Under which conditions does small uniform norm imply 'similarity' of subgradients.
Let $mathcal{S}$ be the set of proper convex functions functions from $X$ to $mathbb{R}$, where $X$ is a open and convex subset of $mathbb{R}^{n}$. I was wondering under which conditions on $mathcal{S}$ we have
begin{gather}
forall epsilon>0 exists delta>0 text{ such that for } f,h in mathcal{S} , ||f-h||_{infty} < delta \
Rightarrow underset{ x in X}{sup} underset{ v in partial f(x), w in partial h(x)}{sup}||v-w||_2 <epsilon
end{gather}
Motivation: Intuitively, the fact that $||f-h||_{infty}$ is small, means that the shape of the graphs of the functions are similar and hence also their suporting hyperplanes might be similar. Of course this is just a picture that I have in mind for the 1-dimensional case.
Any help or suggestions for possible references to similar results would be great.
From where the problemm comes: I have a function $F$ that maps convex functions to elements of their subgradient at any given point. I would like to show that if the mapped functions are similar, i.e. $||f-h||_{infty}$ is sufficiently small, then we can say that $||F(h)-F(f)||_{2}$ is small.
functional-analysis convex-analysis
asked Nov 18 at 14:06
sigmatau
1,7551924
add a comment |
Let $mathcal{S}$ be the set of proper convex functions functions from $X$ to $mathbb{R}$, where $X$ is a open and convex subset of $mathbb{R}^{n}$. I was wondering under which conditions on $mathcal{S}$ we have
begin{gather}
forall epsilon>0 exists delta>0 text{ such that for } f,h in mathcal{S} , ||f-h||_{infty} < delta \
Rightarrow underset{ x in X}{sup} underset{ v in partial f(x), w in partial h(x)}{sup}||v-w||_2 <epsilon
end{gather}
Motivation: Intuitively, the fact that $||f-h||_{infty}$ is small, means that the shape of the graphs of the functions are similar and hence also their suporting hyperplanes might be similar. Of course this is just a picture that I have in mind for the 1-dimensional case.
Any help or suggestions for possible references to similar results would be great.
From where the problemm comes: I have a function $F$ that maps convex functions to elements of their subgradient at any given point. I would like to show that if the mapped functions are similar, i.e. $||f-h||_{infty}$ is sufficiently small, then we can say that $||F(h)-F(f)||_{2}$ is small.
functional-analysis convex-analysis
asked Nov 18 at 14:06
sigmatau
1,7551924
add a comment |
Let $mathcal{S}$ be the set of proper convex functions functions from $X$ to $mathbb{R}$, where $X$ is a open and convex subset of $mathbb{R}^{n}$. I was wondering under which conditions on $mathcal{S}$ we have
begin{gather}
forall epsilon>0 exists delta>0 text{ such that for } f,h in mathcal{S} , ||f-h||_{infty} < delta \
Rightarrow underset{ x in X}{sup} underset{ v in partial f(x), w in partial h(x)}{sup}||v-w||_2 <epsilon
end{gather}
Motivation: Intuitively, the fact that $||f-h||_{infty}$ is small, means that the shape of the graphs of the functions are similar and hence also their suporting hyperplanes might be similar. Of course this is just a picture that I have in mind for the 1-dimensional case.
Any help or suggestions for possible references to similar results would be great.
From where the problemm comes: I have a function $F$ that maps convex functions to elements of their subgradient at any given point. I would like to show that if the mapped functions are similar, i.e. $||f-h||_{infty}$ is sufficiently small, then we can say that $||F(h)-F(f)||_{2}$ is small.
functional-analysis convex-analysis
asked Nov 18 at 14:06
sigmatau
1,7551924
Let $mathcal{S}$ be the set of proper convex functions functions from $X$ to $mathbb{R}$, where $X$ is a open and convex subset of $mathbb{R}^{n}$. I was wondering under which conditions on $mathcal{S}$ we have
begin{gather}
forall epsilon>0 exists delta>0 text{ such that for } f,h in mathcal{S} , ||f-h||_{infty} < delta \
Rightarrow underset{ x in X}{sup} underset{ v in partial f(x), w in partial h(x)}{sup}||v-w||_2 <epsilon
end{gather}
Motivation: Intuitively, the fact that $||f-h||_{infty}$ is small, means that the shape of the graphs of the functions are similar and hence also their suporting hyperplanes might be similar. Of course this is just a picture that I have in mind for the 1-dimensional case.
Any help or suggestions for possible references to similar results would be great.
From where the problemm comes: I have a function $F$ that maps convex functions to elements of their subgradient at any given point. I would like to show that if the mapped functions are similar, i.e. $||f-h||_{infty}$ is sufficiently small, then we can say that $||F(h)-F(f)||_{2}$ is small.
functional-analysis convex-analysis
functional-analysis convex-analysis
asked Nov 18 at 14:06
sigmatau
1,7551924
asked Nov 18 at 14:06
sigmatau
1,7551924
edited Nov 18 at 16:02
asked Nov 18 at 14:06
sigmatau
1,7551924
asked Nov 18 at 14:06
sigmatau
1,7551924
asked Nov 18 at 14:06
sigmatau
1,7551924
1,7551924
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Assuming any conditions on $S$ if $S$ still contains at least one non-smooth function then your claim does not hold.
Proof: Let $f in S$ such that $f$ is not differentiable at $x_0 in X.$ Hence there exist $v , w in partial f(x_0) $ such that $ v neq w $. Set $ epsilon = | v -w | $
Then observe that for any positive delta , your statement does not holds for two functions $f,~ h$ where $ f = h $ and $v in partial f(x_0) $ and $ w in partial h(x_0) $.
P.S:
But I feel like your claim would be true if you switch the second $sup$ to $inf$.
answered Nov 18 at 18:29
Red shoes
4,706621
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
1
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003575%2funder-which-conditions-does-small-uniform-norm-imply-similarity-of-subgradient%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Assuming any conditions on $S$ if $S$ still contains at least one non-smooth function then your claim does not hold.
Proof: Let $f in S$ such that $f$ is not differentiable at $x_0 in X.$ Hence there exist $v , w in partial f(x_0) $ such that $ v neq w $. Set $ epsilon = | v -w | $
Then observe that for any positive delta , your statement does not holds for two functions $f,~ h$ where $ f = h $ and $v in partial f(x_0) $ and $ w in partial h(x_0) $.
P.S:
But I feel like your claim would be true if you switch the second $sup$ to $inf$.
answered Nov 18 at 18:29
Red shoes
4,706621
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
1
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
add a comment |
Assuming any conditions on $S$ if $S$ still contains at least one non-smooth function then your claim does not hold.
Proof: Let $f in S$ such that $f$ is not differentiable at $x_0 in X.$ Hence there exist $v , w in partial f(x_0) $ such that $ v neq w $. Set $ epsilon = | v -w | $
Then observe that for any positive delta , your statement does not holds for two functions $f,~ h$ where $ f = h $ and $v in partial f(x_0) $ and $ w in partial h(x_0) $.
P.S:
But I feel like your claim would be true if you switch the second $sup$ to $inf$.
answered Nov 18 at 18:29
Red shoes
4,706621
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
1
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
add a comment |
Assuming any conditions on $S$ if $S$ still contains at least one non-smooth function then your claim does not hold.
Proof: Let $f in S$ such that $f$ is not differentiable at $x_0 in X.$ Hence there exist $v , w in partial f(x_0) $ such that $ v neq w $. Set $ epsilon = | v -w | $
Then observe that for any positive delta , your statement does not holds for two functions $f,~ h$ where $ f = h $ and $v in partial f(x_0) $ and $ w in partial h(x_0) $.
P.S:
But I feel like your claim would be true if you switch the second $sup$ to $inf$.
answered Nov 18 at 18:29
Red shoes
4,706621
Assuming any conditions on $S$ if $S$ still contains at least one non-smooth function then your claim does not hold.
Proof: Let $f in S$ such that $f$ is not differentiable at $x_0 in X.$ Hence there exist $v , w in partial f(x_0) $ such that $ v neq w $. Set $ epsilon = | v -w | $
Then observe that for any positive delta , your statement does not holds for two functions $f,~ h$ where $ f = h $ and $v in partial f(x_0) $ and $ w in partial h(x_0) $.
P.S:
But I feel like your claim would be true if you switch the second $sup$ to $inf$.
answered Nov 18 at 18:29
Red shoes
4,706621
edited Nov 18 at 18:40
answered Nov 18 at 18:29
Red shoes
4,706621
answered Nov 18 at 18:29
Red shoes
4,706621
answered Nov 18 at 18:29
Red shoes
4,706621
4,706621
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
1
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
add a comment |
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
1
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
I found this ( math.stackexchange.com/questions/976913/…). Hence, there exists a notion of convergence of function which guarantees that the subdifferentials converge in some sense, but I have to give it a look.
– sigmatau
Nov 18 at 20:45
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
are you confident that the result holds if we switch $sup$ with $inf$. Do you have an idea how to prove that?
– sigmatau
Nov 19 at 8:33
1
1
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
@sigmatau I'm not 100 % sure about. I just guessed , that's what my intuition tells me. I need to pick up pen I spend time to prove it (or reject it). I suggest you write a separate question for that.
– Red shoes
Nov 19 at 17:17
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
Already did it today math.stackexchange.com/questions/3004735/… . Thank you for your help. .
– sigmatau
Nov 19 at 17:22
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003575%2funder-which-conditions-does-small-uniform-norm-imply-similarity-of-subgradient%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
