Is $ max_{xinmathbb{R}^n} { f(x)+g(x) } = max_{xinmathbb{R}^n} f(x)+max_{xinmathbb{R}^n} g(x) $ if $f$ and...












0












$begingroup$


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.










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$endgroup$








  • 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
















0












$begingroup$


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.










share|cite|improve this question









$endgroup$








  • 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














0












0








0





$begingroup$


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.










share|cite|improve this question









$endgroup$




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






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share|cite|improve this question











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asked Dec 8 '18 at 11:27









ex.nihilex.nihil

215111




215111








  • 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




    $begingroup$
    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










1 Answer
1






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oldest

votes


















1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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













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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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


















1












$begingroup$

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.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    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
















1












1








1





$begingroup$

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.






share|cite|improve this answer









$endgroup$



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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 8 '18 at 11:55









BigbearZzzBigbearZzz

8,49221652




8,49221652












  • $begingroup$
    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






$begingroup$
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




















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