A condition for 'similarity' of subgradients of convex proper functions











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Motivated by the answer in Under which conditions does small uniform norm imply 'similarity' of subgradients. I have the following question.



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)}{inf}||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.



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.



Some references: A similar question was asked here [Hausdorff Distance between Subdifferential sets. The problem is that the theory defines more general types of convergences and analyzes the convergence of the subdifferentials in that framework. I have almost no background in functional analysis so I would really just like to know under which conditions my type of 'convergence'; holds, as defined above.






functional-analysis convex-analysis convex-optimization







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    up vote
    0
    down vote

    favorite












    Motivated by the answer in Under which conditions does small uniform norm imply 'similarity' of subgradients. I have the following question.



    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)}{inf}||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.



    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.



    Some references: A similar question was asked here [Hausdorff Distance between Subdifferential sets. The problem is that the theory defines more general types of convergences and analyzes the convergence of the subdifferentials in that framework. I have almost no background in functional analysis so I would really just like to know under which conditions my type of 'convergence'; holds, as defined above.






    functional-analysis convex-analysis convex-optimization







    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Motivated by the answer in Under which conditions does small uniform norm imply 'similarity' of subgradients. I have the following question.



      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)}{inf}||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.



      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.



      Some references: A similar question was asked here [Hausdorff Distance between Subdifferential sets. The problem is that the theory defines more general types of convergences and analyzes the convergence of the subdifferentials in that framework. I have almost no background in functional analysis so I would really just like to know under which conditions my type of 'convergence'; holds, as defined above.






      functional-analysis convex-analysis convex-optimization







      share|cite|improve this question













      Motivated by the answer in Under which conditions does small uniform norm imply 'similarity' of subgradients. I have the following question.



      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)}{inf}||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.



      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.



      Some references: A similar question was asked here [Hausdorff Distance between Subdifferential sets. The problem is that the theory defines more general types of convergences and analyzes the convergence of the subdifferentials in that framework. I have almost no background in functional analysis so I would really just like to know under which conditions my type of 'convergence'; holds, as defined above.






      functional-analysis convex-analysis convex-optimization




      functional-analysis convex-analysis convex-optimization






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      asked 15 hours ago









      sigmatau

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