The conical hull , sometimes also called positive hull , is a special hull operator that assigns the smallest convex cone that contains this set to each subset of a vector space . The conical shell is used in the theory of mathematical optimization , especially in linear optimization .
definition
Let a - vector space and an arbitrary subset of . Then is called
R.
{\ displaystyle \ mathbb {R}}
V
{\ displaystyle V}
X
{\ displaystyle X}
V
{\ displaystyle V}
pos
(
X
)
: =
⋂
X
⊆
K
K
is convex cone
K
{\ displaystyle \ operatorname {pos} (X): = \ bigcap _ {X \ subseteq {\ mathcal {K}} \ atop {\ mathcal {K}} {\ text {is a convex cone}}} {\ mathcal { K}}}
the conical shell or positive shell of . It is the smallest convex cone that contains.
X
{\ displaystyle X}
X
{\ displaystyle X}
The definition is equivalent to this
pos
(
X
)
: =
{
∑
i
=
1
n
λ
i
x
i
|
n
∈
N
;
x
i
∈
X
;
λ
i
≥
0
}
{\ displaystyle \ operatorname {pos} (X): = \ left \ {\ sum _ {i = 1} ^ {n} \ lambda _ {i} x_ {i} \, | \, n \ in \ mathbb { N}; x_ {i} \ in X; \ lambda _ {i} \ geq 0 \ right \}}
.
Remarks
More generally, the cone hull can be defined for any vector spaces as long as it is an ordered body .
K
{\ displaystyle \ mathbb {K}}
K
{\ displaystyle \ mathbb {K}}
The notation is not used uniformly in the literature, and the designation is also sometimes found . This notation also sometimes denotes the smallest (ordinary) cone that contains and is then called the cone envelope .
pos
(
X
)
{\ displaystyle \ operatorname {pos} (X)}
cone
(
X
)
{\ displaystyle \ operatorname {cone} (X)}
X
{\ displaystyle X}
properties
The conical shell is the smallest amount that is closed with respect to conical combinations of the elements of . This follows directly from the second characterization.
X
{\ displaystyle X}
pos
{\ displaystyle \ operatorname {pos}}
is an envelope operator , so it holds for
X
,
Y
⊂
V
{\ displaystyle X, Y \ subset V}
X
⊂
pos
(
X
)
{\ displaystyle X \ subset \ operatorname {pos} (X)}
,
X
⊂
Y
⟹
pos
(
X
)
⊂
pos
(
Y
)
{\ displaystyle X \ subset Y \ implies \ operatorname {pos} (X) \ subset \ operatorname {pos} (Y)}
,
pos
(
pos
(
X
)
)
=
pos
(
X
)
{\ displaystyle \ operatorname {pos} (\ operatorname {pos} (X)) = \ operatorname {pos} (X)}
.
It applies . Here is the cone envelope and the convex envelope .
pos
(
X
)
=
cone
(
conv
(
X
)
)
=
conv
(
cone
(
X
)
)
{\ displaystyle \ operatorname {pos} (X) = \ operatorname {cone} (\ operatorname {conv} (X)) = \ operatorname {conv} (\ operatorname {cone} (X))}
cone
{\ displaystyle \ operatorname {cone}}
conv
{\ displaystyle \ operatorname {conv}}
Finally generated cone
A cone is called a finitely generated cone if there is a finite set such that
K
{\ displaystyle K}
X
{\ displaystyle X}
K
=
pos
(
X
)
{\ displaystyle K = \ operatorname {pos} (X)}
is. A cone im is finite if and only if it is a polyhedral cone .
R.
n
{\ displaystyle \ mathbb {R} ^ {n}}
Examples
Are the two vectors
R.
2
{\ displaystyle \ mathbb {R} ^ {2}}
v
1
=
(
0
1
)
,
v
2
=
(
1
0
)
{\ displaystyle v_ {1} = {\ begin {pmatrix} 0 \\ 1 \ end {pmatrix}}, \, v_ {2} = {\ begin {pmatrix} 1 \\ 0 \ end {pmatrix}}}
.
given so is
pos
(
v
1
,
v
2
)
=
{
x
∈
R.
2
,
x
1
≥
0
,
x
2
≥
0
}
{\ displaystyle \ operatorname {pos} (v_ {1}, v_ {2}) = \ {x \ in \ mathbb {R} ^ {2} \ ,, \, x_ {1} \ geq 0, \, x_ {2} \ geq 0 \}}
,
since each element of this set (the first quadrant ) can be represented as a positive combination of or .
v
1
{\ displaystyle v_ {1}}
v
2
{\ displaystyle v_ {2}}
If the monomials are given, then is
x
2
,
x
,
1
{\ displaystyle x ^ {2}, x, 1}
pos
(
x
2
,
x
,
1
)
=
{
λ
2
x
2
+
λ
1
x
+
λ
0
}
{\ displaystyle \ operatorname {pos} (x ^ {2}, x, 1) = \ {\ lambda _ {2} x ^ {2} + \ lambda _ {1} x + \ lambda _ {0} \}}
for . These are then exactly all polynomials of maximum degree 2 with positive coefficients.
λ
i
≥
0
{\ displaystyle \ lambda _ {i} \ geq 0}
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">