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:

[math]\displaystyle{ J_{\alpha} = (\mathrm{id} + \alpha A)^{-1}; }[/math]

and let A_α denote the Yosida approximation to A:

[math]\displaystyle{ A_{\alpha} = \frac1{\alpha} ( \mathrm{id} - J_{\alpha} ). }[/math]

For each α > 0 and x ∈ H, let

[math]\displaystyle{ \varphi_{\alpha} (x) = \inf_{y \in H} \frac1{2 \alpha} \| y - x \|^{2} + \varphi (y). }[/math]

Then

[math]\displaystyle{ \varphi_{\alpha} (x) = \frac{\alpha}{2} \| A_{\alpha} x \|^{2} + \varphi (J_{\alpha} (x)) }[/math]

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

• Showalter, Ralph E. (1997). Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. pp. 162–163. ISBN 0-8218-0500-2.  MR1422252 (Proposition IV.1.8)