Show that the support function of a set and its convex hull are equal.
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The support function of set $A$ is defined as the following
$S_A(x)=sup_{y in A} x^Ty$, where $x in mathbb{R}^n$.
Show that
$$S_A(x)=S_{conv(A)}(x) ,,,, forall x in mathbb{R}^n$$
convex-analysis convex-optimization
edited Nov 29 at 19:04
amWhy
191k28224439
asked Nov 29 at 4:14
Saeed
550110
add a comment |
up vote
-1
down vote
favorite
The support function of set $A$ is defined as the following
$S_A(x)=sup_{y in A} x^Ty$, where $x in mathbb{R}^n$.
Show that
$$S_A(x)=S_{conv(A)}(x) ,,,, forall x in mathbb{R}^n$$
convex-analysis convex-optimization
edited Nov 29 at 19:04
amWhy
191k28224439
asked Nov 29 at 4:14
Saeed
550110
@lntls: Can you help me for this question?
– Saeed
Nov 29 at 4:15
Perhaps you could answer the question yourself using the maximum principle, which states that a convex function attains its maximum value at an extreme point of the set.
– LinAlg
Nov 29 at 4:53
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
The support function of set $A$ is defined as the following
$S_A(x)=sup_{y in A} x^Ty$, where $x in mathbb{R}^n$.
Show that
$$S_A(x)=S_{conv(A)}(x) ,,,, forall x in mathbb{R}^n$$
convex-analysis convex-optimization
edited Nov 29 at 19:04
amWhy
191k28224439
asked Nov 29 at 4:14
Saeed
550110
The support function of set $A$ is defined as the following
$S_A(x)=sup_{y in A} x^Ty$, where $x in mathbb{R}^n$.
Show that
$$S_A(x)=S_{conv(A)}(x) ,,,, forall x in mathbb{R}^n$$
convex-analysis convex-optimization
convex-analysis convex-optimization
edited Nov 29 at 19:04
amWhy
191k28224439
asked Nov 29 at 4:14
Saeed
550110
edited Nov 29 at 19:04
amWhy
191k28224439
asked Nov 29 at 4:14
Saeed
550110
edited Nov 29 at 19:04
amWhy
191k28224439
edited Nov 29 at 19:04
amWhy
191k28224439
edited Nov 29 at 19:04
amWhy
191k28224439
191k28224439
asked Nov 29 at 4:14
Saeed
550110
asked Nov 29 at 4:14
Saeed
550110
asked Nov 29 at 4:14
Saeed
550110
550110
@lntls: Can you help me for this question?
– Saeed
Nov 29 at 4:15
Perhaps you could answer the question yourself using the maximum principle, which states that a convex function attains its maximum value at an extreme point of the set.
– LinAlg
Nov 29 at 4:53
add a comment |
@lntls: Can you help me for this question?
– Saeed
Nov 29 at 4:15
Perhaps you could answer the question yourself using the maximum principle, which states that a convex function attains its maximum value at an extreme point of the set.
– LinAlg
Nov 29 at 4:53
@lntls: Can you help me for this question?
– Saeed
Nov 29 at 4:15
@lntls: Can you help me for this question?
– Saeed
Nov 29 at 4:15
Perhaps you could answer the question yourself using the maximum principle, which states that a convex function attains its maximum value at an extreme point of the set.
– LinAlg
Nov 29 at 4:53
Perhaps you could answer the question yourself using the maximum principle, which states that a convex function attains its maximum value at an extreme point of the set.
– LinAlg
Nov 29 at 4:53
add a comment |
1 Answer
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If $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. Take $E={x^{T}y: y in A}$. [Convex hull of $E$ is nothing but ${x^{T}y: y in convex(A)}$.] Proof of the fact that if $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. If a number $x$ is an upper bound for $E$ then $sum_{i=1}^{n} c_i e_i leq x sum_{i=1}^{n} c_i =x$ for all choices of non-negative $c_i$'s adding up to $1$. So $x$ is also an upper bound for convex hull of $E$. The converse is obvious since $E$ is a subset of convex hull of $E$. Hence the least upper bound of $E$ and convex hull of $E$ are same.
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
add a comment |
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1 Answer
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1 Answer
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oldest
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up vote
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If $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. Take $E={x^{T}y: y in A}$. [Convex hull of $E$ is nothing but ${x^{T}y: y in convex(A)}$.] Proof of the fact that if $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. If a number $x$ is an upper bound for $E$ then $sum_{i=1}^{n} c_i e_i leq x sum_{i=1}^{n} c_i =x$ for all choices of non-negative $c_i$'s adding up to $1$. So $x$ is also an upper bound for convex hull of $E$. The converse is obvious since $E$ is a subset of convex hull of $E$. Hence the least upper bound of $E$ and convex hull of $E$ are same.
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
add a comment |
up vote
0
down vote
If $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. Take $E={x^{T}y: y in A}$. [Convex hull of $E$ is nothing but ${x^{T}y: y in convex(A)}$.] Proof of the fact that if $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. If a number $x$ is an upper bound for $E$ then $sum_{i=1}^{n} c_i e_i leq x sum_{i=1}^{n} c_i =x$ for all choices of non-negative $c_i$'s adding up to $1$. So $x$ is also an upper bound for convex hull of $E$. The converse is obvious since $E$ is a subset of convex hull of $E$. Hence the least upper bound of $E$ and convex hull of $E$ are same.
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
add a comment |
up vote
0
down vote
up vote
0
down vote
If $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. Take $E={x^{T}y: y in A}$. [Convex hull of $E$ is nothing but ${x^{T}y: y in convex(A)}$.] Proof of the fact that if $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. If a number $x$ is an upper bound for $E$ then $sum_{i=1}^{n} c_i e_i leq x sum_{i=1}^{n} c_i =x$ for all choices of non-negative $c_i$'s adding up to $1$. So $x$ is also an upper bound for convex hull of $E$. The converse is obvious since $E$ is a subset of convex hull of $E$. Hence the least upper bound of $E$ and convex hull of $E$ are same.
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
If $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. Take $E={x^{T}y: y in A}$. [Convex hull of $E$ is nothing but ${x^{T}y: y in convex(A)}$.] Proof of the fact that if $E$ is a set of real numbers then supremum of $E$ and supremum of $convex (E)$ are the same. If a number $x$ is an upper bound for $E$ then $sum_{i=1}^{n} c_i e_i leq x sum_{i=1}^{n} c_i =x$ for all choices of non-negative $c_i$'s adding up to $1$. So $x$ is also an upper bound for convex hull of $E$. The converse is obvious since $E$ is a subset of convex hull of $E$. Hence the least upper bound of $E$ and convex hull of $E$ are same.
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
edited Nov 29 at 23:24
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
answered Nov 29 at 5:37
Kavi Rama Murthy
48.1k31854
48.1k31854
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
add a comment |
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Although your first sentence is correct, I think the answer is why that is correct.
– LinAlg
Nov 29 at 14:08
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
Why? Could you please address the question?
– Saeed
Nov 29 at 16:42
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
@Saeed Please see my revised answer.
– Kavi Rama Murthy
Nov 29 at 23:24
add a comment |
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@lntls: Can you help me for this question?
– Saeed
Nov 29 at 4:15
Perhaps you could answer the question yourself using the maximum principle, which states that a convex function attains its maximum value at an extreme point of the set.
– LinAlg
Nov 29 at 4:53