Moreau envelope function

Moreau envelope

Let be a real Hilbert space and let be a lower semi-continuous extended-real-valued function (cf. also Continuous function) such that for a certain ,

is finite. For , the Moreau envelope function is defined by infimal convolution of with , i.e.,

(a1)

This operation amounts geometrically to performing vector addition of the strict epigraphs of and . The Moreau envelopes are usually utilized as approximants of , although regularization was not the purpose of the seminal paper [a12].

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

The convex case.

Suppose is convex (cf. also Convex function (of a real variable)). Then is differentiable and is globally Lipschitz continuous of rate when . Furthermore, is a classical solution of

Let the subdifferential of be defined by

where the first term at the right-hand side is the subdifferential in the sense of convex analysis of the convex function ( means that is finite and for all ). The infimum (a1) is achieved at a unique point , which is denoted by and is given by . Furthermore,

which justifies the alternative term Moreau–Yosida approximation, since , , are the Yosida approximants of . Moreover, in the sense of Kuratowski–Painlevé convergence of graphs in , while converges to the element of of minimal norm unless is empty. Stationary points and values are preserved; as a matter of fact, if , then

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 is always a convex function, so that is smooth when ; in fact, is globally Lipschitz continuous of rate . Explicitly,

for all .

Moreover, and hence pointwise as . The double envelopes , frequently called the Lasry–Lions approximants of , were introduced and investigated by J.-M. Lasry and P.-L. Lions in [a10]; see also [a4], [a13]. One can prove that the equation is true for all exactly when 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, furnishes a classical solution of the initial-value problem

if is a convex function. Let, now, and drop the convexity hypothesis on . While being non-differentiable in general, is nonetheless locally Lipschitz continuous in 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 . 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 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 " 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)