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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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Duality Results for Convex SDP Programming

Suppose we have an SDP program with a convex (but nonlinear) objective $f(X)$, where $X$ is a positive semidefinite matrix. All other constraints are linear. Does there exist a dual program for such a ...
GoemanWilliamson's user avatar
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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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Why is the Cartesian product of second-order cones self-dual?

I came upon a paper regarding conic programming (CP) where a constraint is an inequality defined on Cartesian product of second-order cones, named $K$. And he derived the dual problem of this CP with ...
akio cu's user avatar
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1 answer
102 views

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=-\...
Chris's user avatar
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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 \...
Mani's user avatar
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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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Does this objective function have a dual?

I want to minimize the function $$\begin{aligned} & \underset{x}{\text{minimize}} & & \left[\underset{i}{\Sigma}(x_i - w_i)^2 + (x^TPx - 1)^2\right] \\ & \text{subject to} & & \...
PyRsquared's user avatar
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3 answers
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intuition behind deriving the equation of a double-napped cone

I know the equation of a double-napped cone is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$ but don't fully understand how this is derived. For a right circular cone centered at the origin, ...
John's user avatar
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Cone dual and orthogonal projection

Let $K \subset \mathbb{R}^{n}$ be a closed cone (not necessarily convex) and $y \in K^{*}$, then the orthogonal projection of $y$ onto $K$ is unique and equal to zero. $K^{*} = \{d \in \mathbb{R}^{n}\...
Zacarias89.'s user avatar
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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 $K$. ...
hanava331's user avatar
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Geometric interpretation of dual generalized inequalities in 2D using proper cone and its dual cone.

This question is based on section 2.6.2 of the textbook Convex Optimization by Boyd. The specific mathematical statement I am referring to is the following: $$ x \prec_{K} y \iff \lambda^{T}x < \...
viv's user avatar
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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 ...
John Smith's user avatar
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322 views

Describe Dual Cones in $R^2$

I'm trying to find the dual cones for each of the following cones: $K=\left\{\left(x_{1}, x_{2}\right)\mid \left| x_{1}\right| \leq x_{2}\right\}$ and $K=\left\{\left(x_{1}, x_{2}\right) \mid x_{1} +...
Harry556's user avatar
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Farkas' Lemma and Theorem of Alternatives

Let's define the DUAL CONE of a subset Y of $\mathbb{R}^n$ as follows: $$Y^*=\{x\in\mathbb{R}^n\mid \langle x,y\rangle \ge 0\quad\forall y\in Y\}.$$ If $Y$ is a finite subset of $\mathbb{R}^n$, i.e. ...
Grace53's user avatar
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1 answer
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Finding dual cone for a set of copositive matrices

This is a question from the textbook Convex Optimization by Stephen Boyd and Lieven Vandenberghe (2.35). I did read through the solution manual but I couldn't figure out why it is written the way it ...
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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 ...
generic properties's user avatar
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2 answers
949 views

Dual cone's dual cone is the closure of primal cone's convex hull

Assume $K$ is a cone and its dual cone is $K^* = \{y:x^Ty \geq 0,\, \forall x \in K\}$. Then we have $K^{**} = \text{cl}(\text{conv}\ K)$, where cl means closure, conv means convex hull. How to prove ...
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Dual problem to SDP problem

I'm having problem with formulating dual problem to Semidefinite programing problem: $$\max\;\;tr(X)$$ $$s.t.\;\; \left[ \begin{array}{cc} A & X \\ X & B \end{array} \right]\succeq0$$ where ...
ImrichUljaky's user avatar
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1 answer
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Uniqueness of dual variable for convex optimization problem

recently, I have the following problem when designing the generalized benders decomposition. Given the primal solution of a strict convex (nonlinear) optimization, is the dual variable computed from ...
Stevie Lyh's user avatar
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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 $\Leftarrow$ ...
Britta's user avatar
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1 answer
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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 ...
Britta's user avatar
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5 votes
2 answers
363 views

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). ...
Asaf Shachar's user avatar
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3 answers
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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 ...
honzaik's user avatar
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Volume cut off from the sphere by the cone using cylindrical polar coordinates

Find the volume cut off from the sphere ${x^2}+{y^2}+{z^2}={a^2} $ by the cone ${x^2 }+{y^2} ={z^2}$ using cylindrical polar coordinates? I know there will be a dual cone but I am not able to write ...
AltusXe's user avatar
2 votes
1 answer
744 views

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. ...
CEP's user avatar
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4 votes
3 answers
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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 \...
rims's user avatar
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6 votes
1 answer
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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 ...
rims's user avatar
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0 votes
1 answer
789 views

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}_\...
rims's user avatar
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1 vote
0 answers
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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 \...
strncmp's user avatar
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4 votes
1 answer
223 views

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 ...
Ariel Serranoni's user avatar
2 votes
1 answer
175 views

Is the dual of a conic program $\min_{x\in K} c^T x$ subject to $Ax=b$ also a 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 ...
dstivd's user avatar
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1 answer
352 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=...
Marko Duda's user avatar
1 vote
1 answer
670 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 $...
user avatar
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1 answer
151 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 ...
Ariel Serranoni's user avatar
2 votes
0 answers
249 views

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 ...
venrey's user avatar
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0 answers
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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^{...
VHarisop's user avatar
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2 votes
1 answer
296 views

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 ...
Walter Simon's user avatar
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0 answers
124 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)\...
Ariel Serranoni's user avatar
3 votes
1 answer
496 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{...
Ariel Serranoni's user avatar
0 votes
1 answer
195 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 $A(K)^\ast$...
Ariel Serranoni's user avatar
1 vote
0 answers
155 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 ...
Ariel Serranoni's user avatar
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0 answers
56 views

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} ...
Ariel Serranoni's user avatar
1 vote
1 answer
178 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 ...
jstowell's user avatar
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1 answer
79 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 ...
zainudin saputra's user avatar
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1 answer
1k views

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 ...
maple's user avatar
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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....
InquisitiveInquirer's user avatar
1 vote
1 answer
431 views

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 ...
InquisitiveInquirer's user avatar
4 votes
0 answers
63 views

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 ...
John's user avatar
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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....
yaodao vang's user avatar