How this is possible in the solution of problem 3.40 of Convex Optimization (S. Boyd, L. Vandenberghe)
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In the solution manual, the solution of problem 3.41 starts with the description of "Implicit Function Theorem" which I write below:
Suppose $F:mathbbR^ntimesmathbbR^mtomathbbR$ (where I think it should be $F:mathbbR^ntimesmathbbR^mtomathbbR^m text or n$ please clarify this also whether it should be $n$ or $m$ because this point is also critical in my view in understanding the solution) satisfies
1- $F(barmathbbu,barmathbbv)=0$
2- $F$ is continuously differentiable function and $D_mathbbvF(mathbbu,v)$ is nonsingullar in a neighborhood of $(barmathbbu,barmathbbv)$.
Then there exists a continuously diffrentiable function $phi:mathbbR^ntomathbbR^m$, that satisfies $barmathbbv=phi(barmathbbu)$ and $$F(mathbbu,phi(mathbbu))=0$$ in the neighborhood of $barmathbbu$.
After this description the solution manual says:
Applying this to $mathbbu=mathbby,mathbbv=mathbbx$ and $F(mathbbu,v)=triangledown f(mathbbx)-mathbby$, we see ....
Now, I do not understand how $F(mathbbu,v)=F(mathbby,x)=triangledown f(mathbbx)-mathbby$ is a right description of a function when $n neq m$ since $F(mathbby,x)$ has either dimensions $n$ or $m$ and $triangledown f(mathbbx)$ and $mathbby$ have different dimensions if $n neq m$ and therefore their subtraction is not a valid thing. Please clarify how to understand this in solution manual. Thanks in advance.
optimization convex-analysis convex-optimization
asked Aug 31 at 0:09
Frank Moses
1,115317
add a comment |Â
up vote
1
down vote
favorite
In the solution manual, the solution of problem 3.41 starts with the description of "Implicit Function Theorem" which I write below:
Suppose $F:mathbbR^ntimesmathbbR^mtomathbbR$ (where I think it should be $F:mathbbR^ntimesmathbbR^mtomathbbR^m text or n$ please clarify this also whether it should be $n$ or $m$ because this point is also critical in my view in understanding the solution) satisfies
1- $F(barmathbbu,barmathbbv)=0$
2- $F$ is continuously differentiable function and $D_mathbbvF(mathbbu,v)$ is nonsingullar in a neighborhood of $(barmathbbu,barmathbbv)$.
Then there exists a continuously diffrentiable function $phi:mathbbR^ntomathbbR^m$, that satisfies $barmathbbv=phi(barmathbbu)$ and $$F(mathbbu,phi(mathbbu))=0$$ in the neighborhood of $barmathbbu$.
After this description the solution manual says:
Applying this to $mathbbu=mathbby,mathbbv=mathbbx$ and $F(mathbbu,v)=triangledown f(mathbbx)-mathbby$, we see ....
Now, I do not understand how $F(mathbbu,v)=F(mathbby,x)=triangledown f(mathbbx)-mathbby$ is a right description of a function when $n neq m$ since $F(mathbby,x)$ has either dimensions $n$ or $m$ and $triangledown f(mathbbx)$ and $mathbby$ have different dimensions if $n neq m$ and therefore their subtraction is not a valid thing. Please clarify how to understand this in solution manual. Thanks in advance.
optimization convex-analysis convex-optimization
asked Aug 31 at 0:09
Frank Moses
1,115317
1
Are we looking at the same question 3.41? The conjugate of the negative entropy function?
– LinAlg
Aug 31 at 1:50
@LinAlg oh Sorry, It is actually 3.40. I corrected it. It is about the gradient and Hessian of conjugate function
– Frank Moses
Aug 31 at 1:59
1
Since $D_v F((u,v))$ is invertible, $f$ must map to $mathbbR^m$. To get some intuition, consider a simple $F$ such as $F((u,v)) = Au +Bv$. Then if $B$ is invertible, the equation $F((u,v)) = 0$ has a local (in fact global here) solution $v = -B^-1 Au$.
– copper.hat
Aug 31 at 3:36
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In the solution manual, the solution of problem 3.41 starts with the description of "Implicit Function Theorem" which I write below:
Suppose $F:mathbbR^ntimesmathbbR^mtomathbbR$ (where I think it should be $F:mathbbR^ntimesmathbbR^mtomathbbR^m text or n$ please clarify this also whether it should be $n$ or $m$ because this point is also critical in my view in understanding the solution) satisfies
1- $F(barmathbbu,barmathbbv)=0$
2- $F$ is continuously differentiable function and $D_mathbbvF(mathbbu,v)$ is nonsingullar in a neighborhood of $(barmathbbu,barmathbbv)$.
Then there exists a continuously diffrentiable function $phi:mathbbR^ntomathbbR^m$, that satisfies $barmathbbv=phi(barmathbbu)$ and $$F(mathbbu,phi(mathbbu))=0$$ in the neighborhood of $barmathbbu$.
After this description the solution manual says:
Applying this to $mathbbu=mathbby,mathbbv=mathbbx$ and $F(mathbbu,v)=triangledown f(mathbbx)-mathbby$, we see ....
Now, I do not understand how $F(mathbbu,v)=F(mathbby,x)=triangledown f(mathbbx)-mathbby$ is a right description of a function when $n neq m$ since $F(mathbby,x)$ has either dimensions $n$ or $m$ and $triangledown f(mathbbx)$ and $mathbby$ have different dimensions if $n neq m$ and therefore their subtraction is not a valid thing. Please clarify how to understand this in solution manual. Thanks in advance.
optimization convex-analysis convex-optimization
asked Aug 31 at 0:09
Frank Moses
1,115317
In the solution manual, the solution of problem 3.41 starts with the description of "Implicit Function Theorem" which I write below:
Suppose $F:mathbbR^ntimesmathbbR^mtomathbbR$ (where I think it should be $F:mathbbR^ntimesmathbbR^mtomathbbR^m text or n$ please clarify this also whether it should be $n$ or $m$ because this point is also critical in my view in understanding the solution) satisfies
1- $F(barmathbbu,barmathbbv)=0$
2- $F$ is continuously differentiable function and $D_mathbbvF(mathbbu,v)$ is nonsingullar in a neighborhood of $(barmathbbu,barmathbbv)$.
Then there exists a continuously diffrentiable function $phi:mathbbR^ntomathbbR^m$, that satisfies $barmathbbv=phi(barmathbbu)$ and $$F(mathbbu,phi(mathbbu))=0$$ in the neighborhood of $barmathbbu$.
After this description the solution manual says:
Applying this to $mathbbu=mathbby,mathbbv=mathbbx$ and $F(mathbbu,v)=triangledown f(mathbbx)-mathbby$, we see ....
Now, I do not understand how $F(mathbbu,v)=F(mathbby,x)=triangledown f(mathbbx)-mathbby$ is a right description of a function when $n neq m$ since $F(mathbby,x)$ has either dimensions $n$ or $m$ and $triangledown f(mathbbx)$ and $mathbby$ have different dimensions if $n neq m$ and therefore their subtraction is not a valid thing. Please clarify how to understand this in solution manual. Thanks in advance.
optimization convex-analysis convex-optimization
optimization convex-analysis convex-optimization
asked Aug 31 at 0:09
Frank Moses
1,115317
asked Aug 31 at 0:09
Frank Moses
1,115317
edited Aug 31 at 1:58
asked Aug 31 at 0:09
Frank Moses
1,115317
asked Aug 31 at 0:09
Frank Moses
1,115317
asked Aug 31 at 0:09
Frank Moses
1,115317
1,115317
1
Are we looking at the same question 3.41? The conjugate of the negative entropy function?
– LinAlg
Aug 31 at 1:50
@LinAlg oh Sorry, It is actually 3.40. I corrected it. It is about the gradient and Hessian of conjugate function
– Frank Moses
Aug 31 at 1:59
1
Since $D_v F((u,v))$ is invertible, $f$ must map to $mathbbR^m$. To get some intuition, consider a simple $F$ such as $F((u,v)) = Au +Bv$. Then if $B$ is invertible, the equation $F((u,v)) = 0$ has a local (in fact global here) solution $v = -B^-1 Au$.
– copper.hat
Aug 31 at 3:36
add a comment |Â
1
Are we looking at the same question 3.41? The conjugate of the negative entropy function?
– LinAlg
Aug 31 at 1:50
@LinAlg oh Sorry, It is actually 3.40. I corrected it. It is about the gradient and Hessian of conjugate function
– Frank Moses
Aug 31 at 1:59
1
Since $D_v F((u,v))$ is invertible, $f$ must map to $mathbbR^m$. To get some intuition, consider a simple $F$ such as $F((u,v)) = Au +Bv$. Then if $B$ is invertible, the equation $F((u,v)) = 0$ has a local (in fact global here) solution $v = -B^-1 Au$.
– copper.hat
Aug 31 at 3:36
1
1
Are we looking at the same question 3.41? The conjugate of the negative entropy function?
– LinAlg
Aug 31 at 1:50
Are we looking at the same question 3.41? The conjugate of the negative entropy function?
– LinAlg
Aug 31 at 1:50
@LinAlg oh Sorry, It is actually 3.40. I corrected it. It is about the gradient and Hessian of conjugate function
– Frank Moses
Aug 31 at 1:59
@LinAlg oh Sorry, It is actually 3.40. I corrected it. It is about the gradient and Hessian of conjugate function
– Frank Moses
Aug 31 at 1:59
1
1
Since $D_v F((u,v))$ is invertible, $f$ must map to $mathbbR^m$. To get some intuition, consider a simple $F$ such as $F((u,v)) = Au +Bv$. Then if $B$ is invertible, the equation $F((u,v)) = 0$ has a local (in fact global here) solution $v = -B^-1 Au$.
– copper.hat
Aug 31 at 3:36
Since $D_v F((u,v))$ is invertible, $f$ must map to $mathbbR^m$. To get some intuition, consider a simple $F$ such as $F((u,v)) = Au +Bv$. Then if $B$ is invertible, the equation $F((u,v)) = 0$ has a local (in fact global here) solution $v = -B^-1 Au$.
– copper.hat
Aug 31 at 3:36
add a comment |Â
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1
Are we looking at the same question 3.41? The conjugate of the negative entropy function?
– LinAlg
Aug 31 at 1:50
@LinAlg oh Sorry, It is actually 3.40. I corrected it. It is about the gradient and Hessian of conjugate function
– Frank Moses
Aug 31 at 1:59
1
Since $D_v F((u,v))$ is invertible, $f$ must map to $mathbbR^m$. To get some intuition, consider a simple $F$ such as $F((u,v)) = Au +Bv$. Then if $B$ is invertible, the equation $F((u,v)) = 0$ has a local (in fact global here) solution $v = -B^-1 Au$.
– copper.hat
Aug 31 at 3:36