replace them with a new integer having value equal to difference ...
- 2,821
4
votes
0
answers
539
views
Using Linear Programming as an iterative procedure
Suppose, we have a linear program and an optimal solution to it. Suppose now, we get a new constraint. We want to obtain an optimal solution to the given linear program extended by that new constraint....
- 391
4
votes
1
answer
2k
views
Under what conditions does an Integer Programming problem run in polynomial time?
Given $AX\leq B$ where $A\in\Bbb Z^{m\times n}$,$B\in\Bbb Z^m$ finding $X\in\Bbb Z^n$ where $m\geq n$ is the integer programming problem. If $A$ is totally unimodular then the problem is solvable in ...
CommunityBot
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1
vote
1
answer
185
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Complexity Measures for Mathematical Programming
Question:
Are there any complexity measures in use, that allow one to compare mathematical programming formulations of optimization problems on basis of the number of variables that must be subjected ...
CommunityBot
- 1
3
votes
1
answer
328
views
LP Constraints for Connected Subgraphs of Fixed Size
Question:
how can the connectedness-constraint for a subgraph, that is induced by a proper subset $W\subset V$ of the vertices of $G(V,E),\ |V|=n,\ |W|=m$, be formulated in a $LP$ or $ILP$?
...
- 2,966
1
vote
0
answers
42
views
Computation of sub-gradient for a concave envelope
Let $x_1<\cdots<x_n$ be $n$ points on real line and $g=(g_1,\cdots, g_n)\in\mathbb R^n$ be the scattered data. Let $u_g: [x_1,x_n]\to\mathbb R$ be the linear interpolation of $g_1,\cdots, g_n$, ...
- 345
1
vote
0
answers
81
views
Maximizing sum of homogeneous functions of order one over a polytope
Let $f_i: \mathbb{R}^n\rightarrow \mathbb{R}$ be
concave, increasing (i.e., if $x\geq y$ where the inequality is entry wise, we have $f_i(x)\geq f_i(y)$), and a
homogeneous function of order one for ...
- 393
13
votes
2
answers
664
views
Complexity of a weirdo two-dimensional sorting problem
Please forgive me if this is easy for some reason.
Suppose given $S$, a set of $n^2$ points in $\mathbb{R}^2$.
I want to choose a bijective map $f$ from $S$ to the set of lattice points in $\lbrace ...
- 1,083
3
votes
0
answers
105
views
Are there scenarios under which feasibility bilinear programming is easy?
Given $c\in\Bbb R^{n_1},d\in\Bbb R^{n_2}$, $E\in\Bbb R^{n_1\times n_2}$, $A\in\Bbb R^{m_1\times n_1}$, $B\in\Bbb R^{m_2\times n_2}$ $a\in\Bbb R^{m_1}$, $b\in\Bbb R^{m_2}$ and $t\in\Bbb R$ we know ...
- 13.9k
6
votes
1
answer
761
views
Checking if one polytope is contained in another
I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other.
At the moment I am ...
- 4,713
3
votes
3
answers
349
views
Sensitivity analysis in conic optimization
I have a conic optimization of the form:
$$\min_x \langle c, x \rangle,\ \text{s.t.}\ Ax = b,\ x \in K.$$
where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a ...
CommunityBot
- 1
10
votes
1
answer
682
views
Building a polyhedron from areas of its faces
Is there a known algorithm which, given a finite multiset (unordered list) of integers $A$, returns a yes/no answer for ...
- 151k
2
votes
0
answers
80
views
Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector
I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...
- 123
1
vote
0
answers
20
views
Calculating Cost-Optimal 1-Factors in Digraphs
I need to find a cost-optimal 1-factor in a positively weighted, directed, regular graph $G(V,A)$ without antiparallel arcs, i.e. given $$\text{deg}_{\text{in}}(u)=\text{deg}_{\text{in}}(v)=\text{deg}...
CommunityBot
- 1
2
votes
0
answers
105
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Optimization over a convex cone generated by a set is equal to optimization over the set
Within my research I found an important doubt and that prevents me from advancing, the context of my doubt is as follows:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \...
- 159
8
votes
2
answers
215
views
“Totally transitive” polytopes which are not regular
Is it possible to have an abstract polytope which is vertex-transitive, edge-transitive, face-transitive, etc. (individually transitive on faces of each particular dimension) and yet not flag-...
- 18.4k
1
vote
1
answer
111
views
Optimal "Generalization" of Polylines
This question is inspired by a lossy compression technique for polylines, namely to identify a subset of the points of polyline $\mathcal{P}$, whose removal yields a polyline $\mathcal{Q}$ within a ...
- 151k
10
votes
0
answers
722
views
Fractional Matching version of Hall's Marriage theorem
Let $G=(S,T,E)$ be a bipartite graph, $|S|=|T|$. Then the following are equivalent:
1) there exist a perfect matching in $G$;
2) there exist non-negative weights on edges such that the sum of ...
- 109k
2
votes
0
answers
344
views
Linear programming with an infinite matrix
I would like to solve the following infinite linear system subject to $x_i \ge 0$ that minimizes $x_3$.
The third column contains no additional nonzero values beyond what is shown. Though the first ...
2
votes
1
answer
148
views
Fast algorithm for large-scale, asymmetric transportation linear program
I have a large-ish instance of a transportation problem that is very asymmetric, say of dimensions $100\times10000$. I am currently solving it with a stock LP solver, but obviously something like the ...
CommunityBot
- 1
4
votes
0
answers
124
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Reference for this fact about perturbed polytopes?
Let $K \subset \mathbb{R}^n$ be a polytope (i.e., an intersection of finitely many halfspaces that has finite volume) and consider $F(K) := \int_K \|x\|^2\, {\rm d}x$, where $\|\cdot\|$ is the ...
- 1,016
1
vote
2
answers
83
views
Algorithm for a linear optimization problem
For the vectors $X=(x_1,\cdots, x_n),~ Y=(y_1,\cdots, y_n)$ and $\alpha=(\alpha_1,\cdots,\alpha_n),~ \beta=(\beta_1,\cdots, \beta_n)\in\mathbb R^n_+$ s.t. $\sum_{k=1}^n\alpha_k~~=~~\sum_{k=1}^n\beta_k~...
2
votes
0
answers
64
views
Finding orthogonal basis with constraint
Is there any fast algorithm that output an orthogonal basis $e_i,i\leq n$ of $R^n$
with $e_i\in V_i$? Where $V_i,i\leq n$ are given linear subspaces of $R^n$.
And is there any condition on $V_i,i\leq ...
- 19.6k
1
vote
0
answers
78
views
Family of functions which satisfies $f(\boldsymbol{x}) = 0$ if $\nabla f(\boldsymbol{x})=0$? [closed]
I have a Lagrangian of which I want to find the supremum in the primal variable $\boldsymbol{x}$:
$\mathscr{L}(\boldsymbol{x},\boldsymbol{\lambda})=f(\boldsymbol{x})^T\boldsymbol{a} + \boldsymbol{\...
1
vote
0
answers
47
views
Linear programs [closed]
Can the optimal value of the primal problem of a linear program ever be less then zero?
An example is: minimize $C=2x_1 +3x_2$ Subject to: $3x_1+4x_2 \leq 5$. Obviously, $x_1$ and $x_2$ are free ...
- 993
1
vote
0
answers
93
views
quick hull algorithm detail
When using quick hull algorithm to find the polytope for half space intersection, we are required to provide an interior point to the solver qhalf.
In other words, providing
$$Ax \le b$$
is not ...
- 867
3
votes
2
answers
1k
views
Equality constraints in mixed-integer optimization
Suppose I have a linear mixed-integer optimization problem of the form
$$MIP: min_{(x,y) \in \mathbb{R}^n \times \mathbb{Z}^m} c^\top x + d^\top y \hspace{0.2cm} \text{s.t.} \hspace{0.1cm} Ax+By \leq ...
- 342
1
vote
0
answers
43
views
a question about probabilities on spaces of digraphs
Let $G$ be a directed graph with fixed nodes $s$ and $t$. Assume that each edge $e$ in the graph comes with a number $n(e)\in[0,1]$.
We consider probability spaces $S$ whose points are directed ...
- 61