# The Moreau-Yosida Regularization

The **Moreau-Yosida regularization** is a technique used to approximate lower semicontinuous functions by Lipschitz functions. An important application of this result is to prove Portmanteau's Theorem, which states that integration against a lower semicontinuous and bounded below function is lower semicontinuous with respect to the narrow convergence in the space of probability measures.

## Contents

## Definitions

Let be a metric space, and let denotes the collection of probability measures on . is said to be a **Polish space** if it is complete and separable.

A function is said to be **proper** ^{[1]} if it is not identically equal to , that is, if there exists such that . The **domain** of is the set

- .

For a given function and , its **Moreau-Yosida regularization** ^{[1]} is given by

```
```

The distance term may often be raised to a positive exponent , in particular . For example, when is a Hilbert space ^{[2]} ^{[3]}, is taken to be

```
```

This particular variant in a Hilbert space setting is explored in more detail below.

The dependence on the parameter may also be written instead as

for .

Note that

- .

## Examples

- If , then by definition is constant and .
- If is
*not*proper, then for all .

Take . If is finite-valued and differentiable, we can write down an expression for . For a fixed , the map is continuous everywhere and differentiable everywhere except for when , where the derivative does not exist due to the absolute value. Thus we can apply standard optimization techniques from Calculus to solve for : find the critical points of and take the infimum of evaluated at the critical points. One of these values will always be the original function evaluated at , since this corresponds to the critical point for .

- Let . Then

## Approximating Lower Semicontinuous Functions by Lipschitz Functions

**Proposition.** ^{[1]}^{[4]} Let be a Polish space and let .

- If is proper and bounded below, so is . Furthermore, is Lipschitz continuous for all .
- If, in addition, is lower semicontinuous, then for all .
- In this case, is continuous and bounded and for all .

**Proof.**

- Since is proper, there exists such that . Then for any

Thus is proper and bounded below. Next, for a fixed , let . Then as

- ,

the family is uniformly Lipschitz and hence equicontinuous. Thus is Lipschitz continuous.

- Suppose that is also lower semicontinuous. Note that for all , . Thus it suffices to show that . This inequality is automatically satisfied when the left hand side is infinite, so without loss of generality assume that . By definition of infimum, for each there exists such that

- .

Then

is bounded below by assumption, while the only way to be finite in the limit is for to vanish in the limit. Thus converges to in , and by lower semicontinuity of ,

- .

- By definition, . Since for all , for all .

## Portmanteau Theorem

**Theorem (Portmanteau).** ^{[1]} ^{[4]} Let be a Polish space, and let be lower semicontinuous and bounded below. Then the functional is lower semicontinuous with respect to narrow convergence in , that is

```
.
```

**Proof.** By the Moreau-Yosida approximation, for all ,

- .

Taking , Fatou's Lemma ensures that

- .

## Etymology of Portmanteau Theorem

The curious epithet attached to the above theorem is due to Billingsley ^{[5]}, with a citation to a Jean-Pierre Portmanteau's *Espoir pour l'ensemble vide?* published in *Annales de l'Université de Felletin* in 1915. This is believed to be a fictional citation made as a play on words ^{[6]}.

- The publication date is far too early; Kolmogorov's probability axioms were published in 1933.
^{[7]} - Felletin is a small town in central France with no university, and there is no record of a Jean-Pierre Portmanteau aside from this citation.
- "Espoir pour l'ensemble vide" translates to "hope for the empty set" (translation was by Google, please confirm or amend if you speak French!)

## Generalizations

The Moreau-Yosida regularization is a specific case of a type of convolution, and many of the above results follow from this generalization. This material is adapted from Bauschke-Combettes Chapter 12 ^{[2]}, where the setting is over a Hilbert space instead of a more general Polish space.

Let be a Hilbert space, and let . The **infimal convolution** or **epi-sum** of and is

```
.
```

is said to be **exact** at a point if this infimum is attained. is said to be exact if it is exact at every point of its domain, and in this case it is denoted by .

**Remark.** Bauschke-Combettes uses a box with a dot in the middle for to be exact. Due to technical difficulties, we will use instead.

For an example, let be nonempty. Then is exact, and .

**Proposition.** Let be proper, , and for , let be given by

- .

Then the following hold for all and :

- ,
- for , ,
- ,
- as , and
- is bounded above on every ball in .

**Remark.** The convention given above differs slightly from Bauschke-Combettes to fit the convention in this article. The Moreau-Yosida regularization is the special case where , and is called the **Pasch-Hausdorff Envelope** in Bauschke-Combettes.

**Proposition.** Let be lower semicontinuous and convex, let , and let . Then the infimal convolution is convex, proper, continuous, and exact. Moreover, for every , the infimum

is uniquely attained.

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Craig, Katy C. Lower Semicontinuity in the Narrow Topology. Math 260J. Univ. of Ca. at Santa Barbara. Winter 2022. - ↑
^{2.0}^{2.1}Bauschke, Heinz H. and Patrick L. Combettes.*Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd Ed.*Ch. 12. Springer, 2017. - ↑ Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré.
*Gradient Flows in Metric Spaces and in the Space of Probability Measures.*Ch. 3.1. Birkhäuser, 2005. - ↑
^{4.0}^{4.1}Santambrogio, Filippo.*Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling*Ch. 1.1. Birkhäuser, 2015. - ↑ Billingsley, Patrick.
*Convergence of Probability Measures, 2nd Ed.*John Wiley & Sons, Inc. 1999. - ↑ Pagès, Gilles.
*Numerical Probability: An Introduction with Applications to Finance.*Ch. 4.1. Springer, 2018. - ↑ Kolmogorov, Andrey (1950) [1933]. Foundations of the theory of probability. New York, USA: Chelsea Publishing Company.