All Questions
Tagged with dual-cone convex-cone
33 questions
0
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41
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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} \...
- 881
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1
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73
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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, ...
- 1
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0
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37
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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 ...
- 6,032
1
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1
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179
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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
$$K^*=\{y \in \mathbb{R}^n \...
- 11
1
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1
answer
129
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Polar cone of a closed convex cone in $R^4$ defined by a convex inequality constraint
Let $c>1$ be a constant. Consider points in four dimension with coordinates $(x,y,z,p)\in R_{\ge 0} \times R_{\ge 0} \times R \times R_{\ge 0}$
and the cone
$$K = \{ (x,y,z,p)\in R_{\ge 0} \times ...
- 2,207
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40
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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 $K$.
...
- 169
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473
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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 ...
0
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1
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109
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For full-dimensional cone $K$, $x\in int(K)$, can you take arb. vector $y$ and for some $t$ small enough have that $x-ty\in int(K)$
I'm working on the proof of the following theorem:
$K$ full-dimensional, closed, convex cone. $x\in int(K) \iff y^Tx>0 \quad \forall y\in K^*-{0}$
And we're pretty set with the $\Leftarrow$ ...
- 446
0
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1
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319
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Given a closed, convex, full-dimensional cone $K$, how do I show that $u\in int(K) \iff u^tx>0 \quad \forall x\in K^*-\{0\}$?
Given a closed, convex, full-dimensional cone $K$, how do I show that $x\in int(K) \iff y^Tx>0 \quad \forall y\in K^*- \{0\} $ ?
I've thought about applying the Hahn-Banach separation theorem
...
- 446
1
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3
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419
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A cone in $\mathbb{R}^n$ containing n linearly independent vectors has a non empty interior
I need a help with proving, that if a cone $K \subseteq \mathbb{R}^n$ contains $n$ linearly independent vectors, then the interior of $K$ is non empty.
Lets say $b_1,\dots,b_n \in K$ are the ...
- 323
2
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1
answer
811
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Find the dual cone $K^*_{m+}$ of $K_{m+} = \{ x \in \mathbb{R}^n \mid x_1 \ge x_2 \ge \dots \ge x_n \ge 0\}$
We define the monotone nonnegative cone as
$$K_{m+} = \{ x \in \mathbb{R}^n \mid x_1 \ge x_2 \ge \dots \ge x_n \ge 0\}$$
i.e. all nonnegative vectors with components sorted in nonincreasing order.
...
- 441
0
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1
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868
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Closure of a cone
Suppose a set $\mathcal{X}$ is closed and bounded, and define $\mathcal{K}_\mathcal{X} = \{(x,t) : t > 0, \frac{x}{t} \in \mathcal{X} \}$. Show that:
$$\bar{\mathcal{K}}_\mathcal{X} = \mathcal{K}_\...
- 2,728
1
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64
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Prooving the property of a polyhedral cone
Let me introduce some definitions. Poly$(v_1, v_2,\ldots, v_k) = \{ x \in \mathbb{R}^n \ | \ (x, v_i) \geq 0 \ \ \forall i \}$ called a polyhedral cone. For any cone $C$, $ \ C^{\lor} = \{ x \in \...
- 53
1
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1
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743
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Prove that a cone C is full-dimensional if and only if its dual cone $C^*$ is pointed.
A cone with apex $0$ is said to be pointed if it does not contain any non-trivial subspace. Let C be a closed convex cone with apex $0$. Show that $C$ is full-dimensional if and only if its dual cone $...
user533068
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1
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154
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Minkowski sum of duals
I'm really struggling to prove the following statement:
Let $\mathbb{E}$ be an Euclidean space, let $K,K_p,S\subseteq\mathbb{E}$ be a proper cone, a polyhedral cone and a subspace, respectively. If ...
- 504
1
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0
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254
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Tangent cone to $\ell_1$ norm constraint
We are given the $\ell_1$-constrained convex set $\mathcal{C} = \{ x \in \mathbb{R}^n : \| x \|_1 \leq 1 \}$, involved in a convex optimization problem. Moreover, we know that the optimal solution $x^{...
- 4,025
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130
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How to fix this dual cone?
Consider the following cone:
$$\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)\...
- 504
3
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1
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511
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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{...
- 504
1
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0
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179
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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 ...
- 504
0
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1
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85
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P normal cone of a cone metric space, given $\epsilon > 0$, can we choose c interior point of P ($c \gg 0$) s.t $\|c\| < \epsilon/K$ [closed]
I get this statment from paper "Cone metric spaces and fixed point theorems
of contractive mappings
Huang Long-Guang, Zhang Xian", i failed to understand why there is a guarantee that we can choose an ...
1
vote
1
answer
453
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Different forms of primal-dual second-order cone programs
I'm trying to understand the difference between the following two definitions of a SOCP (second-order cone program). The first way I've seen a primal-dual SOCP define is as follows:
The primal ...
4
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0
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75
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When is a homogeneous cone a Jordan Banach algebra?
A (closed) positive cone $C$ in a vector space $V$ is called homogeneous if for for all $a$ and $b$ in the interior of $C$ there exists an order isomorphism $\Phi: V\rightarrow V$ (i.e. a linear ...
- 877
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24
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is this sets $A(x) $ ,$B(x)$cone?
$A(x)=\{d\in \mathbb{R^n}: \nabla f(x)*d < 0\}$
$B(x)=\{d\in \mathbb{R^n}: \forall i $ $s.t. g_i(x)=0 , \nabla g_i(x)*d < 0\}$
A,B is a set related below optimization problem
$\min f(x)$
s....
- 307
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0
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69
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Does Gordan's Lemma hold in infinite dimensional vector spaces?
Gordan's Lemma: Let $A \in \mathbb{R}^{m \times n}$. Then exactly one of the following two statements is true:
There exists $x \in \mathbb{R}^n$ with $Ax > 0$, or
There exists nonzero $y ...
- 4,216
1
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1
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2k
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Polar cone of the Polar cone of $K$ a closed convex cone is again $K$
Let $\mathcal{H}$ be a Hilbert space and $K\subset \mathcal{H}$ a subset. We define a cone $C$ in $\mathcal{H}$ to be a set which satisfies $x\in C\implies \alpha x\in C$ for all nonnegative $\alpha$. ...
- 7,804
3
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2
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646
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Closed cones and exposed faces
I was wondering if a closed cone $C$ in a Banach space $X$ of dimension at least two always has an exposed face, that is, a face $F$ such that $F=C\cap\ker\phi$ for some positive $\phi\in X^*\setminus\...
- 315
6
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1
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228
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Characterizing duals of cones that are linear images of the positive semidefinite cone
Let $M_n$ denote the space of $n\times n$ matrices over complex numbers. The space of self-adjoint matrices is denoted
$$
M_n^{sa} = \{A\in M_n\, :\, A^*=A \},
$$
where $A^*$ denotes the conjugate ...
- 1,952
3
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2
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2k
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Convex cone generated by extreme rays
Let $X$ be a vector space and $K \subseteq X$ be a pointed convex cone. Let $L$ denote the set of extreme rays of $K.$ The questions are: under which condition can I guarantee that $$K= cone(conv(L))?$...
- 1,950
1
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0
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851
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How does the cone look from front view?
I have a customized cone and I only have information about top and side view of that cone.
Top and Side view of Cone
Questions:
How does the cone look from the front view?
How can I plot it?
Thank ...
- 111
0
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0
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1k
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Finding dual cone of general cones
Given that cones $K_1 = \{ (x_1, x_2) \in \mathbb{R}^2 \mid |x_1| \leq x_2\}$ and $K_2 = \{ (x_1, x_2) \in \mathbb{R}^2 \mid x_1 + x_2 = 0 \}$..
I think that $K$ will be like the below picture:
...
- 603
2
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1
answer
2k
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Cone and Dual Cone in $\mathbb{R}^2$ space
Boyd's book, my understanding of cone and dual cone for 2-space is:
If we think of a circle in $\mathbb{R}^2$ space, cone $K$ and dual cone $K^*$ would be like this:
Here, $K^* = y | x^Ty \geq 0 \...
- 603
2
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1
answer
534
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Proof of closure, convex hull and minimal cone of dual set
In my studies of convexity I have recently come across the following:
Let $ V=\mathbb{R}^d ; d \geq 1 $ be a Euclidean space and $ S \neq \emptyset $ be a non empty set of the vector space $ V $, ...
- 3,889
2
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1
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214
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The dual of a circular cone
In my studies of cones and convexity I have recently come across the following unexplained piece of information presented:
We consider a Euclidean space $ R^d $ for $ d \geq 1 $ we look at the ...
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