Questions tagged [convex-cone]
This tag should be used with convex cones defined as usual within the context of Euclidean spaces or topological vector spaces, and their applications within geometry and topology, optimization, combinatorics and any other related area of mathematics using the classic definition of convex cone.
247
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If ${\rm ker}T\cap C_\infty=\{0\}$, prove that $T(C_\infty)=T(C)_\infty$
Let $\mathbb{E_1,E_2}$ be two finite dimension Euclidean spaces. $T\in\mathcal{L}(\mathbb{E_1,E_2})$ is a linear map. Given a closed set $C\subset\mathbb{E_1}$. Let $C_\infty$ denote the asymptotic ...
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Does every pointed polyhedral cone in $\mathbb{R}^n$ generated by its $n+1$ extreme rays contain two vectors whose sum is in the interior?
This might be a stupid question, but I have been thinking and googling for some time now and I still cannot seem to find a response.
I am interested in how many extreme rays a finitely generated (...
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"Slicing off" the cone by a linear inequality leaves only vectors generated by the "remaining" cone generators (Lemma 1 from Gleeson & Ryan 1990)
Let $ Ax \le b $ be a infeasible set of linear inequalities. Gleeson & Ryan (1990) derive a polyhedron with vertices being in a one-to-one mapping with the set of irreducible inconsistent subsets. ...
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Show that $S^{\star} + C$ is a finitely generated cone whose generators span $\mathbb{R}^m$
Let $C = \{A \lambda \, | \, \lambda \geq 0\}$ be a finitely generated cone for some $A \in \mathbb{R}^{m \times n}$. Suppose the columns of $A$ do not span $\mathbb{R}^m$. Let $S$ denote the column ...
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Connected components of complement of double napped cone
How may connected components does the complement of the conic surface (extending on both sides) have in the three dimensional space?
I think the connected components is two, but am not convinced about ...
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What is the isotropic cone of det in $M_2(K)$?
Consider the application $\text{det}: M_2(K) \to K$
What is the isotropic cone of det in $M_2(K)$?
Consider the vector subspace $D$ of the diagonal matrices of
$M_2(K)$. Is it true that $M_2(K)=D \...
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What does the term 'convex' mean, when we say 'convex cone'.
According to Wikipedia:
A convex cone is a a subset of a vector space that is closed under linear combinations with positive coefficients.
I wonder if the term 'convex' has a special meaning or ...
- 103
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Weyl-Minkowski V-H representation of a cone, with kronecker products, reference request
Do you know if someone has established this conjecture to be true: "if the V cone generated by the points in the matrix A has an H representation captured by matrix B, and $A=A_1\otimes A_2$, $B= ...
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Solid Angles Beyond Dimension Three
There is a theorem by Ribando (Measuring Solid Angles Beyond Dimension Three, Discrete Comput Geom 36:479–487 (2006)) https://link.springer.com/content/pdf/10.1007/s00454-006-1253-4.pdf?pdf=button:
...
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convex polyhedral cone interpretation
According to my text, a convex polyhedral cone in $N_{\mathbb{R}}$, is a set of the form $\sigma = \big \lbrace \: a_1n_1+\cdots +a_rn_r \mid a_i\in \mathbb{R}_{\geq 0} \: \big \rbrace$ generated by a ...
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Convex optimization problem not expressible as a conic program
I've been reading Boyd & Vandenberghe and it says that conic programming is a subclass of convex optimization. I haven't been able to find an example of a convex optimization problem that I cannot ...
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Open discrete cones covering cover sphere?
This seems to be true, but I can't prove it. Define a cone to be $K\subset \mathbb{R}^n$ such that if $x\in K$ iff $\lambda x\in K$ for all $\lambda>0$. A discrete open cone $K'$ is the ...
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Asymptotic Cone of Product Sets [closed]
We denote the asymptotic cone of a set $S$ by $A(S)$: given $S\subset \mathbb{R}^n$ and a natural number $k$, let $\Gamma(S^k)$ be the smallest closed cone at $0$ containing $S^k=\{x\in S: ||x||\ge k\}...
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Subset relation between two polyhedral cones
Suppose we are given two vectors $b_1, b_2 \in \mathbb{R}^n$ and $m \leq n$ vectors $a_1, \dotsc, a_m \in\mathbb{R}^n$ with
$ a_1, \dotsc, a_m \notin \text{cone}\, (b_1,b_2) $
$ \text{cone}(a_1, \...
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Every closed convex cone is contained in a closed convex cone generated by a basis
Let $C$ be a closed convex cone in a Banach space $X$ such that $C\cap (-C)=\{0\}$. Is it true that there exists a Hamel basis $\mathscr{B}$ of $X$ such that
$$
C\subseteq \mathrm{cone}(\mathscr{B})\,\...
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Fixed point of the isotropy group in a symmetric one
Let $V$ be Eucldiean vector space of finite dimension. Let $C$ a symetric cone, that is to say:
open
convex
self-dual
homogeneous: $\forall x,y\in C, \exists M\in G, Mx=y$ where $G=\lbrace M| M\in GL(...
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Intuitively, what is the matrix core, and how has it proven useful?
I have begun reading some of the (few) references I can find on the concept of a "core" of a non-negative matrix, e.g. here.
In brief, my question is as stated in the title. I would like to ...
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Preparing for oral exam in convex analysis help to understand question
I am preparing for my oral exam in convex analysis and I just wonder what my teacher exactly means with the following question?
"Elaborate on the precise meaning of separation between a (closed) ...
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why the inner product with an element that does not belong to a convex cone is negative
Let us consider a set with an infinite number of vectors $\{v_k \mid k \in \mathbb{N} \}$ with $n$ cordinates, and if we consider the conned convex set containing all theses vectors, denoted by $K$.
...
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Are the examples to polar cones correct?
I have the following definition of a polar cone:
$$K(C) = \{ y | y^t x \leq 0 \forall x\in C\}$$
I am experimenting with some examples and wondering if I am right:
C = {2,5} means that K(C) = $\{y \...
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Is the asymptotic cone $\{0\}$?
I am studying convex analysis at home for self-learning. I focus especially on the structure of closed convex sets. I came across a weird definition of an asymptotic cone in a French (Sorbonne) ...
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28
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Decomposition of closed convex sets which don't contain any affine lines in a finite demension.
I am studying convex analysis especially the structure of convex closed sets in a finite demension. I am trying to digest this is theorem below, and yes I understand the demonstration and all but I ...
- 125
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37
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Is the closure of a positive convex cone positive?
I am studying convex analysis especially the structure of closed convex sets. I need a clarification on something that sounds quite easy but I can't put my fingers on it.
Let $E$ be a normed VS of a ...
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What is an Augmented Vector Space?
I am studying a little bit of Convex Analysis, especially the structure of the closed convex sets (in French). I came across a term that kind of confused me which is "espace vectoriel augmenté&...
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What is the correct representation of the generalized gamma function?
The NIST Digital Library of Mathematical Functions defines the multivariate gamma function as
$$
\Gamma_{m}\left(s_{1},\dots,s_{m}\right)=\int_{\boldsymbol{\Omega}}\mathrm{etr%
}\left(-\mathbf{X}\...
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A closed, convex set has a recession direction if and only if it is unbounded.
Let $C\subseteq \Bbb R^n$ be a nonempty closed, convex set. Prove that $C$ has a recession direction if and only if $C$ is unbounded.
If $C$ has a recession direction then trivially it is unbounded.
...
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closed, convex cone C $\in \mathbb{R}^n$ whose linear hull is the entire $\mathbb{R}^2$
I am trying to construct the following example:
let $A \in \mathbb{R}^{2x2}$ be symmetric matrix and C $\in \mathbb{R^n}$ a closed, convex cone whose linear hull is the entire $\mathbb{R}^2$ and
$x^...
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$K$ is polyhedral cone iff its dual $K^\ast$ is polyhedral
Show that $K$ is polyhedral cone in $\mathbb{R}^n$ iff its dual $K^\ast$ is polyhedral cone, where $K^\ast = \{x \in \mathbb{R}^n | $<x, y> $\geq$ 0$ \ \ \forall y \in K \}$.
I referred to a ...
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Does the closed convex hull of a compact set in the interior of a convex cone is still contained in the interior of the cone?
Let $C$ be a convex cone in a Banach space $X$ with nonempty interior. The set $A\subset {\rm Int}C$ is a compact subset, where ${\rm Int}C$ means the interior of $C$. Denote the closed convex hull of ...
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Is conic programs with exponential cone solvable in polynomial time?
I'm trying to find out some theoretical guarantees of time complexity for my problems.
My problem is to minimise a log-sum-exp function.
I found that the minimisation of log-sum-exp function can be ...
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Describe the cone generated by vectors $(1,0,0,1), (0,2,3,4), (0,0,3,1), (1,3,2,4), (2,4,6,4)$ by its irredundant linear inequalities.
I'm really not sure on how to tackle this. I have tried considering all matrices comprised of three of the five vectors -- as rows -- and row reducing them, but that hasn't really lead anywhere (so ...
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Dual of a norm Cone.
Problem
[Example 2.25 Taken from Convex Optimization By Stephen Boyd, Lieven Vandenberghe]
In this example, it proves that the dual of a norm cone is the cone of ...
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What are the extreme rays of this cone?
I have given the following cone:
$P=\lbrace x | Ax \geq 0\rbrace$ where $A=\begin{pmatrix}
-1& 1 & 0 & 0 & 0&0\\
0& -1 & 1 & 0 & -1 & 0\\
0 & 0 & -1 &...
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Prove that the angle between an outer support vector and a unit vector of a cone is minimized on its extreme ray
Let $C$ be a closed convex cone and $\nu$ be an outer support vector, i.e. $\langle\nu, x\rangle\le 0$ for all $x\in C$. Assume $\nu$ is maximized uniquely at some point $y\in C$. I am trying to show ...
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Is half-space a convex cone?
I understand from Show that halfspace is not affine. that halfspace is not an affine set.
Would like to ask is half-space a convex cone? How do you prove it mathematically? I am unsure if the ...
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Pappus centroid theorem and Hypercones.
The volume of a straight cone in $\mathbb R^3$ is usually find adding the circular sections orthogonal to the height. If the base has radius $R$ and the height is $h$ we have:
$$
V_{C3}=\int_0^h \pi r^...
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A matrix $A$ is $K$-irreducible if and only if no eigenvector lies on $\partial K$.
I'm studying the properties of nonnegative matrices, and I encountered a theorem for which I can not understand its proof. The theorem can be found in "Nonnegative matrices in the Mathematical ...
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Is the convex cone of sums of self tensor products of vectors in a given vector space full-dimensional in the space of symmetric matrices?
Let $V$ be a finite-dimensional real vector space. Let $S_n(V)$ denote the real vector space of all symmetric $n\times n$ matrices with entries in $V$.
Consider the convex cone $C\subset S_n(V) $ ...
user630227
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Minimal spanning set ("conical basis") for 2x2 Hermitian PSD (positive semi-definite) cone?
A linear combination $ax + by$ is called conic(al) if $a, b \ge 0$ (cf. section 2.1.5 of Boyd, Vandenberghe). I.e. conic(al) combinations are just linear combinations where the coefficients are ...
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Cone in spherical coordinates after rotation
A volume of a $3$-d cone with the apex at the origin of a Cartesian coordinate system $\mathbf{x} = [x, y, z]^T$ and with its axis of symmetry (given by a unit vector $\mathbf{e}_c$) aligned with the $...
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Correctness of QCQP formulation of an SDP problem
Consider the following semidefinite program:
\begin{split}
\max_{X,Y} \; & X_{12}^\top B + \mathrm{Tr}[(X_{11} + Y_{11})A]\\
\mbox{s.t.}\; & \mathrm{Tr}[X_{11} + Y_{11} - 2X_{12} E^\top - 2Y_{...
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Open cones in $\mathbb{R}^n$
Is any proper open cone in $\mathbb{R}^n$ of the form $$C(x, e, \alpha) = \left\{ y\in\mathbb{R}^n : \arccos \frac{e\cdot(y-x)}{|y-x|} < \alpha\right\}$$ for unit vector $e$, angle $0<\alpha<\...
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Tangent cone: don't understand
A cone is a set that is closed under multiplication by positive scalars. The tangent
cone of $f$ at $x$ is defined as the set of descent directions of $f$
at $x$
$$\mathcal{T}_f(\boldsymbol{x})=\{\...
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SemiDefinite representable set that is not Conic Quadratic representable.
Let $X \subset \mathbb{R}^n$.
We say that $X$ is Conic Quadratic representable (CQr for short) if
$$X=\left\{x \in \mathbb{R}^n: \exists u, A_j\begin{pmatrix}x \\ u\end{pmatrix}-b_j \in L^{m_j}, \ j=1,...
- 1,055
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Locally polyhedral and locally finitely generated cones
Let $V$ be a finite dimensional real vector space. The following definitions are taken from a paper.
A closed convex cone $K \subset V$ is locally polyhedral at $v \in \partial K$ if there exists a ...
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Example of the equivalence of two desrciption of polyhedra
This is an extract from a linear optimization book:
However, I do not understand why both of these conditions describe the same polyhedra? The linear equation results in a set
$$\left\{\begin{bmatrix}...
- 870
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78
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Hyperbolic Coxeter groups, Humphreys' book
Let $(W,S)$ be an irreducible Coxeter system with non-degenerate bilinear form $B$ on the Euclidean vector space $V$. The simple root attached to $s\in S$ is denoted by $\alpha_s\in V$. Let $\{\...
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Why is the support function of a convex cone the indicator function of its polar cone?
Let $K ⊆ \mathbb{R}^{d}$ be a non-empty, closed, convex cone. Consider the support function $\sigma_K(x):= \sup_{y \in K} \langle x, y \rangle$. This function describes the (signed) distances of ...
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Prove that normal cone of a closed convex set is different from the set has only element 0
Problem: Suppose that $E$ is Euclidean space. Given a closed convex set $C \subset E,$ the normal
cone $N_C$ to $C$ at $x \in C$ is defined by $$N_C(x) := \left\{v \in
E^{*}:\forall y \in C, \langle ...
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0
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Cone described by quadratic form
Let $P,Q$ be two real symmetric matrices of dimension $n \times n$. Define two sets $A=\{x \in \mathbb{R}^n: x^T P x \le 0 \}$ and $B = \{ x \in \mathbb{R}^n: x^T Q x \le 0 \}$. If $A \subset B$, ...
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