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 &phi; : H &rarr; R &cup; {+&infin;} be a proper, convex and lower semi-continuous extended real-valued functional on H. Let A stand for &part;&phi;, the subderivative of &phi;; for &alpha; &gt; 0 let J&alpha; denote the resolvent:


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

and let A&alpha; denote the Yosida approximation to A:


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

For each &alpha; &gt; 0 and x &isin; 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 &phi;&alpha; is convex and Fréchet differentiable with derivative d&phi;&alpha; = A&alpha;. Also, for each x &isin; H (pointwise), &phi;&alpha;(x) converges upwards to &phi;(x) as &alpha; &rarr; 0.