Mixing
Determine the normal and tangent cones $N_C (x)$ and $T_C (x)$ for all $x in C$.
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GIVEN
Let $C = { x in mathbb{R}^n : Ax=b }$, where $A$ is an $m times n$ matrix and $b in mathbb{R}^m$.
Determine the normal cone $N_C(x)$ and $T_C(x)$ for all $x in C$.
USEFUL DEFINITIONS
Let $C$ be a nonempty, closed and convex set and let $x in C$.
Normal Cone
The normal cone of $C$ at $x$ is denoted by $N_C(x)$, and is defined by:
$$z in N_C(x) Longleftrightarrow langle z, c-xrangle leq 0, ; forall c in C$$
If $x in text{int}(C)$ then $N_C(x) = { 0 }$, and if $x in text{bdry}(C)$ then then $N_C(x) neq { 0 }$.
($text{int}$, $text{cl}$ and $text{bdry}$ refer to the interior, closure and the boundary).
Tangent Cone
It is defined to be the polar cone of the normal cone.
$$T_C(x) = big(N_C(x)big)^circ = { u in mathbb{R}^n : langle u,v rangle leq 0, ; forall v in N_C(x) }$$
It can also be expressed as,
$$T_C(x) = text{cl}{ lambda (c-x) : c in C text{ and } lambda geq 0}$$
ATTEMPT
I do not clearly understand what $C = { x in mathbb{R}^n : Ax=b }$ is, nor do I know how to use its $Ax=b$ property. I am not even sure how to prove that it is closed and convex to use the above definitions.
I first interpreted $Ax=b$ as being a collection of hyperplanes $langle a^i ,x rangle = b_i$ with $i={1,ldots,m}$ and $a^i$ being the $i$th row of $A$. This gives me the impression that $x$ is the intersection of hyperplanes.
I am very confused.
How might I be able to calculate the normal and tangent cones of $C$?
Any help is immensely appreciated.
vector-spaces convex-analysis convex-geometry
asked 2 days ago
ex.nihil
16710
|
show 8 more comments
up vote
1
down vote
favorite
GIVEN
Let $C = { x in mathbb{R}^n : Ax=b }$, where $A$ is an $m times n$ matrix and $b in mathbb{R}^m$.
Determine the normal cone $N_C(x)$ and $T_C(x)$ for all $x in C$.
USEFUL DEFINITIONS
Let $C$ be a nonempty, closed and convex set and let $x in C$.
Normal Cone
The normal cone of $C$ at $x$ is denoted by $N_C(x)$, and is defined by:
$$z in N_C(x) Longleftrightarrow langle z, c-xrangle leq 0, ; forall c in C$$
If $x in text{int}(C)$ then $N_C(x) = { 0 }$, and if $x in text{bdry}(C)$ then then $N_C(x) neq { 0 }$.
($text{int}$, $text{cl}$ and $text{bdry}$ refer to the interior, closure and the boundary).
Tangent Cone
It is defined to be the polar cone of the normal cone.
$$T_C(x) = big(N_C(x)big)^circ = { u in mathbb{R}^n : langle u,v rangle leq 0, ; forall v in N_C(x) }$$
It can also be expressed as,
$$T_C(x) = text{cl}{ lambda (c-x) : c in C text{ and } lambda geq 0}$$
ATTEMPT
I do not clearly understand what $C = { x in mathbb{R}^n : Ax=b }$ is, nor do I know how to use its $Ax=b$ property. I am not even sure how to prove that it is closed and convex to use the above definitions.
I first interpreted $Ax=b$ as being a collection of hyperplanes $langle a^i ,x rangle = b_i$ with $i={1,ldots,m}$ and $a^i$ being the $i$th row of $A$. This gives me the impression that $x$ is the intersection of hyperplanes.
I am very confused.
How might I be able to calculate the normal and tangent cones of $C$?
Any help is immensely appreciated.
vector-spaces convex-analysis convex-geometry
asked 2 days ago
ex.nihil
16710
1
You are missing $leq0$ in the definition of $N_C$
– Federico
2 days ago
1
Notice that $C={x:Ax=b}$ is an affine space. It is of the form $x_0+ker A$. It is closed and convex
– Federico
2 days ago
1
Since $N$ and $T$ are invariant by translation, meaning $N_{x_0+C}(x_0+x)=N_C(x)$, you just have to study the case of a vector space $ker A$
– Federico
2 days ago
1
Moreover, since $y+ker A=ker A$ if $yinker A$, you can just study what happens at the origin: $N_{ker A}(0)$ and $T_{ker A}(0)$.
– Federico
2 days ago
1
Going with your hints, doesn't that mean that we obtain $langle z, crangle leq 0$ for all $c in ker(A)$? How would this characterize the normal cone?
– ex.nihil
2 days ago
|
show 8 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
GIVEN
Let $C = { x in mathbb{R}^n : Ax=b }$, where $A$ is an $m times n$ matrix and $b in mathbb{R}^m$.
Determine the normal cone $N_C(x)$ and $T_C(x)$ for all $x in C$.
USEFUL DEFINITIONS
Let $C$ be a nonempty, closed and convex set and let $x in C$.
Normal Cone
The normal cone of $C$ at $x$ is denoted by $N_C(x)$, and is defined by:
$$z in N_C(x) Longleftrightarrow langle z, c-xrangle leq 0, ; forall c in C$$
If $x in text{int}(C)$ then $N_C(x) = { 0 }$, and if $x in text{bdry}(C)$ then then $N_C(x) neq { 0 }$.
($text{int}$, $text{cl}$ and $text{bdry}$ refer to the interior, closure and the boundary).
Tangent Cone
It is defined to be the polar cone of the normal cone.
$$T_C(x) = big(N_C(x)big)^circ = { u in mathbb{R}^n : langle u,v rangle leq 0, ; forall v in N_C(x) }$$
It can also be expressed as,
$$T_C(x) = text{cl}{ lambda (c-x) : c in C text{ and } lambda geq 0}$$
ATTEMPT
I do not clearly understand what $C = { x in mathbb{R}^n : Ax=b }$ is, nor do I know how to use its $Ax=b$ property. I am not even sure how to prove that it is closed and convex to use the above definitions.
I first interpreted $Ax=b$ as being a collection of hyperplanes $langle a^i ,x rangle = b_i$ with $i={1,ldots,m}$ and $a^i$ being the $i$th row of $A$. This gives me the impression that $x$ is the intersection of hyperplanes.
I am very confused.
How might I be able to calculate the normal and tangent cones of $C$?
Any help is immensely appreciated.
vector-spaces convex-analysis convex-geometry
asked 2 days ago
ex.nihil
16710
GIVEN
Let $C = { x in mathbb{R}^n : Ax=b }$, where $A$ is an $m times n$ matrix and $b in mathbb{R}^m$.
Determine the normal cone $N_C(x)$ and $T_C(x)$ for all $x in C$.
USEFUL DEFINITIONS
Let $C$ be a nonempty, closed and convex set and let $x in C$.
Normal Cone
The normal cone of $C$ at $x$ is denoted by $N_C(x)$, and is defined by:
$$z in N_C(x) Longleftrightarrow langle z, c-xrangle leq 0, ; forall c in C$$
If $x in text{int}(C)$ then $N_C(x) = { 0 }$, and if $x in text{bdry}(C)$ then then $N_C(x) neq { 0 }$.
($text{int}$, $text{cl}$ and $text{bdry}$ refer to the interior, closure and the boundary).
Tangent Cone
It is defined to be the polar cone of the normal cone.
$$T_C(x) = big(N_C(x)big)^circ = { u in mathbb{R}^n : langle u,v rangle leq 0, ; forall v in N_C(x) }$$
It can also be expressed as,
$$T_C(x) = text{cl}{ lambda (c-x) : c in C text{ and } lambda geq 0}$$
ATTEMPT
I do not clearly understand what $C = { x in mathbb{R}^n : Ax=b }$ is, nor do I know how to use its $Ax=b$ property. I am not even sure how to prove that it is closed and convex to use the above definitions.
I first interpreted $Ax=b$ as being a collection of hyperplanes $langle a^i ,x rangle = b_i$ with $i={1,ldots,m}$ and $a^i$ being the $i$th row of $A$. This gives me the impression that $x$ is the intersection of hyperplanes.
I am very confused.
How might I be able to calculate the normal and tangent cones of $C$?
Any help is immensely appreciated.
vector-spaces convex-analysis convex-geometry
vector-spaces convex-analysis convex-geometry
asked 2 days ago
ex.nihil
16710
asked 2 days ago
ex.nihil
16710
edited 2 days ago
asked 2 days ago
ex.nihil
16710
asked 2 days ago
ex.nihil
16710
asked 2 days ago
ex.nihil
16710
16710
1
You are missing $leq0$ in the definition of $N_C$
– Federico
2 days ago
1
Notice that $C={x:Ax=b}$ is an affine space. It is of the form $x_0+ker A$. It is closed and convex
– Federico
2 days ago
1
Since $N$ and $T$ are invariant by translation, meaning $N_{x_0+C}(x_0+x)=N_C(x)$, you just have to study the case of a vector space $ker A$
– Federico
2 days ago
1
Moreover, since $y+ker A=ker A$ if $yinker A$, you can just study what happens at the origin: $N_{ker A}(0)$ and $T_{ker A}(0)$.
– Federico
2 days ago
1
Going with your hints, doesn't that mean that we obtain $langle z, crangle leq 0$ for all $c in ker(A)$? How would this characterize the normal cone?
– ex.nihil
2 days ago
|
show 8 more comments
1
You are missing $leq0$ in the definition of $N_C$
– Federico
2 days ago
1
Notice that $C={x:Ax=b}$ is an affine space. It is of the form $x_0+ker A$. It is closed and convex
– Federico
2 days ago
1
Since $N$ and $T$ are invariant by translation, meaning $N_{x_0+C}(x_0+x)=N_C(x)$, you just have to study the case of a vector space $ker A$
– Federico
2 days ago
1
Moreover, since $y+ker A=ker A$ if $yinker A$, you can just study what happens at the origin: $N_{ker A}(0)$ and $T_{ker A}(0)$.
– Federico
2 days ago
1
Going with your hints, doesn't that mean that we obtain $langle z, crangle leq 0$ for all $c in ker(A)$? How would this characterize the normal cone?
– ex.nihil
2 days ago
1
1
You are missing $leq0$ in the definition of $N_C$
– Federico
2 days ago
You are missing $leq0$ in the definition of $N_C$
– Federico
2 days ago
1
1
Notice that $C={x:Ax=b}$ is an affine space. It is of the form $x_0+ker A$. It is closed and convex
– Federico
2 days ago
Notice that $C={x:Ax=b}$ is an affine space. It is of the form $x_0+ker A$. It is closed and convex
– Federico
2 days ago
1
1
Since $N$ and $T$ are invariant by translation, meaning $N_{x_0+C}(x_0+x)=N_C(x)$, you just have to study the case of a vector space $ker A$
– Federico
2 days ago
Since $N$ and $T$ are invariant by translation, meaning $N_{x_0+C}(x_0+x)=N_C(x)$, you just have to study the case of a vector space $ker A$
– Federico
2 days ago
1
1
Moreover, since $y+ker A=ker A$ if $yinker A$, you can just study what happens at the origin: $N_{ker A}(0)$ and $T_{ker A}(0)$.
– Federico
2 days ago
Moreover, since $y+ker A=ker A$ if $yinker A$, you can just study what happens at the origin: $N_{ker A}(0)$ and $T_{ker A}(0)$.
– Federico
2 days ago
1
1
Going with your hints, doesn't that mean that we obtain $langle z, crangle leq 0$ for all $c in ker(A)$? How would this characterize the normal cone?
– ex.nihil
2 days ago
Going with your hints, doesn't that mean that we obtain $langle z, crangle leq 0$ for all $c in ker(A)$? How would this characterize the normal cone?
– ex.nihil
2 days ago
|
show 8 more comments
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1
You are missing $leq0$ in the definition of $N_C$
– Federico
2 days ago
1
Notice that $C={x:Ax=b}$ is an affine space. It is of the form $x_0+ker A$. It is closed and convex
– Federico
2 days ago
1
Since $N$ and $T$ are invariant by translation, meaning $N_{x_0+C}(x_0+x)=N_C(x)$, you just have to study the case of a vector space $ker A$
– Federico
2 days ago
1
Moreover, since $y+ker A=ker A$ if $yinker A$, you can just study what happens at the origin: $N_{ker A}(0)$ and $T_{ker A}(0)$.
– Federico
2 days ago
1
Going with your hints, doesn't that mean that we obtain $langle z, crangle leq 0$ for all $c in ker(A)$? How would this characterize the normal cone?
– ex.nihil
2 days ago