Moreau's theorem

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.

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:

$J_{\alpha }=(\mathrm{i}\mathrm{d}+\alpha A{)}^{-1};$

and let A_α denote the Yosida approximation to A:

$A_{\alpha }=\frac{1}{\alpha }(\mathrm{i}\mathrm{d}-J_{\alpha }).$

For each α > 0 and x ∈ H, let

${\phi }_{\alpha }(x)=\underset{y\in H}{\inf }\frac{1}{2\alpha }\Vert y-x{\Vert }^2+\phi (y).$

Then

${\phi }_{\alpha }(x)=\frac{\alpha }{2}\Vert A_{\alpha }x{\Vert }^2+\phi (J_{\alpha }(x))$

and φ_α is convex and Fréchet differentiable with derivative dφ_α = A_α. Also, for each x ∈ H (pointwise), φ_α(x) converges upwards to φ(x) as α → 0.

References

• Template:Cite book Template:MathSciNet (Proposition IV.1.8)