Little bit of convex analysis on a Hilbert space

In summary, the problem asks to show that for a bounded below, convex, and lower semi-continuous function f on a Hilbert space H, there exists a unique x_lambda in H such that the expression \lambda f(x_{\lambda})+||x-x_{\lambda}||^2 is minimized. The attempt at a solution involves showing that a sequence of points {y_n} converges to the minimum, but it is unclear if there is enough information to prove this. Other methods, such as using the fact that a continuous function on a compact set attains its minimum or that the distance of a point to a closed convex set in a Hilbert space is minimized by a point of the convex, do not seem applicable
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quasar987
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[SOLVED] Little bit of convex analysis on a Hilbert space

Homework Statement


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]

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?
 
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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.

...
 

Related to Little bit of convex analysis on a Hilbert space

1. What is a convex set?

A convex set is a set where any line segment connecting two points in the set lies entirely within the set. In other words, it is a set where every point within the set can be reached by a straight line drawn between any two points in the set.

2. What is a convex function?

A convex function is a function where the line segment connecting any two points on the graph of the function lies above or on the graph. In other words, the function is always "curving upwards" and never has any "dips".

3. What is a Hilbert space?

A Hilbert space is a mathematical concept that is used to describe an infinite-dimensional vector space that has a well-defined notion of distance and angle. It is a generalization of Euclidean space to an infinite number of dimensions and is commonly used in functional analysis and other areas of mathematics.

4. How is convex analysis used in optimization?

Convex analysis is used in optimization to find the minimum or maximum value of a convex function. This is done by finding the point where the gradient of the function is equal to zero, which is a necessary condition for a local minimum or maximum.

5. What are some real-world applications of convex analysis on a Hilbert space?

Convex analysis on a Hilbert space has many real-world applications, such as in economics for modeling consumer preferences, in signal processing for image reconstruction, and in machine learning for optimization problems. It is also used in physics and engineering for modeling physical systems and in statistics for data analysis and prediction.

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