In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. 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.
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:
J\alpha=(id+\alphaA)-1;
and let Aα denote the Yosida approximation to A:
A\alpha=
| 1{\alpha} | |
| ( |
id-J\alpha).
For each α > 0 and x ∈ H, let
\varphi\alpha(x)=infy
| 1{2 | |
| \alpha} |
\|y-x\|2+\varphi(y).
Then
\varphi\alpha(x)=
| \alpha | |
| 2 |
\|A\alphax\|2+\varphi(J\alpha(x))
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.