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.
276 questions
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Could you review my algorithm to assess its conceptual and practical validity for maintaining primal feasibility?
Problem Formulation
We aim to solve the following convex quadratic optimization problem:
Objective Function:
Minimize:
$$
f(x) = \frac{1}{2} x^\top Q x + q^\top x,
$$
where:
( Q ) is a positive ...
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hyperplane intersecting a polyhedral cone
$\newcommand{\R}{\mathbb R}$
Consider a family $V$ of non-null vectors in $\R^n$. Consider the (open) polyhedral cone
$$C(V) = \{x\in\R^n\;:\;v\cdot x> 0\,,\;\forall v\in V\}\,.$$
We will call $C(V)...
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Are there only finitely many faces that contain an extremal ray of a closed convex cone?
Let $C$ be a closed convex cone in $\mathbb{R}^n$, and let $l$ be an extremal ray of $C$. Are there only finitely many (exposed)faces containing $l$? (Actually, I care more about the Mori cone.)
...
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Let $C$ be a convex cone, $z \in X$, and $K = C + z$. Let $x \in X$ and $y_0 \in K$.
Let $C$ be a convex cone, $z \in X$, and $K = C + z$.
Let $x \in X$ and $y_0 \in K$.
Verify that $y_0 = P_K(x)$ if and only if
$$
\langle x - y_0, y \rangle \leq 0 \quad \text{for all} \quad y \in C
$...
user1392151
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Linearizing cone (constrained optimization)
I have a problem in understanding the linearizing cone. Given an admissible set $\Omega$, characterized by equality and inequality constraints $c_i(x)$, the linearizing cone at x is given by $L_\Omega(...
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Two definitions of normal cone of a closed convex cone
I am studying problems of the form
$$\text{minimize } \quad f(x) \quad \text{subject to} \quad G (x) \in \mathcal{K},\tag{1}$$
where $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ and $G:\mathbb{R}^{n} \...
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If $C $ is a closed and convex subset of a Hilbert space $H$ then $y= P_Cx$ is the unique point characterized by
Just to understand answer OP gave, can someone please provide a link (or even a book) to the theorem (and of course it's prove) OP have pointed out in the answer of this link: Show that $C$ is a ...
user1389572
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Let $C$ be a convex cone in this space, $z \in C$, and $K = C + z$.
Let $(X, \langle \cdot, \cdot \rangle)$ be a vector space over $\mathbb{R}$ with an inner product. Let $C$ be a convex cone in this space, $z \in C$, and $K = C + z$. If $x \in X$ and $y_0 \in K$, ...
user1348309
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Show that $C$ is a Chebyshev convex cone in $l_2(I)$.
Let $C = \{ x \in l_2(I) \mid x(i) \geq 0 \text{ for all } i \in I \}$. Show that $C$ is a Chebyshev convex cone in $l_2(I)$ and
$$
P_C(x) = x^+ := \max \{x, 0\}
$$
for every $x \in l_2(I)$.
My ...
user1351058
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When $n = 0$, for every $x \in C_2[a,b]$, $P_C(x) = \max \left\{ 0, \frac{1}{b-a} \int_a^b x(t) \, dt \right\}.$ What about any $n$?
Let $C = \{ p \in C_2[a,b] \mid p \in P_n, p \geq 0 \}$.
I first show that $C$ is a Chebyshev convex cone. Now I want to show: When $n = 0$, for every $x \in C_2[a,b]$,
$$
P_C(x) = \max \left\{ 0, \...
user1351058
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Curve starting at vertex of a cone with a given tangent direction stays inside the cone for a short period of time?
While working on some algebraic stuff, I was led to the following situtation:
Assume we have two points $x_0\neq y_0$ in $\mathbb{R}^n$. The vector starting at $x_0$ and ending in $y_0$ is denoted by $...
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Prove that if $K$ is a convex cone , then its closure $\overline{K}$ is also a convex cone.
Prove that if $K$ is a convex cone , then its closure $\overline{K}$ is also a convex cone.
Attempt: I start by noting that $K$ being a convex cone means that for any $x, y \in K$ and any non-negative ...
user1339458
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If a polyhedral cone has no extremal rays, then it is a linear space
Let $C \subset \mathbb{R}^n$. We say that $C$ is a polyhedral cone if there exist $v_1, \ldots, v_k \in \mathbb{R}^n$ such that:
$$ C = \{ \lambda_1 v_1 + \cdots + \lambda_k v_k : \lambda_1, \ldots, \...
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Normal Cones on a Circular Arc
Let $C = \{(x, y) \in \mathbb{R}^2: x^2 + y^2 \leq 1, y \geq 0, y \leq x\}.$ I wish to determine the normal cone $N_C(x_0, y_0)$, for any $(x_0, y_0)$ on the boundary of $C$.
There are six cases to ...
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Proofing that the interior of a convex cone united with the origin is a convex cone also (does the proof suffice?)
(One of my colleagues said that it's just clever renaming and I've tried to adapt someone else's proof; but I'm not sure if I'm showing what I should. And I'm not sure I know what I should actually ...
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Verifying that a second-order cone is proper.
I recently came across the mention that a second-order cone
$$\mathcal{K}= \{ x\in\mathbb{R}^n \big\vert x^TKx\leq0, c^Tx\geq 0 \}$$
with $K\in\mathbb{R}^{n\times n}$ and $c\in\mathbb{R}^n$ could be ...
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Minimization of smooth objective with conic constraint
I am interested in deriving first-order optimality conditions for
\begin{equation}
\min_{x\in\mathbb{R}^{n}}f(x)\\
\text{s.t. }x\in\mathcal{K}
\end{equation}
where $f$ is a smooth function and $\...
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Tangent cone to a convex set is a convex cone
Let $Y \subseteq \mathbb{R}^n$ be a convex set and let $\bar{y} \in Y$. The tangent cone to $Y$ at $\bar{y}$, denoted $T(Y, \bar{y})$, is the set of all limits of the form $h = \lim t_{l}(y_{l} - \bar{...
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Describe all cones in $\mathbb R^n$ that are simplicial and cosimplicial.
For any finite set of vectors $(v_1, \ldots, v_k \in \mathbb{R}^n)$, let's define:
$Cone(v_1,…,v_k)= \{λ_1v_1+…+λ_kv_k,∣,λ_1,…,λ_k \ge 0\}$ and $Poly(v_1,…,v_k)= \{ x \in \mathbb{R}^n ∣(v_i, x) \ge 0, ...
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The normal cone to a ball at a boundary point of the ball
I got stuck in a problem that in the end I needed to know what is the normal cone to a ball $B = \left\{ x \in \mathbb{R}^n : \lVert x \rVert \leq 1 \right\}$ (here $\lVert \cdot \rVert$ is an ...
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Prove if the polyhedron $A x \leq b$ is bounded, then the cone hull of all rows of $A$ is $\mathbb{R}^d$
Let $ P = \{x \in \mathbb{R}^d:Ax \leq b\}$ be a polyhedron, where $A = \begin{bmatrix} a_1^T\\a_2^T\\...\end{bmatrix}$, then the cone hull of $(a_1,a_2,\cdots)$ is $\mathbb{R}^d$. How to prove this ...
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Basis of the intersection of a cone and its dual.
Let $a_1,\dots, a_m$ be vectors in $\mathbb R^n$, let $A\in\mathbb R^{n\times m}$ whose columns are $a_i$s. The cone generated by $A$ (denoted $\operatorname{cone}A$) is $C=\{ Aw: 0\leq w \}$, its ...
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Questions related to Cones and Subspaces of Euclidean Space
Cone: A subset $ S \subseteq \mathbb{R}^n$ is a cone if $\alpha \geq 0 \implies \alpha S \subseteq S.$
Polar: A Polar $K^*$ of a cone $K$ is a closed convex cone such that
$$K^*=\{y \in \mathbb{R}^n \...
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If a convex cone K and a linear subspace L are such that $K\cap L=\{0\}$, is it true that the intersection of their polars has a non-zero vector?
A bit more details: I'm trying to prove that if the first condition holds, then the following statement is also true:
$$\exists z\neq 0 \text{ such that }z\in K°\cap L^\perp.$$
That second condition ...
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Can this constraint be cast as a second order cone constraint?
Can someone please explain if it possible to convert the following constraint into a second order cone programming formulation:
$xy \ge ay + b$
Here $x,y$ are non negative decision variables, $a,b$ ...
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A statement regarding tangent cones and polyhedricity and variational inequalities (need help with a proof)
I'm reading this thesis on page 182. We work in a Hilbert space $H$, and $y \in H$ solves the variational inequality:
$$y \in K : \langle Ay-f, v-y \rangle \geq 0 \; \forall v \in K$$
for some given ...
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Polar cone of a closed convex cone in $R^4$ defined by a convex inequality constraint
Let $c>1$ be a constant. Consider points in four dimension with coordinates $(x,y,z,p)\in R_{\ge 0} \times R_{\ge 0} \times R \times R_{\ge 0}$
and the cone
$$K = \{ (x,y,z,p)\in R_{\ge 0} \times ...
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Closedness of a cone
Let $N\in\mathbb{N}$ and $P$ be a symmetric pattern; i.e. $P$ is a subset of $\{1,\dots,N\}\times\{1,\dots,N\}$ such that $(i,i)\in P$, for all $i\in\{1,\dots,N\}$, and $(i,j)\in P$ if and only if $(j,...
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GM and AM convex cone
Let $\displaystyle K_\alpha = \{x \in \mathbb{R}^n| \sqrt[n]{x_1x_2...x_n} \ge \alpha\frac{x_1 + x_2 + ... + x_n}{n}, x_i \ge 0\}$ and $\alpha \in [0, 1]$. Show that $K_\alpha$ is a convex cone.
I ...
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Intuition for the Cone Lemma
The Cone Lemma.
If a system of homogenous linear equations with integer coefficients has a
positive real solution, then it also has a positive integer solution.
This is proved in Proofs from THE BOOK,...
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Tangent cone of boundary points of polyhedron
I am reading the paper Positive Invariance Condition for Continuous Dynamical Systems Based on Nagumo Theorem, and specifically I am concerning Theorem 3.1, where the tangent cone at boundary points ...
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Does the vector space of differences of quantile functions have a neat characterization?
Consider the convex cone of quantile functions of random variables on the real line (with finite second moment), that is
$$
C := \{ Q_{\mu}: \mu \in \mathcal P_{(2)}(\mathbb R) \},
$$
where $Q_{\mu}(p)...
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the case of separating N disjoint convex cones [closed]
I am learing about the separation theorem :
separation between a (closed) convex set $X$ in $\mathbb R^n$ and a vector outside $X$.
I know about it but what are some examples where separation is used ...
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How could I check to see if a point is inside a cone projected by another point?
I use CesiumJS, a javascript library, to render entities in a 3-dimensional space. I need to detect if one entity is "viewing" another entity. My initial thought is to use a cone to ...
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A question about ordered vector spaces
Relevant Definitions:
Definition:
Let $X$ be a set. A partial order $\leq$ on $X$ is a relation that is reflexive, anti-symmetric, and transitive.
Definition:
Let $\leq$ be partial order on a real ...
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basis of monoid of integral vectors
Suppose that $M\in\mathbb{Z}^{n\times k}$ is a matrix of rank $k<n$. How can I obtain a set of vectors $b_1,\ldots,b_k\in\mathbb{Z}^k$ (if exists) such that each row of $M$ is a non-negative ...
- 4,405
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The number of faces of a pointed convex polyhedral cone
Let C be an n-dimensional pointed convex polyhedral cone with a uniqe frame {a1.a2.......,ar}, where ais are extrem half-lines.
Is there a formula for the number of r-faces in C?
Let me state the ...
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Symmetric cones and symmetric spaces
I start by stating what I think I understood on symmetric cones (https://en.wikipedia.org/wiki/Symmetric_cone). Let $\mathcal{C}$ be a symmetric cone in a vector space $V$. There are Riemannian ...
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Equivalence of polar cone and polar set
The Wikipedia page on polar cones contains the following statement: "For a closed convex cone C in X, the polar cone is equivalent to the polar set for C".
I'm not sure why this is true, ...
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Bouligand and Clarke Tangent cones
Reffering to this question Clarke's tangent cone, Bouligand's tangent cone, and set regularity I'm asking myself if may exist a closed bounded set $S\in\mathbb{R}^2$ and a point $x\in\partial S$ such ...
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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 extreme ray 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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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 \...
- 285
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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
Conjecture. If the $V$ cone generated by the points in the matrix $A$ has an $H$ representation captured by matrix $B$, and $A = A_1 \otimes A_2$, $B = B_1 \otimes B_2$, where $B_1$ is the $H$ ...
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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 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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