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:

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

For each α > 0 and x ∈ 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 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.

| last = Showalter

| first = Ralph E.

| title = Monotone operators in Banach space and nonlinear partial differential equations

| series = Mathematical Surveys and Monographs 49

| publisher = American Mathematical Society

| location = Providence, RI

| year = 1997

| pages = 162–163

| isbn = 0-8218-0500-2

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