General question about Convexity of Multivariate Functions (Convexity in only some (i.e. not all) of the...
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Let a general multivariate function of $n$ variables, $f : mathbb{R}^n to mathbb{R}$ say, be given.
Suppose we want to prove that $f$ is convex (concave) in just some of the $n$ variables, not all.
In general, if one wants to prove that a function is convex (concave) in all variables, one should use the Hessian of the function.
So, I suspect that if one wants to prove that a function is convex (concave) in only some of the variables, one should again use a matrix of second-order partial derivatives of the function but in this case ONLY with respect to the variables for one wishes to confirm convexity (concavity).
Is my supposition correct?
In my case I am dealing with a function $f : mathbb{R}^4 to mathbb{R}, f(t,x,u,p) = 1 + x - u^2 + p(x + u)$ where I (only) want to show that it is concave in $(x,u)$.
convex-analysis
asked Dec 16 '18 at 16:48
AnnaAnna
7619
$endgroup$
add a comment |
$begingroup$
Let a general multivariate function of $n$ variables, $f : mathbb{R}^n to mathbb{R}$ say, be given.
Suppose we want to prove that $f$ is convex (concave) in just some of the $n$ variables, not all.
In general, if one wants to prove that a function is convex (concave) in all variables, one should use the Hessian of the function.
So, I suspect that if one wants to prove that a function is convex (concave) in only some of the variables, one should again use a matrix of second-order partial derivatives of the function but in this case ONLY with respect to the variables for one wishes to confirm convexity (concavity).
Is my supposition correct?
In my case I am dealing with a function $f : mathbb{R}^4 to mathbb{R}, f(t,x,u,p) = 1 + x - u^2 + p(x + u)$ where I (only) want to show that it is concave in $(x,u)$.
convex-analysis
asked Dec 16 '18 at 16:48
AnnaAnna
7619
$endgroup$
1
$begingroup$
Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture.
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– Giuseppe Negro
Dec 16 '18 at 17:02
$begingroup$
@GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that).
$endgroup$
– Anna
Dec 16 '18 at 17:07
add a comment |
$begingroup$
Let a general multivariate function of $n$ variables, $f : mathbb{R}^n to mathbb{R}$ say, be given.
Suppose we want to prove that $f$ is convex (concave) in just some of the $n$ variables, not all.
In general, if one wants to prove that a function is convex (concave) in all variables, one should use the Hessian of the function.
So, I suspect that if one wants to prove that a function is convex (concave) in only some of the variables, one should again use a matrix of second-order partial derivatives of the function but in this case ONLY with respect to the variables for one wishes to confirm convexity (concavity).
Is my supposition correct?
In my case I am dealing with a function $f : mathbb{R}^4 to mathbb{R}, f(t,x,u,p) = 1 + x - u^2 + p(x + u)$ where I (only) want to show that it is concave in $(x,u)$.
convex-analysis
asked Dec 16 '18 at 16:48
AnnaAnna
7619
$endgroup$
Let a general multivariate function of $n$ variables, $f : mathbb{R}^n to mathbb{R}$ say, be given.
Suppose we want to prove that $f$ is convex (concave) in just some of the $n$ variables, not all.
In general, if one wants to prove that a function is convex (concave) in all variables, one should use the Hessian of the function.
So, I suspect that if one wants to prove that a function is convex (concave) in only some of the variables, one should again use a matrix of second-order partial derivatives of the function but in this case ONLY with respect to the variables for one wishes to confirm convexity (concavity).
Is my supposition correct?
In my case I am dealing with a function $f : mathbb{R}^4 to mathbb{R}, f(t,x,u,p) = 1 + x - u^2 + p(x + u)$ where I (only) want to show that it is concave in $(x,u)$.
convex-analysis
convex-analysis
asked Dec 16 '18 at 16:48
AnnaAnna
7619
asked Dec 16 '18 at 16:48
AnnaAnna
7619
edited Dec 16 '18 at 16:55
Anna
asked Dec 16 '18 at 16:48
AnnaAnna
7619
asked Dec 16 '18 at 16:48
AnnaAnna
7619
asked Dec 16 '18 at 16:48
AnnaAnna
7619
7619
1
$begingroup$
Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture.
$endgroup$
– Giuseppe Negro
Dec 16 '18 at 17:02
$begingroup$
@GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that).
$endgroup$
– Anna
Dec 16 '18 at 17:07
add a comment |
1
$begingroup$
Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture.
$endgroup$
– Giuseppe Negro
Dec 16 '18 at 17:02
$begingroup$
@GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that).
$endgroup$
– Anna
Dec 16 '18 at 17:07
1
1
$begingroup$
Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture.
$endgroup$
– Giuseppe Negro
Dec 16 '18 at 17:02
$begingroup$
Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture.
$endgroup$
– Giuseppe Negro
Dec 16 '18 at 17:02
$begingroup$
@GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that).
$endgroup$
– Anna
Dec 16 '18 at 17:07
$begingroup$
@GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that).
$endgroup$
– Anna
Dec 16 '18 at 17:07
add a comment |
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$begingroup$
Just fix $t, p$ and consider the function $$g(x, u)=f(t,x,u,p).$$ You want to prove that $g$ is concave on $mathbb R^2$. Then, study its second derivatives etc etc... This amounts to considering only the derivatives in $x, u$, as you conjecture.
$endgroup$
– Giuseppe Negro
Dec 16 '18 at 17:02
$begingroup$
@GiuseppeNegro Thank you, that is precisely that I wanted to know (whether you could do that).
$endgroup$
– Anna
Dec 16 '18 at 17:07