Moreau envelope function

Moreau envelope

Let $H$ be a real Hilbert space and let $f : H \rightarrow ( - \infty , + \infty ]$ be a lower semi-continuous extended-real-valued function (cf. also Continuous function) such that for a certain $T > 0$,

\begin{equation*} \operatorname { inf } _ { x \in H } \left( f ( x ) + ( 2 T ) ^ { - 1 } \| x \| ^ { 2 } \right) \end{equation*}

is finite. For $t > 0$, the Moreau envelope function $f _ { t }$ is defined by infimal convolution of $f$ with $( 2 t ) ^ { - 1 } \| . \| ^ { 2 }$, i.e.,

\begin{equation} \tag{a1} f _ { t } ( x ) = \operatorname { inf } _ { y \in H } \left( f ( y ) + \frac { 1 } { 2 t } \| x - y \| ^ { 2 } \right) , \quad x \in H. \end{equation}

This operation amounts geometrically to performing vector addition of the strict epigraphs of $f$ and $( 2 t ) ^ { - 1 } \| . \| ^ { 2 }$. The Moreau envelopes are usually utilized as approximants of $f$, although regularization was not the purpose of the seminal paper [a12].

If $t \in ( 0 , T )$, $f _ { t }$ is everywhere finite and Lipschitz continuous (cf. also Lipschitz condition) on bounded sets. Moreover, $f _ { t }$ increases pointwise to $f$ as $t$ decreases to $0$; the convergence is in fact uniform on bounded sets (cf. also Uniform convergence) when $f$ is uniformly continuous on bounded sets (cf. also Uniform continuity). One might expect that under some additional assumptions on $f$ the differentiability properties of the square of the norm in $H$ should to some extent carry over to $f _ { t }$, thus giving rise to a smooth regularization of $f$. This is true in the presence of convexity.

The convex case.

Suppose $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ is convex (cf. also Convex function (of a real variable)). Then $f _ { t }$ is differentiable and $d f _ { t }$ is globally Lipschitz continuous of rate $\operatorname { max } \{ 1 / t , 1 / ( T - t ) \}$ when $t \in ( 0 , T )$. Furthermore, $u ( t , x ) = f _ { t } ( x )$ is a classical solution of

\begin{equation*} \frac { \partial u ( t , x ) } { \partial t } + \frac { 1 } { 2 } \| d _ { x } u ( t , x ) \| ^ { 2 } = 0 , \quad ( t , x ) \in ( 0 , T ) \times H. \end{equation*}

Let the subdifferential of $f$ be defined by

\begin{equation*} \partial f ( x ) = \partial _ { c } \left( f + ( 2 T ) ^ { - 1 } \| \cdot \| ^ { 2 } \right) ( x ) - T ^ { - 1 } x , \quad x \in H, \end{equation*}

where the first term at the right-hand side is the subdifferential in the sense of convex analysis of the convex function $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ ($\xi \in \partial _ { c } g ( x )$ means that $g ( x )$ is finite and $g ( y ) \geq g ( x ) + \langle y - x , \xi \rangle$ for all $y$). The infimum (a1) is achieved at a unique point $y$, which is denoted by $R _ { t } ( x )$ and is given by $R _ { t } ( x ) = ( I + t \partial f ) ^ { - 1 } ( x )$. Furthermore,

\begin{equation*} d f _ { t } = t ^ { - 1 } ( I - R _ { t } ) = ( ( \partial f ) ^ { - 1 } + t I ) ^ { - 1 }, \end{equation*}

which justifies the alternative term Moreau–Yosida approximation, since $( ( \partial f ) ^ { - 1 } + t l ) ^ { - 1 }$, $t > 0$, are the Yosida approximants of $\partial f$. Moreover, $d f _ { t } \rightarrow \partial f$ in the sense of Kuratowski–Painlevé convergence of graphs in $H \times H$, while $d f _ { t } ( x )$ converges to the element of $\partial f ( x )$ of minimal norm unless $\partial f ( x )$ is empty. Stationary points and values are preserved; as a matter of fact, if $( t , x ) \in ( 0 , T ) \times H$, then

\begin{equation*} d f _ { t } ( x ) = 0 \Leftrightarrow \partial f ( x ) \ni 0 \Leftrightarrow f _ { t } ( x ) = f ( x ). \end{equation*}

For various applications and further properties, consult [a3], [a4], [a1], [a2].

Regularization in the non-convex case.

If one insists on a smooth regularization, the Moreau envelopes cannot be used for arbitrary functions. This is however not a serious drawback. It is easy to see that $- f _ { t } + ( 2 t ) ^ { - 1 } \| . \| ^ { 2 }$ is always a convex function, so that $f _ { t , s } : = - ( - f _ { t } ) _ { s }$ is smooth when $0 < s < t < T$; in fact, $d f _ { t ,\, s }$ is globally Lipschitz continuous of rate $\operatorname { max } \{ 1 / s , 1 / ( t - s ) \}$. Explicitly,

\begin{equation*} f _ { t , s } ( x ) = \operatorname { sup } _ { z \in H } \operatorname { inf } _ { y \in H } \left( f ( y ) + \frac { 1 } { 2 t } \| z - y \| ^ { 2 } - \frac { 1 } { 2 s } \| x - z \| ^ { 2 } \right) \end{equation*}

for all $x \in H$.

Moreover, $f _ { t - s } \leq f _ { t , s } \leq f$ and hence $f _ { t , s } \rightarrow f$ pointwise as $0 < s < t \rightarrow 0$. The double envelopes $f _ { t , s }$, frequently called the Lasry–Lions approximants of $f$, were introduced and investigated by J.-M. Lasry and P.-L. Lions in [a10]; see also [a4], [a13]. One can prove that the equation $f _ { t , s } = f _ { t - s }$ is true for all $0 < s < t < T$ exactly when $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ is a convex function [a13], in which case therefore the approximation method reduces to the previous one.

Connections with the Hamilton–Jacobi equation.

As stated above, $u ( t , x ) = f _ { t } ( x )$ furnishes a classical solution of the initial-value problem

\begin{equation*} \frac { \partial u ( t , x ) } { \partial t } + \frac { 1 } { 2 } \| d _ { x } u ( t , x ) \| ^ { 2 } = 0 , \quad ( t , x ) \in ( 0 , T ) \times H, \end{equation*}

\begin{equation*} \operatorname { lim } _ { t \downarrow 0 } u ( t , x ) = f ( x ) \quad \text { for all } x \in H, \end{equation*}

if $f + ( 2 T ) ^ { - 1 } \| . \| ^ { 2 }$ is a convex function. Let, now, $H = {\bf R} ^ { n }$ and drop the convexity hypothesis on $f$. While being non-differentiable in general, $u ( t , x ) = f _ { t } ( x )$ is nonetheless locally Lipschitz continuous in $( 0 , T ) \times \mathbf{R} ^ { n }$ and is known to be the unique viscosity solution of the above initial-value problem [a11], [a14]. (This notion of a generalized solution allows merely continuous functions to be solutions, [a6]; cf. Viscosity solutions). In this context, (a1) is referred to as the Lax formula; the Lax formula is intimately related to the Hopf formula for conservation laws, see [a9], [a7], [a8], [a11].

Extensions to Banach spaces.

The Lasry–Lions regularization scheme has been extended to certain classes of Banach spaces $( X , \| \, .\, \| )$. For the case where $\| .\|$ and the dual norm $\| . \|_{*}$ are simultaneously locally uniformly rotund, an approach by means of the Legendre–Fenchel transformation has been taken in [a13]. Another extension appears in [a5], under the hypothesis that $X$ be super-reflexive (cf. also Reflexive space).

For more on the theme of regularization, see (the editorial comments to) Regularization and Regularization method.

References

[a1]	I. Ekeland, J.M. Lasry, "On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface" Ann. of Math. , 112 (1980) pp. 283–319
[a2]	R.T. Rockafellar, R.J.-B. Wets, "Variational analysis" , Springer (1998)
[a3]	H. Attouch, "Variational convergence for functions and operators" , Applicable Math. , Pitman (1984)
[a4]	H. Attouch, D. Azé, "Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry–Lions method" Ann. Inst. H. Poincaré Anal. Non Lin. , 10 (1993) pp. 289–312
[a5]	M. Cepedello–Boiso, "On regularization in superreflexive Banach spaces by infimal convolution formulas" Studia Math. , 129 (1998) pp. 265–284
[a6]	M.G. Crandall, P.-L. Lions, "Viscosity solutions of Hamilton–Jacobi equations" Trans. Amer. Math. Soc. , 277 (1983) pp. 1–42
[a7]	E. Hopf, "The partial differential equation $u _ { t } + u u _ { x } = \mu u _ { xx }$" Commun. Pure Appl. Math. , 3 (1950) pp. 201–230
[a8]	E. Hopf, "Generalized solutions of non-linear equations of first order" J. Math. Mech. , 14 (1965) pp. 951–973
[a9]	P.D. Lax, "Hyperbolic systems of conservation laws II" Commun. Pure Appl. Math. , 10 (1957) pp. 537–566
[a10]	J.-M. Lasry, P.-L. Lions, "A remark on regularization in Hilbert spaces" Israel J. Math. , 55 (1986) pp. 257–266
[a11]	P.-L. Lions, "Generalized solutions of Hamilton–Jacobi equations" , Res. Notes Math. , 69 , Pitman (1982)
[a12]	J-J. Moreau, "Proximité et dualité dans un espace hilbertien" Bull. Soc. Math. France , 93 (1965) pp. 273–299
[a13]	T. Strömberg, "On regularization in Banach spaces" Ark. Mat. , 34 (1996) pp. 383–406
[a14]	T. Strömberg, "Hopf's formula gives the unique viscosity solution" Math. Scand. (submitted)