Moreau's Theorem

Moreau's theorem is a result in convex analysis . It shows that sufficiently good 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 .

Statement of the theorem

Let be $\varphi :H\to \mathbb {R} \cup \{+\infty \}$ ${\ displaystyle \ varphi: H \ to \ mathbb {R} \ cup \ {+ \ infty \}}$ is a proper convex lower semicontinuous functional in the Hilbert space H with values in the extended number line . Let A mean $\partial \varphi$ ${\ displaystyle \ partial \ varphi}$ subdifferential $\varphi$ ${\ displaystyle \ varphi}$ . For $\alpha >0$ ${\ displaystyle \ alpha> 0}$ let be $J_{\alpha }$ ${\ displaystyle J _ {\ alpha}}$ means resolvent:

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

{\ displaystyle J _ {\ alpha} = (\ mathrm {id} + \ alpha A) ^ {- 1};}

but $A_{\alpha }$ ${\ displaystyle A _ {\ alpha}}$ means the approximation of Yosida for A :

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

{\ displaystyle A _ {\ alpha} = {\ frac {1} {\ alpha}} (\ mathrm {id} -J _ {\ alpha}).}

For each $\alpha >0$ ${\ displaystyle \ alpha> 0}$ and $x\in H$ ${\ displaystyle x \ in H}$ put

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

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

Then

\varphi _{\alpha }(x)={\frac {\alpha }{2}}\|A_{\alpha }x\|^{2}+\varphi (J_{\alpha }(x))

{\ displaystyle \ varphi _ {\ alpha} (x) = {\ frac {\ alpha} {2}} \ | A _ {\ alpha} x \ | ^ {2} + \ varphi (J _ {\ alpha} (x ))}

,

$\varphi _{\alpha }$ ${\ displaystyle \ varphi _ {\ alpha}}$ convex and Frechet differentiable with derivative $d\varphi +\alpha =A_{\alpha }$ ${\ displaystyle d \ varphi + \ alpha = A _ {\ alpha}}$ . Also for any $x\in H$ ${\ displaystyle x \ in H}$ (dotwise) $\varphi _{\alpha }(x)$ ${\ displaystyle \ varphi _ {\ alpha} (x)}$ converges to $\varphi (x)$ ${\ displaystyle \ varphi (x)}$ at $\alpha \to 0$ ${\ displaystyle \ alpha \ to 0}$ .

Literature

Ralph E. Showalter. Monotone operators in Banach space and nonlinear partial differential equations. - Providence, RI: American Mathematical Society, 1997. - S. 162–163. - (Mathematical Surveys and Monographs 49). - ISBN 0-8218-0500-2 . MR : 1422252 (Proposition IV.1.8)

Moreau's Theorem

Statement of the theorem

Literature

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