Moreau's theorem

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 J*α* denote the resolvent:

:J_{\alpha} = (\mathrm{id} + \alpha A)^{-1};

and let A*α* denote the Yosida approximation to A:

:A_{\alpha} = \frac1{\alpha} ( \mathrm{id} - J_{\alpha} ).

For each α 0 and x ∈ H, let

:\varphi_{\alpha} (x) = \inf_{y \in H} \frac1{2 \alpha} | y - x |^{2} + \varphi (y).

Then

:\varphi_{\alpha} (x) = \frac{\alpha}{2} | A_{\alpha} x |^{2} + \varphi (J_{\alpha} (x))

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

References

• {{cite book