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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.

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20 views

what is the relation between projection of a polytope and this polytope?

suppose we have a polytope $P$ in $R^{4}$ and $-1\leq x_{3}\leq 4$ and $0\leq x_{4}\leq 6$, if I replace the upper and lower bound of $x_{3}$ and $x_{4}$ (it depends on the sign of variables $x_{4}$ ...
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1answer
11 views

What is the characteristic cone of a system with infinite linear inequalities?

Suppose that I have a system of linear equations for a variable $x\in\mathbb{R}^n$: $$a_i^Tx\ge b_i,\quad i\in I,\quad (1)$$ where $a_i\in\mathbb{R}^n$ and $b_i\in\mathbb{R}$ for all $i\in I$, and $...
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36 views

Minimal nonnegative combinations of vectors

Fact: Start with $m$ points $u_1,...,u_m\in \mathbb R^n$. Now pick any $u$ in the cone of $u_1,...,u_m$; that is, $$ u = \sum_{i=1}^m \alpha_i u_i \text{ for some } \alpha_i \geq 0, \; i = 1,...,m....
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1answer
34 views

How to prove this inequality in convex optimization?

Suppose that $C$ is a proper convex cone, and $C^0$ is the dual cone. Then we have the following inequality $$\sup _{\|z\|_{2}=1, z \in C} h^{T} z \leq \inf _{z \in C^{\circ}}\|h-z\|_{2}.$$ I tried to ...
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10 views

Is a unimodular cone generated by unimodular vectors?

Let $A$ be an $m\times n $ matrix and $r(A)=n$. $A$ is unimodular if the determinant of every basis of $A$ is $\pm 1$. A vector $x \in \mathbb R^n$ is unimodular if each entry is either $0$ or $\...
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17 views

Propriety of cone function

Ok, I've got this exercise: Let $K$ be a convex body in $\mathbb{R}^n$ and let $\phi: K \to \mathbb{R}$ be a cone function. Set $M := \text{max}_{x\in K}\phi (x)$. Prove that, for $0<t<M$, it ...
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28 views

The measures which are non-negative on all convex non-negative functions

Let $X$ be a subset of $\mathbb{R}^n_{++}$ (vectors with non-negative coordiantes), and let $M$ be the set of regular Borel measures with finite variation on $X$ and finite first moment: $$M:=\{\mu;\ |...
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1answer
33 views

How is a nonnegative orthant a cone?

My textbook says the following: The nonnegative orthant is the set of points with nonnegative components, i.e., $$\mathbb{R}_+^n = \{ x \in \mathbb{R}^n \mid x_i \ge 0, i = 1, \dots, n\} = \{ ...
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21 views

Cone in higher dimensions

Let $\Omega$ be a bounded open convex set of $R^{n}$, $u \in C^{0}(\bar{\Omega})$ a convex function, and $v$ a convex function whose graph is the upside down cone with vertex $(x_{0},u(x_{0}))$ and ...
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16 views

Finitely generated cone definition clarification

The definition for a finetley generated cone says that a set $C$ is a finetly generated cone if there exists {$x^{(1)},x^{(2)},...,x^{(k)}$} such that $C$ = {$\sum\lambda_ix^{(i)}, \lambda \geq 0$}. ...
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47 views

Deriving the geodesic equations on a cone. Are these equations correct?

So I'd like to derive the geodesic equations of a cone which I call $\mathcal{C}$. I believe I've done this correctly but would like a second opinion. $\mathcal{C}$ can be described by taking the line ...
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1answer
29 views

Missing constraint

While reading this paper, I stumbled over the statement (1, 8) about linear independence in trees. Just to make sure, that I understand it correctly: He means conical independence and forces $a_{i, k}...
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1answer
33 views

Cone differentiation question help!

A right circular cone has a constant volume. The height h and the base radius r can both vary. Find the rate at which h is changing with respect to r at the instant when r and h are equal. For this ...
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1answer
192 views

Prove the condition $a+b+c=0$ for three mutually perpendicular generators of a cone.

Question: Prove that for the cone $$ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0$$ having three mutually perpendicular generators, $a+b+c=0$. Proceed: If the cone has three mutually perpendicular generators, then ...
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58 views

Perspective of log-sum-exp as exponential cone

According to the Mosek documentation, Geometric Programming constraints of form log-sum-exp can be formulated with exponential cones. If the constraint is of form $$t \geq \log(\exp(x_1)+\ldots + \...
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80 views

Decomposing a polyhedron in $\mathbb{R}^3$ into a lineality space, cone, and polytope.

Consider the set in $\mathbb{R}^3$ given by $$\{(x,y,z) : x+y+z\ge 3, x \ge 0\}$$ I can picture this set; it is simply the intersection of two half-spaces. A theorem states that every polyhedron can ...
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13 views

Supporting hyperplanes for closed convex cones

Well I've been working on it for 3 days without results, it is not a question from any textbook and seems like it is not an important issue. But I really want to know the answers. The definition of ...
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1answer
45 views

Inner product identity for cones

Let $\emptyset \neq C\subseteq \mathbb R^n$ be a convex, open cone with the property that $\operatorname{int }C^* \neq \emptyset$, where $C^*$ denotes the dual cone defined by $$C^* = \{x \in \mathbb ...
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47 views

generators of a cone

Let $P=\{x\in R^d: Ax\leq 0\}$ be a polyhedral cone (possibly non-pointed). Let $I$ be the index set of inequalities in $P$. Let $I^{=} (x)$ be the index set in $I$ for which the corresponding ...
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1answer
55 views

Prove that a cone C is full-dimensional if and only if its dual cone $C^*$ is pointed.

A cone with apex $0$ is said to be pointed if it does not contain any non-trivial subspace. Let C be a closed convex cone with apex $0$. Show that $C$ is full-dimensional if and only if its dual cone $...
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2answers
145 views

Slant cone volume problem

I was given a a problem to solve, I thought I solved it but my answers don't look like the ones provided. The Problem A cone with radius of base r and height <...
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19 views

Notation for the cone corresponding to a set

Is there a standard notation for the operator which returns the conic hull w.r.t to the first orthant of a set, i.e. the operator $V\mapsto\{x\in \mathbb{R}^s_+ : x=y-z,\ y\in\operatorname{co}(V),\ z\...
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41 views

Why cones are represented by matrices

I see there are multiple definitions of cones: 1) Cone $K$ is defines as a set of vertices $[x_1, x_2, x_3, ...]$ with $[0]$ as the base (starting point) 2) Cone $K$ is defined as intersection of ...
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44 views

Intersection of two conic hulls is equal to the conic hull of the intersection

since the above conjecture is wrong in general, I would like to know (and maybe prove) that the following Statement holds: Let $A,B$ be two closed, convex sets in $\mathbb{R}^n$ such that $A\cap B=\...
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1answer
28 views

Why minimum of a differentiable function over a cone is in the interior of a cone?

Let $K \subseteq \mathbb{R}^n$ be a convex cone and $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be a differentiable function. Now consider the following optimization problem $$ \min_{x \in K} f(x) $$ ...
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2answers
50 views

Understanding Cones in general and the Ice cream Cone

Definitions Let $\mathbb{R}^n$ be the n dimensional Eucledean space. With $S \subseteq \mathbb{R}^n$, let $S^G$ be the set of all finite nonnegative linear combinations of elements of $S$. A set $K$ ...
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0answers
25 views

Area of the intersection between a sphere and a cone (located in the center of the sphere)

Please, how do I calculate the area of the intersection between a sphere and a cone, as shown in the image below? The beginning of the cone is located in the center of the sphere, and both geometric ...
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134 views

About Tangent Cone and Normal Cone

Suppose that $C$ and $D$ are polyhedral convex sets in $R^n $and x is in $C \cap D$. show (a) $T_{C \cap D}(x)=T_C(x) \cap T_D(x)$ (b) $N_{C \cap D}(x)=N_C(x) + N_D(x)$ (c) compute the tangent and ...
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35 views

How to prove there exists a second-order cone that is retractive

Can you help me to prove there exists a second-order cone that is retractive. Second-order cones are sets of the form $\{(x,z)\in R^n \times R: ||x||_2\leq z\}$. Can you help me? Thanks
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1answer
63 views

Minkowski sum of duals

I'm really struggling to prove the following statement: Let $\mathbb{E}$ be an Euclidean space, let $K,K_p,S\subseteq\mathbb{E}$ be a proper cone, a polyhedral cone and a subspace, respectively. If ...
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1answer
84 views

Show that finitely generated cone in Hilbert space is closed.

Let $X$ be a real Hilbert space. Let $\{c_i\}_{i=1}^m\subset X$. Denote $$ Y=\left\{\sum\limits_{i=1}^m\lambda_ic_i\Big|\lambda_i\in[0,+\infty),\quad 1\leq i\leq m\right\}. $$ Show that $Y$ is a ...
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1answer
111 views

Calculating The First Fundamental Form of a Generalized Cone

$\newcommand{\bs}{\boldsymbol{\sigma}} \newcommand{\bg}{\boldsymbol{\gamma}}\newcommand{\bp}{\textbf{p}}$ Let $\bs(u,v)=(1-v)\bp+v\bg(u)$ be a generalized cone, where $\bg$ is unit speed curve and $\...
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1answer
95 views

Does cone associated with PSD matrix always convex?

Is it true that PSD-cone always convex? (If not, please provide an example). If this is the case, then set of PSD matrices always convex or it could happen that such set might not be convex for some ...
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81 views

Tangent cone to $\ell_1$ norm constraint

We are given the $\ell_1$-constrained convex set $\mathcal{C} = \{ x \in \mathbb{R}^n : \| x \|_1 \leq 1 \}$, involved in a convex optimization problem. Moreover, we know that the optimal solution $x^{...
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3answers
224 views

Angle of sector formed by cutting a cone

A cone has a height of 10cm and circular base with radius 4.it is slit and cut open to form a sector.find the angle formed by the two radii.which is the simplest method to solve this?
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99 views

How to fix this dual cone?

Consider the following cone: $$\mathbb{G}_n=\Bigg\{\,(x\oplus\theta\oplus\kappa) \in\mathbb{R}^n\oplus\mathbb{R}_+\oplus\mathbb{R}_+\,\colon \theta\sum_{i\in [n]}\exp\bigg(\frac{-x_i}{\theta}\bigg)\...
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1answer
83 views

Show that any plane whose normal lies on cone $(b+c)x^2+(c+a)y^2+(a+b)z^2=0$ cuts the surface $ax^2+by^2+cz^2=1$ is rectangular hyperbola

Show that any plane whose normal lies on cone $(b+c)x^2+(c+a)y^2+(a+b)z^2=0$ cuts the surface $ax^2+by^2+cz^2=1$ is rectangular hyperbola My attempt: let $\frac {x}{l} = \frac {y}{m} = \frac {z}{n}$ ...
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1answer
185 views

Dual of the relative entropy cone

I've been trying to calculate the dual cone of the relative entropy cone, which is given by: $$\mathbb{H}_n = \Bigg\{\,(\theta\oplus \kappa\oplus x)\in\mathbb{R}^n_+\oplus\mathbb{R}_+^n\oplus\mathbb{...
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1answer
52 views

About theory of convex cones

Working with positive semidefinite matrices I discovered the concept of convex cone. That is, a subset $C$ of a vector space $V$ that is closed under positive linear combinations. I had never read ...
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46 views

Dual of epigraph-type cones

I am trying to calculate the dual of some cones that I don't know 'a priori'. For example, looking at MOSEK https://docs.mosek.com/MOSEKModelingCookbook-letter.pdf it seems that he already know the ...
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1answer
30 views

Defining conic problems over some specific cone

I am working with the following definition for conic optimization problems (based on Alexander Barvinok's book "A Course in Convexity"): Let $\mathbb{E}$ and $\mathbb{Y}$ be Euclidean spaces, let $K\...
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1answer
54 views

Definition of a bounded subset of the cone of positive semidefinite matrices

I cannot understand what it formally means when one says that a subset of the cone of positive semidefinite (PSD) matrices is bounded. I found a pretty general definition of a bounded set, i.e., ...
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1answer
109 views

the connection between matrix and convex cone

I'm trying to understand the connection between convex cone and matarix. according to Boyd as you can see in the pic: X is a p.s.d matrix but how this matrix represent a convex cone? and why the ...
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0answers
37 views

why in this chance constraint we have convexity?

I havea chance constraint of the form $$ \mathbb{P}[a^\text{T}x\leqslant b]\geqslant\alpha$$ where $b\in\mathbb{R}$ is fixed, $a\in\mathbb{R}^n$ is a vector whose entries are Independent and ...
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1answer
65 views

In $R^n,$ convex cone of any compact set is closed? cone of any compact set is closed? [closed]

In $R^n,$ convex cone of a compact set is closed? cone of a compact set is closed?
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44 views

Cone of feasible directions of equality constraint $y=x^5$ at $(0,0)$

So let $$D_x = \{ d \in \mathcal{R} \mid d \neq 0,x+ad \in S \} $$ We usually have $S$ as an inequality but this time we have it $$ S =\{(x, y) \mid y = x^5\}$$ So my thoughts are $d = (d_1,d_2) = (...
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0answers
18 views

How does the geometrical definition of irreduciblity implies the combinotarial definition?

Recently I have come across the geometrical definition of irreducible matrix, as goes follow. A matrix in $\pi(K)$ is $K-$irreducible if the only faces of $K$ that it leaves invariant are $\{0\}$ ...
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35 views

If $lk(v;K)$ is collapsible, then $K$ collapses to $del(v;K)$.

I'm doing exercises about simplicial complexes and I'm stuck with one for which I'll first give some definitions. Let $K$ a simplicial complex and $v\in K^0$ a $0$-simplex (vertex). The star of $...
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0answers
35 views

Find points in a multi-dimensional half cone

By knowing the opening angle, a point on a multi-dimensional half cone and the cone axis, is it possible to find any other point in that hypersurface? For $2D$ half cones, it is enough to multiply ...
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0answers
62 views

Formulate as Second-order Cone Programming $minimize \frac{\|Ax-b\|^2_1}{1-\|x\|_\infty}$

I have this as problem and do not know how to solve it: Formulate the following problem as SOCP: $minimize \frac{\|Ax-b\|^2_1}{1-\|x\|_\infty}$ Anyone can help me or give me guidelines.