The cones tag has no wiki summary.

**5**

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**2**answers

182 views

### Decomposing polyhedral cones into “direct sums” and a polynomial

This question consists of two parts. I'm not breaking it up into two separate ones because posing the second question would essentially require me two rewrite the first one. Also, to some extent, the ...

**2**

votes

**1**answer

82 views

### How to (efficiently) find intersection of two polyhedral cones?

I have two polyhedral cones represented by their rays. I am looking to find their intersection, which would also be a polyhedral cone, hopefully efficiently. Does anybody know a way to do that?
...

**2**

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**2**answers

104 views

### sensitivity analysis in conic optimization

I have a conic optimization of the form:
$\min_x \langle c, x \rangle$, s.t. $Ax = b$, $x \in K$.
Where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a self ...

**-2**

votes

**1**answer

98 views

### Is dual cone unique? [closed]

Suppose we have the following relationship, note that $A,B,C$ are closed convex matrix cones,
$A^\ast=C,$
$B^\ast=C,$
can we state that $A=B$? Is the dual cone of a cone is unique?
the definition ...

**0**

votes

**1**answer

271 views

### Kleiman's and Nakai-Moishezon's ampleness criteria

I would like to work out a simple example to understand the relation between Kleiman ampleness criterion and Nakai-Moishezon ampleness criterion.
Namely, let $X$ be the blow-up of $\mathbb{P}^{2}$ at ...

**2**

votes

**2**answers

105 views

### Representation of Banach spaces partially ordered by solid, normal, minihedral cones

I've been using the representation result below, from Krasnosel'skij/Lifshits/Sobolev; Positive Linear Systems---The Method of Positive Linear Operators. Heldermann Verlag, 1989.
Theorem. Let $E$ be ...

**1**

vote

**0**answers

66 views

### Suprema and infima in spaces ordered by non-normal cones

Background
We shall call a subset $V_+ \subseteq V$ of a Banach space $V$ a cone if
$V_+$ is closed,
$\alpha V_+ \subseteq V_+$ for all $\alpha \geqslant 0$, and
$V_+ \cap (-V_+) = \{0\}$.
Cones ...

**0**

votes

**0**answers

99 views

### Normal cones and the geometry of closed subschemes

Let $S$ be a closed subscheme of a smooth variety $M$ and suppose its ideal sheaf factors as $\mathscr{I}_S=\mathscr{I}_{S_1}\cdot \mathscr{I}_{S_2}$ for closed subschemes $S_1$ and $S_2$. Then what ...

**1**

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**0**answers

92 views

### Linearization of cones

Suppose that $K$ is a closed convex cone in $R^{n}$. Is there a "nice" function $f:R^{n} \rightarrow R^{m}$ so that $f(K)$ is a subspace? What about an approximate subspace?

**10**

votes

**1**answer

391 views

### When is a matrix similar to a non-negative matrix?

Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...

**1**

vote

**0**answers

60 views

### Which matrix/operator in a cone has the largest negative spectral part?

Background:
Let $\mathcal{K}$ be set (convex cone, if you like) of symmetric matrices of order $n$. Each matrix $A \in \mathcal{K}$ can be decomposed in a unique way as $A=A_{+}-A_{-}$, where ...

**2**

votes

**0**answers

200 views

### Is it true that a solid, minihedral cone in infinite dimensions cannot be regular?

Background
Consider a real Banach$^1$ space $V$. We'll call a subset $V_+ \subseteq V$ a cone if
$V$ is closed,
$\alpha V_+ \subseteq V_+$ for every $\alpha \geqslant 0$ and
$V_+ \cap (-V_+) = ...

**1**

vote

**1**answer

217 views

### Covering the cone of positive semidefinite matrices by intervals

Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices?
How about a general convex cone?
For the finite case the ...

**3**

votes

**1**answer

189 views

### When are cones of matrices “generated” by vectors?

The cone $P_{n}$ of positive semidefinite matrices of order $n$ can be represented in this form: $P_{n}=\{A|\forall x\geq 0: \langle A,xx^{T}>0 \rangle \}$ with $x$ running over ...