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Little bit of convex analysis on a Hilbert space

  1. Feb 2, 2008 #1

    quasar987

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    [SOLVED] Little bit of convex analysis on a Hilbert space

    1. The problem statement, all variables and given/known data
    Let H be a Hilbert space over R and f:H-->R a function that is bounded below, convex and lower semi continuous (i.e., f(x) [tex]\leq \liminf_{y\rightarrow x}f(y)[/tex] for all x in H).

    (a) For all x in H and lambda>0, show that there exists a unique x_lambda in H such that

    [tex]\lambda f(x_{\lambda})+||x-x_{\lambda}||^2=\min_{y\in H}(\lambda f(y)+||x-y||^2)[/tex]

    3. The attempt at a solution

    Let {y_n} be a sequence such that [tex]\lambda f(y_{n})+||x-y_{n}||^2\rightarrow \inf_{y\in H}(\lambda f(y)+||x-y||^2)[/tex]. If I could show that {y_n} is Cauchy, then by continuity of the norm and lower semi continuity of f, I could conclude that the limit of {y_n} minimizes [tex]\lambda f(y)+||x-y||^2)[/tex].

    But do I have enough information to achieve that?
     
    Last edited: Feb 3, 2008
  2. jcsd
  3. Feb 3, 2008 #2

    quasar987

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    What other methods are there to show a function attains it's minimum?

    I know a continuous function on a compact set attains its min. But this is not useful here.

    I know the distance of a point to a closed convex set in a hilbert space is minimized by a point of the convex. But this is not useful here either as far as i can see.

    ...
     
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