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Sum of convex functions
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Let $f: R rightarrow R$ be a convex function. Define the function $g$ to be the sum of $f(x)$ taking on different values, i.e. $g(1,2)=f(1)+f(2)$. Does $g$ possess any interesting/special properties other than the fact that it is also convex? Suggestions or references are greatly appreciated.
calculus convex-optimization
edited Jul 27 at 16:30
Cornman
2,37021027
asked Jul 27 at 16:17
David D.
32
 |Â
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up vote
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down vote
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Let $f: R rightarrow R$ be a convex function. Define the function $g$ to be the sum of $f(x)$ taking on different values, i.e. $g(1,2)=f(1)+f(2)$. Does $g$ possess any interesting/special properties other than the fact that it is also convex? Suggestions or references are greatly appreciated.
calculus convex-optimization
edited Jul 27 at 16:30
Cornman
2,37021027
asked Jul 27 at 16:17
David D.
32
2
It is really not clear what you are asking.
– copper.hat
Jul 27 at 16:19
The way you write it, $g$ appears to be a constant, not a function.
– Kusma
Jul 27 at 16:21
In your post $g$ does not seem to be a function, but rather a series (so, if convergent, which is not clear, just one value). So saying "the fact that it is also convex" seems out of place here.
– hardmath
Jul 27 at 16:23
Apart from the difficulty with $g$ can you please tell me what unrealized means? I searched google "unrealized convex function" and none of the results contains unrealized.
– Aritra
Jul 27 at 16:25
Let's say we have another convex function $r$, and define $h(1,1)=f(1)+r(1)$. And let's specify $g(1,1)=f(1)+f(1)$, does $g$ possess some interesting property that $h$ does not have since it sums over the same function?
– David D.
Jul 27 at 16:25
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $f: R rightarrow R$ be a convex function. Define the function $g$ to be the sum of $f(x)$ taking on different values, i.e. $g(1,2)=f(1)+f(2)$. Does $g$ possess any interesting/special properties other than the fact that it is also convex? Suggestions or references are greatly appreciated.
calculus convex-optimization
edited Jul 27 at 16:30
Cornman
2,37021027
asked Jul 27 at 16:17
David D.
32
Let $f: R rightarrow R$ be a convex function. Define the function $g$ to be the sum of $f(x)$ taking on different values, i.e. $g(1,2)=f(1)+f(2)$. Does $g$ possess any interesting/special properties other than the fact that it is also convex? Suggestions or references are greatly appreciated.
calculus convex-optimization
edited Jul 27 at 16:30
Cornman
2,37021027
asked Jul 27 at 16:17
David D.
32
edited Jul 27 at 16:30
Cornman
2,37021027
edited Jul 27 at 16:30
Cornman
2,37021027
edited Jul 27 at 16:30
Cornman
2,37021027
2,37021027
asked Jul 27 at 16:17
David D.
32
asked Jul 27 at 16:17
David D.
32
asked Jul 27 at 16:17
David D.
32
32
2
It is really not clear what you are asking.
– copper.hat
Jul 27 at 16:19
The way you write it, $g$ appears to be a constant, not a function.
– Kusma
Jul 27 at 16:21
In your post $g$ does not seem to be a function, but rather a series (so, if convergent, which is not clear, just one value). So saying "the fact that it is also convex" seems out of place here.
– hardmath
Jul 27 at 16:23
Apart from the difficulty with $g$ can you please tell me what unrealized means? I searched google "unrealized convex function" and none of the results contains unrealized.
– Aritra
Jul 27 at 16:25
Let's say we have another convex function $r$, and define $h(1,1)=f(1)+r(1)$. And let's specify $g(1,1)=f(1)+f(1)$, does $g$ possess some interesting property that $h$ does not have since it sums over the same function?
– David D.
Jul 27 at 16:25
 |Â
show 2 more comments
2
It is really not clear what you are asking.
– copper.hat
Jul 27 at 16:19
The way you write it, $g$ appears to be a constant, not a function.
– Kusma
Jul 27 at 16:21
In your post $g$ does not seem to be a function, but rather a series (so, if convergent, which is not clear, just one value). So saying "the fact that it is also convex" seems out of place here.
– hardmath
Jul 27 at 16:23
Apart from the difficulty with $g$ can you please tell me what unrealized means? I searched google "unrealized convex function" and none of the results contains unrealized.
– Aritra
Jul 27 at 16:25
Let's say we have another convex function $r$, and define $h(1,1)=f(1)+r(1)$. And let's specify $g(1,1)=f(1)+f(1)$, does $g$ possess some interesting property that $h$ does not have since it sums over the same function?
– David D.
Jul 27 at 16:25
2
2
It is really not clear what you are asking.
– copper.hat
Jul 27 at 16:19
It is really not clear what you are asking.
– copper.hat
Jul 27 at 16:19
The way you write it, $g$ appears to be a constant, not a function.
– Kusma
Jul 27 at 16:21
The way you write it, $g$ appears to be a constant, not a function.
– Kusma
Jul 27 at 16:21
In your post $g$ does not seem to be a function, but rather a series (so, if convergent, which is not clear, just one value). So saying "the fact that it is also convex" seems out of place here.
– hardmath
Jul 27 at 16:23
In your post $g$ does not seem to be a function, but rather a series (so, if convergent, which is not clear, just one value). So saying "the fact that it is also convex" seems out of place here.
– hardmath
Jul 27 at 16:23
Apart from the difficulty with $g$ can you please tell me what unrealized means? I searched google "unrealized convex function" and none of the results contains unrealized.
– Aritra
Jul 27 at 16:25
Apart from the difficulty with $g$ can you please tell me what unrealized means? I searched google "unrealized convex function" and none of the results contains unrealized.
– Aritra
Jul 27 at 16:25
Let's say we have another convex function $r$, and define $h(1,1)=f(1)+r(1)$. And let's specify $g(1,1)=f(1)+f(1)$, does $g$ possess some interesting property that $h$ does not have since it sums over the same function?
– David D.
Jul 27 at 16:25
Let's say we have another convex function $r$, and define $h(1,1)=f(1)+r(1)$. And let's specify $g(1,1)=f(1)+f(1)$, does $g$ possess some interesting property that $h$ does not have since it sums over the same function?
– David D.
Jul 27 at 16:25
 |Â
show 2 more comments
1 Answer
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oldest
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0
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accepted
There might be many interesting properties, depending on what you are looking for. Here are some examples:
- The problem $min_x_1, dots, x_n g(x_1, dots, x_n)$ is separable. Can be minimized w.r.t $x$ and $y$ separately.
- The Hessian matrix of $g$, if $f$ are all twice differentiable, is diagonal. It might help if you are minimizing $g(x) + h(x)$, where the Hessian of $h$ has low rank, and you are using Newton's method. Another example where it might help is when you are minimizing $g(x)$ subject to $A x + b$, and $A$ has low rank.
- Using separability, it is also easy, in many cases, to derive a dual. For example, look at minimizing $g(x)$ subject to $|x|_2 leq 1$.
If you tell us what exactly you are looking for, we might be able to find more `interesting' properties.
answered Jul 28 at 12:55
Alex Shtof
532514
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
There might be many interesting properties, depending on what you are looking for. Here are some examples:
- The problem $min_x_1, dots, x_n g(x_1, dots, x_n)$ is separable. Can be minimized w.r.t $x$ and $y$ separately.
- The Hessian matrix of $g$, if $f$ are all twice differentiable, is diagonal. It might help if you are minimizing $g(x) + h(x)$, where the Hessian of $h$ has low rank, and you are using Newton's method. Another example where it might help is when you are minimizing $g(x)$ subject to $A x + b$, and $A$ has low rank.
- Using separability, it is also easy, in many cases, to derive a dual. For example, look at minimizing $g(x)$ subject to $|x|_2 leq 1$.
If you tell us what exactly you are looking for, we might be able to find more `interesting' properties.
answered Jul 28 at 12:55
Alex Shtof
532514
add a comment |Â
up vote
0
down vote
accepted
There might be many interesting properties, depending on what you are looking for. Here are some examples:
- The problem $min_x_1, dots, x_n g(x_1, dots, x_n)$ is separable. Can be minimized w.r.t $x$ and $y$ separately.
- The Hessian matrix of $g$, if $f$ are all twice differentiable, is diagonal. It might help if you are minimizing $g(x) + h(x)$, where the Hessian of $h$ has low rank, and you are using Newton's method. Another example where it might help is when you are minimizing $g(x)$ subject to $A x + b$, and $A$ has low rank.
- Using separability, it is also easy, in many cases, to derive a dual. For example, look at minimizing $g(x)$ subject to $|x|_2 leq 1$.
If you tell us what exactly you are looking for, we might be able to find more `interesting' properties.
answered Jul 28 at 12:55
Alex Shtof
532514
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
There might be many interesting properties, depending on what you are looking for. Here are some examples:
- The problem $min_x_1, dots, x_n g(x_1, dots, x_n)$ is separable. Can be minimized w.r.t $x$ and $y$ separately.
- The Hessian matrix of $g$, if $f$ are all twice differentiable, is diagonal. It might help if you are minimizing $g(x) + h(x)$, where the Hessian of $h$ has low rank, and you are using Newton's method. Another example where it might help is when you are minimizing $g(x)$ subject to $A x + b$, and $A$ has low rank.
- Using separability, it is also easy, in many cases, to derive a dual. For example, look at minimizing $g(x)$ subject to $|x|_2 leq 1$.
If you tell us what exactly you are looking for, we might be able to find more `interesting' properties.
answered Jul 28 at 12:55
Alex Shtof
532514
There might be many interesting properties, depending on what you are looking for. Here are some examples:
- The problem $min_x_1, dots, x_n g(x_1, dots, x_n)$ is separable. Can be minimized w.r.t $x$ and $y$ separately.
- The Hessian matrix of $g$, if $f$ are all twice differentiable, is diagonal. It might help if you are minimizing $g(x) + h(x)$, where the Hessian of $h$ has low rank, and you are using Newton's method. Another example where it might help is when you are minimizing $g(x)$ subject to $A x + b$, and $A$ has low rank.
- Using separability, it is also easy, in many cases, to derive a dual. For example, look at minimizing $g(x)$ subject to $|x|_2 leq 1$.
If you tell us what exactly you are looking for, we might be able to find more `interesting' properties.
answered Jul 28 at 12:55
Alex Shtof
532514
answered Jul 28 at 12:55
Alex Shtof
532514
answered Jul 28 at 12:55
Alex Shtof
532514
answered Jul 28 at 12:55
Alex Shtof
532514
532514
add a comment |Â
add a comment |Â
Â
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2
It is really not clear what you are asking.
– copper.hat
Jul 27 at 16:19
The way you write it, $g$ appears to be a constant, not a function.
– Kusma
Jul 27 at 16:21
In your post $g$ does not seem to be a function, but rather a series (so, if convergent, which is not clear, just one value). So saying "the fact that it is also convex" seems out of place here.
– hardmath
Jul 27 at 16:23
Apart from the difficulty with $g$ can you please tell me what unrealized means? I searched google "unrealized convex function" and none of the results contains unrealized.
– Aritra
Jul 27 at 16:25
Let's say we have another convex function $r$, and define $h(1,1)=f(1)+r(1)$. And let's specify $g(1,1)=f(1)+f(1)$, does $g$ possess some interesting property that $h$ does not have since it sums over the same function?
– David D.
Jul 27 at 16:25