Moreau's theorem

In mathematics, Moreau's theorem is a result in convex analysis. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms of the resolvent operator.

Statement of the theorem

Let H be a Hilbert space and let φ : H  R  {+} be a proper, convex and lower semi-continuous extended real-valued functional on H. Let A stand for φ, the subderivative of φ; for α > 0 let Jα denote the resolvent:

and let Aα denote the Yosida approximation to A:

For each α > 0 and x  H, let

Then

and φα is convex and Fréchet differentiable with derivative dφα = Aα. Also, for each x  H (pointwise), φα(x) converges upwards to φ(x) as α  0.

References

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