In mathematics, Moreau's theorem is a result in

convex analysis Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory. Convex sets A subset C \subseteq X of ...

named after French mathematician Jean-Jacques Moreau. It shows that sufficiently

well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. T ...

convex function In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of poin ...

als on

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...

s 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 In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, r ...

extended real-valued functional on ''H''. Let ''A'' stand for ∂''φ'', the

subderivative In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection ...

of ''φ''; for ''α'' > 0 let ''J''_''α'' denote the resolvent: :

J_ = (\mathrm + \alpha A)^;

and let ''A''_''α'' denote the Yosida approximation to ''A'': :

A_ = \frac1 ( \mathrm - J_ ).

For each ''α'' > 0 and ''x'' ∈ ''H'', let :

\varphi_ (x) = \inf_ \frac1 \,  y - x \, ^ + \varphi (y).

Then :

\varphi_ (x) = \frac \,  A_ x \, ^ + \varphi (J_ (x))

and ''φ''_''α'' is convex and Fréchet differentiable with derivative d''φ''_''α'' = ''A''_''α''. Also, for each ''x'' ∈ ''H'' (pointwise), ''φ''_''α''(''x'') converges upwards to ''φ''(''x'') as ''α'' → 0.

References

* (Proposition IV.1.8) {{Functional analysis Convex analysis Theorems in functional analysis