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Questions tagged [dual-cone]

Use this tag for questions involving dual cones. In convex analysis, the "dual cone" to a set is the collection of all elements that form a "positive angle" with every element in the set. That is, given a set $S$ in a vector space $V$, we define the dual cone by $S^* = {y:\langle x,y \rangle \geq 0 \text{ for all } x \in S}$ (the precise meaning of $\langle \cdot,\cdot\rangle$ depends on the context).

In convex analysis, the "dual cone" to a set is the collection of all elements that form a "positive angle" with every element in the set. That is, given a set $S$ in a vector space $V$, we define the dual cone by $S^* = \{y:\langle x,y \rangle \geq 0 \text{ for all } x \in S\}$ (the precise meaning of $\langle \cdot,\cdot\rangle$ depends on the context).

For more information, see this Wikipedia page.

110 questions
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Dual of a positive semidefinite cone

The PSD cone is the set of all positive semidefinite matrices. The dual is the set of all matrices $A$ such that tr($A^T X$) $\geq 0$ for all positive semidefinite matrices $X$. How to prove that the PSD cone is self-dual,i.e. the dual is also the…
rims
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Examples of tangent cone

In this lecture a tangent cone is defined as the closure of the feasible directions. Definition 9. (Tangent Cone) Let $C ⊆\mathbb R^n$ be a nonempty set, and let $x ∈ C$. Then the tangent cone of $C$ at $x$, denoted by $TC(x)$, is defined as…
Olórin
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Finding the dual cone

Looking at the example here, I'm trying to understand how the author finds the dual cone $K^*$. The question asks to find the dual cone of $\{Ax | x \succeq 0\}$ where $A \in \mathbb R^{m\ \mathrm{x}\ n}$. I know that the dual cone for a cone $K$ is…
strimp099
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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 transpose of $A$, and the cone of positive…
Luftbahnfahrer
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dual cone is closed

I have been trying to understand why dual of a cone is closed, no matter the cone is closed or not. I know the proof is 'It is because dual cone is an intersection of closed halfspaces.'. I just do not understand how it is linked to the definition…
Haesol
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How to project a symmetric matrix onto the cone of positive semidefinite (PSD) matrices?

How would you project a symmetric real matrix onto the cone of all positive semidefinite matrices?
Alice
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Prove that $(C \cap D)^\vee = C^\vee + D^\vee$.

Definitions and notations: Let $M$ to be the $n$-dimensional Euclidean space $\mathbb{R}^n$ and $N$ its dual space $M^\ast = \mathrm{Hom}_\mathbb{R}(M, \mathbb{R}).$ A subset $C \subset N$ is said to be cone if it is spanned by finitely many…
Orat
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Characterizing the dual cone of the squares of skew-symmetric matrices

Let $X$ be the set of all real $n \times n$ diagonal matrices $D$ satisfying $\langle D,B^2 \rangle \le 0$ for any (real) skew-symmetric matrix $B$. (I am using the Frobenius Euclidean product here). $X$ is a convex cone. Can we give an explicit…
Asaf Shachar
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Linear Transformation of Closed Convex Cone

Given a closed convex cone $C \subset \mathbb{R}^n$ and a matrix $M \in \mathbb{R}^{m\times n}$, is the set $S = \{Mx\mid x \in C\}$ also a closed convex cone? Firstly, $S$ must be a convex cone. But how about the closeness? I conjecture that $S$…
PSPACEhard
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What Toric Variety does this fan correspond to?

Let $\Sigma$ be the fan defined by $\{\sigma_1,\sigma_2,\sigma_3,\sigma_4,\star\}$, where $\sigma_1=\operatorname{cone}(e_1)$, $\sigma_2=\operatorname{cone}(e_2)$, $\sigma_3=-\sigma_1$, $\sigma_4=-\sigma_2$, and $\star=\operatorname{cone}((0,0))$.…
Chris
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Dual norm only for a part of matrix

A dual norm $\|\cdot\|^\circ$ of norm $\|\cdot\|$ can be given in terms of inner product $$\|A\|^\circ=\max_B |\text{Tr}(AB)|,$$ with the constraint $\|B\|\leq1$. This can be re-expressed, for unitarily invariant norms,…
generic properties
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Dual of a polyhedral cone

A general polyhedral cone $\mathcal{P} \subseteq \mathbb{R}^n$ can be represented as either $\mathcal{P} = \{x \in \mathbb{R}^n : Ax \geq 0 \}$ or $\mathcal{P} = \{V x : x \in \mathbb{R}_+^k , V \in \mathbb{R}^{n \times k} \}$. I am trying to do…
rims
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Epigraphical Cones, Fenchel Conjugates, and Duality

I'm trying to derive a result relating cones conceived as epigraphs of convex functions, duality, and Fenchel conjungates. Let me state exactly what I'm looking for: Let $\mathbb{E}$ be an Euclidean space and let $f\colon\mathbb{E}\to\mathbb{R}$ be…
Ariel Serranoni
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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 bijection such that $\Phi(C)=C$) such that…
John
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How to find the dual cone?

The similar question is here , but I do not see the desired answer. Assume a cone $K=\{(x,y) |\ x+y=0\}$, find the dual cone of $K$. The definition of dual cone is here: $K^*=\{y|x^{T}y\geq0, \forall x\in K \}$ And the given answer is $K^∗=\{(x,y)|…
stander Qiu
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