Moreau's theorem

In mathematics, Moreau's theorem is a result in convex analysis. 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 Jα denote the resolvent:

and let Aα denote the Yosida approximation to A:

For each α > 0 and x ∈ H, let

Then

and φα is convex and Fréchet differentiable with derivative dφα = Aα. Also, for each x ∈ H (pointwise), φα(x) converges upwards to φ(x) as α → 0.

gollark: What, length terminated strings or yclib?
gollark: One of these years I should probably work out how to apply SIMD to it, for performance.
gollark: I was lazy, so it reads (can read) slightly outside of the LTS buffer.
gollark: I think `lts_length` is technically capable of causing segfaults in some situations too.
gollark: ```c#define REP(x) x x x x#define T1 "a"#define T2 REP(T1)#define T3 REP(T2)#define T4 REP(T3)#define T5 REP(T4)#define T6 REP(T5)#define T7 REP(T6)```This was part of my failed plan to implement hyperoperations in the preprocessor.

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)
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