All Questions
7 questions
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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
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
$$K = \{ (x,y,z,p)\in R_{\ge 0} \times ...
- 2,207
0
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0
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40
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why the inner product with an element that does not belong to a convex cone is negative
Let us consider a set with an infinite number of vectors $\{v_k \mid k \in \mathbb{N} \}$ with $n$ cordinates, and if we consider the conned convex set containing all theses vectors, denoted by $K$.
...
- 169
1
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3
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419
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A cone in $\mathbb{R}^n$ containing n linearly independent vectors has a non empty interior
I need a help with proving, that if a cone $K \subseteq \mathbb{R}^n$ contains $n$ linearly independent vectors, then the interior of $K$ is non empty.
Lets say $b_1,\dots,b_n \in K$ are the ...
- 323
0
votes
1
answer
868
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Closure of a cone
Suppose a set $\mathcal{X}$ is closed and bounded, and define $\mathcal{K}_\mathcal{X} = \{(x,t) : t > 0, \frac{x}{t} \in \mathcal{X} \}$. Show that:
$$\bar{\mathcal{K}}_\mathcal{X} = \mathcal{K}_\...
- 2,728
1
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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
1
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1
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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 ...
