Is $ max_{xinmathbb{R}^n} { f(x)+g(x) } = max_{xinmathbb{R}^n} f(x)+max_{xinmathbb{R}^n} g(x) $ if $f$ and...
$begingroup$
Let $x in mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $mathbb{R}$.
Is the following property true?
$$
max_{xinmathbb{R}^n} { f(x) + g(x) } = max_{xinmathbb{R}^n} f(x) + max_{xinmathbb{R}^n} g(x)
$$
Of course, for arbitrary functions this is $leq$ instead of $=$, but I need this property in a larger proof and I am not sure if it is true or false.
Could anyone verify, and possibly sketch a small proof?
Greatly appreciated.
real-analysis functions proof-verification optimization vector-spaces
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
$endgroup$
add a comment |
$begingroup$
Let $x in mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $mathbb{R}$.
Is the following property true?
$$
max_{xinmathbb{R}^n} { f(x) + g(x) } = max_{xinmathbb{R}^n} f(x) + max_{xinmathbb{R}^n} g(x)
$$
Of course, for arbitrary functions this is $leq$ instead of $=$, but I need this property in a larger proof and I am not sure if it is true or false.
Could anyone verify, and possibly sketch a small proof?
Greatly appreciated.
real-analysis functions proof-verification optimization vector-spaces
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
$endgroup$
1
$begingroup$
Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist?
$endgroup$
– Hermione
Dec 8 '18 at 11:44
add a comment |
$begingroup$
Let $x in mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $mathbb{R}$.
Is the following property true?
$$
max_{xinmathbb{R}^n} { f(x) + g(x) } = max_{xinmathbb{R}^n} f(x) + max_{xinmathbb{R}^n} g(x)
$$
Of course, for arbitrary functions this is $leq$ instead of $=$, but I need this property in a larger proof and I am not sure if it is true or false.
Could anyone verify, and possibly sketch a small proof?
Greatly appreciated.
real-analysis functions proof-verification optimization vector-spaces
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
$endgroup$
Let $x in mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $mathbb{R}$.
Is the following property true?
$$
max_{xinmathbb{R}^n} { f(x) + g(x) } = max_{xinmathbb{R}^n} f(x) + max_{xinmathbb{R}^n} g(x)
$$
Of course, for arbitrary functions this is $leq$ instead of $=$, but I need this property in a larger proof and I am not sure if it is true or false.
Could anyone verify, and possibly sketch a small proof?
Greatly appreciated.
real-analysis functions proof-verification optimization vector-spaces
real-analysis functions proof-verification optimization vector-spaces
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
asked Dec 8 '18 at 11:27
ex.nihilex.nihil
218111
218111
1
$begingroup$
Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist?
$endgroup$
– Hermione
Dec 8 '18 at 11:44
add a comment |
1
$begingroup$
Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist?
$endgroup$
– Hermione
Dec 8 '18 at 11:44
1
1
$begingroup$
Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist?
$endgroup$
– Hermione
Dec 8 '18 at 11:44
$begingroup$
Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist?
$endgroup$
– Hermione
Dec 8 '18 at 11:44
add a comment |
1 Answer
1
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votes
$begingroup$
This cannot be true since the maximum of an affine function $f$ on $Bbb R^n$ is always $+infty$ unless it's a constant function.
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
$endgroup$
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
add a comment |
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$begingroup$
This cannot be true since the maximum of an affine function $f$ on $Bbb R^n$ is always $+infty$ unless it's a constant function.
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
$endgroup$
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
add a comment |
$begingroup$
This cannot be true since the maximum of an affine function $f$ on $Bbb R^n$ is always $+infty$ unless it's a constant function.
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
$endgroup$
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
add a comment |
$begingroup$
This cannot be true since the maximum of an affine function $f$ on $Bbb R^n$ is always $+infty$ unless it's a constant function.
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
$endgroup$
This cannot be true since the maximum of an affine function $f$ on $Bbb R^n$ is always $+infty$ unless it's a constant function.
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
answered Dec 8 '18 at 11:55
BigbearZzzBigbearZzz
8,58421652
8,58421652
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
add a comment |
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
$begingroup$
This is actually very helpful. My specific case is $max_nu { langle b,nu rangle + langle nu, u rangle + Vert u Vert }$. So this is only equal to $max_nu langle b,nu rangle + max_nu { langle nu, u rangle + Vert u Vert }$ if I set the condition that $langle nu, u rangle = - Vert u Vert$, a constant. Correct?
$endgroup$
– ex.nihil
Dec 8 '18 at 11:59
add a comment |
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Take $n=1$ and the functions $f(x)=x$ and $g(x) = -x$. Then the left hand side term is zero, and the right hand side term is not zero. Moreover does the maximum always exist?
$endgroup$
– Hermione
Dec 8 '18 at 11:44