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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).

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Parameterizing the Dual Cone of a Cone defined by a Sublevel Set

While attempting to implement the results of a research paper, I've run across an interesting puzzle where I am seeking to identify an ellipsoidal set $\mathcal{C}$, but am only given the parameters ...
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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 ...
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Dual of conic program

Let $A$ be an $m \times n$ matrix (over $\mathbb{R}$), $b \in \mathbb{R}^m$, $c \in \mathbb{R}^n$ and $K \subseteq \mathbb{R}^n $ is a closed, convex, pointed cone with non-empty interior. We define a ...
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46 views

Derive the implicit cone equation from the implicit circle equation

Is it possible to derive the implicit equation of a cone $x^2+y^2-z^2=0$ from the circle equation $x^2+y^2=1$, which is the intersection between the cone and the hyperplane $\{(x,y,z)\in\Bbb R^3\,|\,z=...
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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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Finding optimal savings with dual formulation (word problem)

We want to build a bridge over a river of width 2`, with a pillar in the middle of the river. The bridge is symmetric and drops (linearly) to a minimum height of h meters below the initial level of ...
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42 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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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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43 views

Question about dual cone

Update: I have solved this problem, thanks to this inspiring post: Polar cone of the Polar cone of $K$ a closed convex cone is again $K$. I will add my solution later. ================================...
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Is the nonnegative orthant isometric to itself under orthogonal mapping?

Problem description: (Informal). I simply want to know if there exists a necessary and (or at least) sufficient condition for an orthogonal matrix to map every point of the nonnegative orthant (say ...
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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^{...
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The dual of a regular polyhedral cone is regular

A rational polyhedral cone in $\mathbb{R}^n$ is a set of the form $$\sigma=\{\lambda_1x_1+\dots+\lambda_kx_k\in \mathbb{R}^n\mid \lambda_i\in \mathbb{R}_{\geq 0} \; \;\forall \, 1\leq i\leq k\}$$ for ...
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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)\...
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186 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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35 views

Linear image of a dual cone

Let $\mathbb{E}$ and $\mathbb{Y}$ are Euclidean spaces, $K\subseteq\mathbb{E}$ is a proper cone and $A\colon\mathbb{E}\to\mathbb{Y}$ is a linear transformation, what is the relation between $A(K)^\ast$...
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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 ...
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Calculating the dual of a conic problem

I am struggling to calculate the dual of the following conic problem: $$\inf\{a\lambda_a+b\lambda_b\,\colon w_1,w_2,\lambda_a,\lambda_b\in\mathbb{R}, (1,\lambda_a,w_2-w_1)\in\mathbb{G}_1 \text{, and} ...
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63 views

What is the graph of a hyperbola where the two cones are split through the middle?

So If you have two cones stacked on top of each other like you see in a normal conic section, and the cones are split perfectly in two (1/2 of the diameter of the cone's base), how would you graph the ...
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42 views

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 ...
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How to prove that the dual of any set is a closed convex cone?

In page 3 of https://link.springer.com/content/pdf/10.1007%2Fs10107-005-0690-4.pdf . It is stated that "The dual of any set is a closed convex cone". I want to know how to prove this. We formulate the ...
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KKT conditions for general conic optimization problem

From reading the literature, it seems that if we have a general conic optimization problem given by, \begin{equation} \begin{aligned} & \underset{x}{\text{min}} & & c^T x \\ & \text{s....
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1answer
93 views

Different forms of primal-dual second-order cone program optimization problems

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 ...
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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 ...
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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....
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41 views

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 ...
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184 views

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 $\alpha$. ...
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148 views

Dual optimization problem for standard linear programming

Let $K$ be a proper convex cone. Let's have a cone problem $$ \text{min } \textbf{c}^T\textbf{x} \\ \text{s.t. } A\textbf{x} =\textbf{b}\\\textbf{x}\succeq_K \textbf{0} $$ How to derive the dual ...
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117 views

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\...
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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 ...
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111 views

Conic programming: optimality conditions

I am trying to derive the optimality conditions of a conic program which is a minimization problem in the form: $$ \mathrm{Minimize}_{x,s} c'x\\ s\in C\\ Ax + s = b. $$ Here $C$ is a cone with ...
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368 views

Convex cone generated by extreme rays

Let $X$ be a vector space and $K \subseteq X$ a pointed convex cone. Let $L$ denote the set of extreme rays of $K.$ The questions are: under which condition can I guarantee that $$K= cone(conv(L))?$$ ...
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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 ...
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81 views

what is the dual of line (x=-y)

I encountered this question: Find the dual cone of $K = \{(x,y)|(x+y=0)\}$. To find the dual ($K^* = \{y \mid x'y \geq 0 \text{ for all }x \in K \}$), I did the following: \begin{align} ...
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Property of Dual Generalized Inequalities

For $K$ a proper cone we define $x<_K y$ if and only if $y-x\in\text{int}K$. Denote $K^*$ as the dual cone of $K$ (also a proper cone). I would like to prove that $$x<_Ky\iff \lambda^T x<\...
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258 views

Showing that the dual of convex cone is closure of its open space

I came across a quite obvious statement but I am stuck in proving that. Let $\Lambda$ be a subset of $R^n$, the dual cone $\Lambda^*=\{w:\forall \lambda\in\Lambda, w\cdot\lambda\geq0\}$, and its open ...
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321 views

Some basic topological properties of dual cones

For a cone $K\subseteq\mathbb{R}^n$ (not necessarily convex nor closed), we define its dual cone as $$K^*=\{y\vert x^Ty\ge 0\,\text{for all }x\in K\}.$$ I know that $K^*$ is a closed, convex cone. I ...
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Proof of Lorentz(seconde grade) cone is convex and self-dual

Given Lorentz cone $L^{n+1}=\{(x,t)\in \mathbb R^{n+1}: ||x||_2 \leq t\}$ How can i proof the convexity and self duality? I tried to do it from definitions, but i couldn't solve it. Do i need more ...
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293 views

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: ...
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302 views

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 \...
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subgroups of regular polytopes that preserve a given face

Say I have a regular polytope (e.g. it is vertex and face transitive). Given a face F, is it true that there are symmetry operations taking every vertex of F to every other that also send F to its ...
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60 views

The extremal points of the dual polytopes of vertex transitive polytopes

Is the following true? I have a convex polytope $P$ that is vertex transitive - roughly speaking all extremal points of $P$ have the same face-sets (the polytopes are isogonal figures). It is known ...
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261 views

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 $, ...
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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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Under what cases of cone $K$ is $x = \text{proj}_K(x) + \text{proj}_{-K^*}(x)$ possible for all $x$?

Here, $K^*$ is the dual cone of $K$: $K^* = \{x \mid x^Ty \geq 0 \forall y\in K\}.$ The property is true if $K$ is the nonnegative cone or the positive semidefinite cone. Does a more general ...
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What mean T in formula?

I was reading a thesis (this one) when I came to some formulas with a T with came from nowhere. ,, . Could someone explain me what mean the T in these formulas ?
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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, \...
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305 views

Why are generalized inequalities defined over proper cone?

Why generalized inequality is defined over a proper cone? What property does not hold if we define it over non-convex cone? Same with `pointed'. For example, generalized inequality makes sense in a ...
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1answer
518 views

Dual polyhedron & dual cone

From Wiki: Def. of dual of polyhedral (polytope): polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. EX: 2. Def. of dual cone: ...
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681 views

General equation of a cone

What is the general equation of a cone in $\mathbb{R}^3$ space? There should be no assumptions about the location of the vertex, direction of the axis or aperture angle, these should all be variable.
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536 views

Dual cone and sum of closed cones

Picture below is from the 35 page of Schneider R.-Convex Bodies_ The Brunn-Minkowski Theory-Cambridge University Press (2013) , I think $C^o$ is always closed no matter $C$ is closed or not. Because ...