# 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 ...
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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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### 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 ...
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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 ...
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### 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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### 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{aligned} & \underset{x}{\text{min}} & & c^T x \\ & \text{s....
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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 ...
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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....