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:

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

and let A_α denote the Yosida approximation to A:

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

For each α > 0 and x ∈ H, let

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

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 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)