Ewa Bednarczuk

Giovanni Bruccola
email: giovanni.bruccola@ibspan.waw.pl

Gabriele Scrivanti

The Hung Tran

Calculus rules for proximal ε-subdifferentials and inexact proximity operators for weakly convex functions

Keywords: weakly convex functions, criticality, proximal operator, inexactness, inexact proximal operator, sum rule for proximal ε-subdifferentials

We investigate inexact proximity operators for weakly convex functions. To this aim we derive sum rules for proximal ε-subdifferentials, by incorporating the moduli of weak convexity of the functions into the respective formulas. This allows us to investigate inexact proximity operators for weakly convex functions in terms of proximal ε-subdifferentials.

I. INTRODUCTION

Proximal operators are a fundamental tool in constructing algorithms for solving large-scale convex optimisation problems [START_REF] Combettes | Proximal splitting methods in signal processing[END_REF]. Responding to the need for solving optimization problems with convex objectives which do not fall into the class of typical convex objective functions appearing in data analysis (see e.g. the webpage [START_REF] Chierchia | The proximity operator repository[END_REF]), a number of inexact (approximate) proximal operators have been introduced, see e.g. [START_REF] Salzo | Inexact and accelerated proximal point algorithms[END_REF], [START_REF] Villa | Accelerated and inexact forward-backward algorithms[END_REF], [START_REF] Rasch | Inexact first-order Primal-Dual algorithms[END_REF].

It is our aim to investigate inexact proximal operators for a class of functions which is larger than the one of convex functions: in the present work, we focus on weakly convex functions, which have been appearing in current models in data science problems in a rapidly growing number.

a) Weak convexity: Weak convexity can be considered as a special case of the general notions of γparaconvexity and αp¨q-paraconvexity that were studied by, among others, Jourani and Rolewicz [START_REF] Jourani | Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions[END_REF], [START_REF] Rolewicz | On paraconvex multifunctions[END_REF], [START_REF] Rolewicz | Paraconvex analysis[END_REF]. For a general characterisation in Hilbert spaces, weakly convex functions can be expressed as the difference between a convex function and a quadratic function. This class includes all the convex functions and all the smooth (but not necessarily convex) functions with a Lipschitz continuous gradient, together with many other interesting non-convex functions. Examples of weakly convex functions appearing in data analysis can be found in [START_REF] Davis | Stochastic model-based minimization of weakly convex functions[END_REF] and in [START_REF] Böhm | Variable smoothing for weakly convex composite functions[END_REF]. The growing interest in the use of this class of function in many fields of applications suggested the necessity of a careful analysis of their properties in terms of subdifferentials and proximal operators, which is the core of the present work.

b) Proximal Subdifferentials: It has already been observed that the concept of subdifferential which is particularly suitable when defining criticality for weakly convex functions is that of proximal subdifferential (see e.g. [START_REF] Davis | Proximally guided stochastic subgradient method for nonsmooth, nonconvex problems[END_REF], [START_REF] Davis | Subgradient methods for sharp weakly convex functions[END_REF]). There exists a vast literature devoted to proximal subdifferentials, see e.g. in the finite dimensional case, the monograph by Rockafellar and Wets [START_REF] Rockafellar | Variational Analysis[END_REF], in Hilbert spaces the work by Bernards and Thibault [START_REF] Bernard | Prox-regular functions in Hilbert spaces[END_REF]. In these monographs and papers, the proximal subdifferential at a given x 0 is defined locally, in the sense that there exists a neighbourhood V of x 0 and a constant C ě 0, such that for every x P V , xx ˚, x ´x0 y ď f pxq ´f px 0 q `C}x ´x0 } 2 .

(1)

In our developments we make use of a property that holds in the class of paraconvex functions, called globalisation property. Precisely, in the class of paraconvex functions, if (1) holds, then it holds globally over the whole space (see Def. 1 and Prop. 1). More on the globalization property can be found in [START_REF] Rolewicz | On a globalization property[END_REF].

We focus our analysis on the more general notion of proximal ε-subdifferentials, which represents a useful tool allowing to take into account inexactness and perturbations in the resolution of optimisation problems. c) Contribution: Our contribution addresses the following issues.

1) We provide sufficient and necessary conditions

for the sum rule of the global proximal ε-subdifferentials for the sum of two ρ-weakly convex functions ( see Theo. 2 and Theo. 3) (Section III). 2) By using the above theorems, in Prop. [START_REF] Bednarczuk | On duality for nonconvex minimization problems within the framework of abstract convexity[END_REF]

and

Prop. 5, we investigate the relationship between the ε-proximal operator of a ρ-weakly convex function f and the ε-proximal subdifferential of f (Section IV).

3) The notion of inexact (approximate) proximal point that we infer can be related to Type-1 and Type-2 approximations proposed in [START_REF] Salzo | Inexact and accelerated proximal point algorithms[END_REF], [START_REF] Rasch | Inexact first-order Primal-Dual algorithms[END_REF] in the convex settings (Section IV). In the present work, we incorporate and make a consistent use of the modulus of proximal subdifferentiability and of the modulus of weak convexity ρ into the calculus rules for proximal ε-subdifferentials.

II. PRELIMINARIES

Before focusing on the class of weakly convex functions, we introduce a more general notion of γ-paraconvexity and the corresponding notion of pγ, Cq-subdifferential as presented in [START_REF] Jourani | Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions[END_REF]. The class of γ-paraconvex functions has been studied by [START_REF] Rolewicz | On paraconvex multifunctions[END_REF]. For γ " 2, we obtain weakly convex functions.

Definition 1 (γ-Paraconvexity): Let X be a normed vector space. A function f : X Ñ p´8, `8s is said to be γ ´paraconvex if there exists a positive constant C such that for λ P r0, 1s, p@px, yq P X 2 q, the following inequality holds:

f pλx`p1´λqyq ď λf pxq`p1´λqf pyq`Cλp1´λq}x´y} γ .
When α : r0, `8q Ñ r0, `8q is a nondecreasing function with lim tÓ0 αptq t " 0, a function f : X Ñ p´8, `8s is called αp¨q-paraconvex if there exists a constant C ą 0 such that for λ P r0, 1s p@px, yq P X 2 q f pλx `p1 ´λqyq ď λf pxq `p1 ´λqf pyq `C mintλ, 1 ´λuαp}x ´y}q This class has been introduced by Rolewicz under the name of αp¨q-strongly paraconvex functions and investigated in a series of papers by Jourani [START_REF] Jourani | Open mapping theorem and inversion theorem for γ-paraconvex multivalued mappings and applications[END_REF], [START_REF] Jourani | Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions[END_REF] and Rolewicz [START_REF] Rolewicz | On paraconvex multifunctions[END_REF], [START_REF] Rolewicz | Paraconvex analysis[END_REF] . When αp}x ´y}q " }x ´y} γ the notion of αp¨q-paraconvexity coincides with the one of γ-paraconvexity, (see [START_REF] Rolewicz | On uniformly approximate convex and strongly αp¨q-paraconvex functions[END_REF]Lemma 5]). In Hilbert spaces, when lim sup tÓ0 αptq t 2 ă `8, then a αp¨q-paraconvex function is a difference of a convex and a quadratic function and is called weakly convex (see [START_REF] Rolewicz | Paraconvex analysis[END_REF]). Definition 2 (pγ, Cq´Subdifferential [16, Def. 3.1]): Let X be a normed vector space. By X ˚we denote the dual space of all continuous linear functionals defined on X. Let γ ą 0 and C ą 0. Let x 0 P X and f : X Ñ p´8, `8s be finite at x 0 . A point x ˚P X is said to be a pγ, Cq-subgradient of f at x 0 if there exists a neighbourhood V of x 0 such that p@x P V q xx ˚, x´x 0 y ď f pxq´f px 0 q`C}x´x 0 } γ .

The set of all pγ, Cq´subgradients of f at x 0 is denoted by B Loc pγ,Cq f px 0 q and it is referred to as pγ, Cq-Subdifferential. Whenever B Loc pγ,Cq f px 0 q ‰ H, we say that f is proximally C-subdifferentiable at x 0 .

Proposition 1 ( [16, Prop. 3.1]): Let X be a normed space. Let f : X Ñ p´8, `8s be γ-paraconvex with γ ą 1. Then there exists C ą 0 such that B Loc pγ,Cq f px 0 q " B pγ,Cq f px 0 q where B pγ,Cq f px 0 q :" tx ˚P X ˚| xx ˚, x ´x0 y ď f pxq ´f px 0 q `C}x ´x0 } γ @ x P X u . The constant C can be chosen equal to the one appearing in the definition of paraconvexity (Def. 1).

Occasionally, we will refer to the constant C appearing in the definition of proximal subdifferentiability as the modulus of proximal subdifferentiability.

In the sequel we will use the global p2, Cq-subdifferential of f at x which will be refferred to as proximal subdifferential: (see e.g. [START_REF] Rockafellar | Variational Analysis[END_REF], [START_REF] Bernard | Prox-regular functions in Hilbert spaces[END_REF]).

Definition 3 (Global Proximal Subdifferential): Let X be a normed vector space. Let f : X Ñ p´8, `8s, ρ ě 0 and let x P dom f . Then the proximal subdifferential of f at x 0 with constant C ą 0 is defined as the set

B p2,Cq f px 0 q :" tx ˚P X ˚| f pxq ě f px 0 q `xx ˚, x ´x0 y ´C}x ´x0 } 2 , @x P X u. (2)
In view of (2), B p2,0q denotes the subdifferential in the sense of convex analysis. For simplicity, in this case we will use the notation B 0 " B p2,0q .

When investigating inexact proximal points, the following concept of proximal ε-subdifferentials is used.

Definition 4 (Global proximal ε-subdifferentials): Let X be a normed vector space and ε ě 0. The global proximal ε-subdifferentials of a function f : X Ñ p´8, `8s at x 0 P X for C ě 0 is defined as follows:

B ε p2,Cq f px 0 q " tv P X | s.t. f pxq ´f px 0 q ě xv, x ´x0 y ´C}x ´x0 } 2 ´ε @ x P X u. Clearly, for every ε 1 ě ε and C 1 ě C we have the following inclusion B ε p2,Cq f px 0 q Ď B ε 1 p2,C 1 q f px 0 q. (3)
In Hilbert spaces, a weakly convex function f in the sense of Def. 1 for γ " 2 and C " ρ{2 can be characterised by the fact that f p¨q `ρ{2} ¨}2 is a convex function. A proof can be obtained by directly adapting the finite-dimensional proof given in [START_REF] Cannarsa | Semiconcave functions[END_REF]Prop. 1.1.3]. Such a function will then be referred to as a ρ-weakly convex and ρ is known as modulus of weak convexity. A variant of Prop. 1 corresponding to γ " 2 and C not necessarily coinciding with the weak convexity parameter can be found in [START_REF] Syga | On global properties of lower semicontinuous quadratically minorized functions[END_REF].

For any set-valued mapping M : X ⇒ X , we will use the notation dom M to indicate the set dom M :" tx P X | M pxq ‰ Hu, while for a function f : X Ñ p´8, `8s, the notation dom f will indicate the set dom f :" tx P X | f pxq ă `8u.

Proposition 2: Let X be a Hilbert space. Let f : X Ñ p´8, `8s be a lower semicontinuous and ρ-weakly convex function with ρ ě 0. Then for every ε ě 0

dom B ε p2,ρ{2q f " dom B ε 0 pf `ρ 2 } ¨}2 q and dom B ε p2,ρ{2q f Ă dom f. Moreover, for every ε ą 0 dom B ε p2,ρ{2q f " dom f. Proof:
We start by showing that for any x 0 P dom B ε 0 pf `ρ 2 } ¨}2 q and ε ě 0, we have that

B ε 0 pf `ρ 2 } ¨}2 qpx 0 q ´ρx 0 " B ε p2,ρ{2q f px 0 q. (4)
Indeed, for any v P B ε 0 pf `ρ 2 } ¨}2 qpx 0 q and x 0 P dom B ε 0 pf `ρ 2 } ¨}2 q, we have f pxq `ρ 2 }x} 2 ´f px 0 q ´ρ 2 }x 0 } 2 ě xv, x ´x0 y ´ε ðñ f pxq ´f px 0 q ě xv ´ρx 0 , x ´x0 y ´ρ 2 }x ´x0 } 2 ´ε which is equivalent to the fact that v ´ρx 0 P B ε p2,ρ{2q f px 0 q and proves (4). Hence, dom B ε 0 pf `ρ 2 } ¨}2 q " dom B ε p2,ρ{2q f . Since B ε 0 corresponds to the ε-subdifferential for convex functions and function pf `ρ 2 qp¨q is convex, we have that for all ε ą 0

dom B ε 0 pf `ρ 2 } ¨}2 q " dom pf `ρ 2 } ¨}2 q
(see [2, Cor. 2.81]). Since dom f " dom pf `ρ 2 } ¨}2 q, the assertion follows.

In Prop. 2, we include the assumption of lower semicontinuity on f because it is required by [2, Cor.

2.81].

Definition 5 (ε-solution): Let X be a normed space. Let f : X Ñ p´8, `8s be a proper function that is bounded from below. Then, for any ε ě 0, the element x ε is said to be an ε-solution to the minimisation problem minimize xPX f pxq if the following condition is satisfied: p@x P X q f px ε q ď f pxq `ε.

Definition 6 (ε-C-critical point): Let X be a normed space and ε ě 0. Let f : X Ñ p´8, `8s be a proper function. A point x P X is said to be a ε-C-critical point of f if 0 P B ε p2,Cq f pxq. The set of ε-C-critical points is identified as

ε -crit C f :" tx P X | 0 P B ε p2,Cq f pxqu. When f is ρ-weakly convex,
it is of particular interest to consider ε-ρ{2-critical points and then we write "ε-critical points" and use the notation ε -crit.

Remark 1 (Fermat's Rule): We highlight that ε-C-criticality is a necessary condition for a point to be a ε-solution. Notice that, by Prop. 2, dom f " dom B ε p2,Cq h.

If x ε P dom f is an ε-solution of f , then p@x P X q f pxq ě f px ε q ´ε ě f px ε q ´C}x ´xε } 2 ´ε
for every C ě 0. This implies 0 P B ε p2,Cq f px ε q. III. CALCULUS RULES In the literature, there exist numerous results providing calculus rules for the Fréchet, the limiting and the proximal subdifferentials, see e.g. [START_REF] Jourani | Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions[END_REF], [START_REF] Kruger | On Fréchet subdifferentials[END_REF], [START_REF] Mordukhovich | On nonconvex subdifferential calculus in Banach spaces[END_REF], [START_REF] Mordukhovich | Exact calculus for proximal subgradients with applications to optimization[END_REF], [START_REF] Thibault | On subdifferentials of optimal value functions[END_REF] and many others. The main result of the present section is stated in Theo. 2, where we provide the conditions for a sum rule for the global proximal ε-subdifferentials (in the sense of Def. 4) of the sum of two weakly convex functions. The proposed result allows to extend the sum rule in [16, Theo. 5.1] -proved for exact proximal subdifferentials in normed spaces -to proximal ε-subdifferentials in Hilbert spaces: the interesting aspect of such rule is that it allows to keep track of the modulus of weak convexity.

The following notion of ρ-conjugate function will be used in the proof of Theo. 2.

Definition 7: Let X be a Hilbert space. Let f : X Ñ p´8, `8s be a proper function. For every ρ ě 0 the function phq ρ : X Ñ r´8, `8s defined as pf q ρ puq :" sup yPX ! ´ρ 2 }y} 2 `xu, yy ´f pyq

)
is called ρ-conjugate of f at u P X (when ρ " 0 we obtain the definition of the conjugate as defined in convex analysis and in this case we omit the subscript).

We recall the following result, which is an important fact in view of the proof of Theo. 2.

Theorem 1 ([22, Theo. 3] ): Let X be a Hilbert space. Let f 0 , f 1 : X Ñ p´8, `8s be proper convex functions. Assume that dom f 0 Xdom f 1 contains a point at which either f 0 or f 1 is continuous. Then, for all s, x P X we have paq pf 0 `f1 q ˚psq " min p0,p1PX s"p 0 `p1 f 0 pp 0 q `f 1 pp 1 q ( pbq B 0 pf 0 `f1 q pxq " B 0 f 0 pxq `B0 f 1 pxq

Remark 2: By [3, Theo. 15.3 (Attouch-Brézis Theorem)], [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF] , paq of Theo. 1 can be proved under the assumption that f 0 and f 1 are convex proper lsc functions such that the conical hull of dom f 0 ´dom f 1 is a closed linear subspace, i.e., 0 P sripdom f 0 ´dom f 1 q, where sri denotes the strong relative interior, see [3, Def. 6.9]. The regularity assumption in [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF] is more general than the one in [START_REF] Rockafellar | Extension of Fenchel's duality theorem for convex functions[END_REF] (see [START_REF] Attouch | Duality for the sum of convex functions in general Banach spaces[END_REF]Remark 1.3]). However, in some cases it is easier to verify the regularity condition requested in [START_REF] Rockafellar | Extension of Fenchel's duality theorem for convex functions[END_REF].

The following proposition provides an important auxiliary fact used in the proof of Theo. 2.

Proposition 3: Let X be a Hilbert space. For i " 0, 1, let function f i : X Ñ p´8, `8s be proper lower semicontinuous and ρ i -weakly convex on X with ρ i ě 0. Assume that dom f 0 X dom f 1 contains a point at which either f 0 or f 1 is continuous. Then the following holds: for any s P dom pf 0 `f1 q ρ , there exist p 0 , p 1 P X such that s " p 0 `p1 and pf 0 `f1 q ρ0`ρ1 psq " pf 0 q ρ0 pp 0 q `pf 1 q ρ1 pp 1 q.

Proof: We have that f 0 p¨q " f 0 p¨q `ρ0 {2} ¨}2 and f 1 p¨q " f 1 p¨q `ρ1 {2} ¨}2 are convex. By Theo. 1, there exist p 0 , p 1 such that s " p 0 `p1 and pf 0 `f 1 q ˚psq " pf 0 q ˚pp 0 q `pf 1 q ˚pp 1 q Notice that for i " 0, 1

f i p¨q :" sup yPX x¨, yy ´f i pyq ( " sup yPX ! x¨, yy ´fi pyq ´ρi 2 }y} 2
) " pf i q ρi p¨q so pf 0 `f 1 q ˚psq " pf 0 q ρ0 pp 0 q `pf 1 q ρ1 pp 1 q and in conclusion pf 0 `f1 q ρ0`ρ1 psq " pf 0 q ρ0 pp 0 q `pf 1 q ρ1 pp 1 q.

Now we are ready to prove the following sum rule for proximal ε-subdifferentials B ε p2,ρ{2q . This result generalises [14, Theo. 3.1.1] and [32, Theo. 2.8.7], which are formulated for convex functions and convex subdifferentials. An important aspect of our resultwhich will be used below in the analysis of proximal operators -is that it allows to keep track of the modulus of proximal subdifferentiability (as related to the modulus of weak convexity of the functions involved).

Theorem 2 (Sum Rule for ε-subdifferential): Let X be a Hilbert space. For i " 0, 1, let function f i : X Ñ p´8, `8s be proper lower semicontinuous and ρ i -weakly convex on X with ρ i ě 0. Then, for all x P dom f 0 X dom f 1 and for all ε 0 , ε 1 ě 0 we have

B ε0
p2,ρ0{2q f 0 pxq `Bε1 p2,ρ1{2q f 1 pxq Ď B ε p2,ρ{2q pf 0 `f1 qpxq (5) for all ε ě ε 0 `ε1 and for all ρ ě ρ 0 `ρ1 . The equality

B ε p2,pρ0`ρ1q{2q pf 0 `f1 qpxq " ď ε0,ε1 | ε0`ε1ďε B ε0 p2,ρ0{2q f 0 pxq `B ε1 p2,ρ1{2q f 1 pxq (6)
holds when dom f 0 X dom f 1 contains a point at which either f 0 `ρ0 {2} ¨}2 or f 1 p¨q `ρ1 {2} ¨}2 is continuous.

Proof:

For x P X , if w P B ε0 p2,ρ0{2q f 0 pxq, v P B ε1
p2,ρ1{2q f 1 pxq, then it is clear that w `v P B ε p2,pρ0`ρ1q{2q pf 0 `f1 qpxq. Hence the inclusion ( 5) is satisfied.

To prove the equality in ( 6), let us consider x P dom f 0 X dom f 1 and u P B ε p2,ρ{2q pf 0 `f1 qpxq, where ρ " ρ 0 `ρ1 . By [4, Theo. 2.4.ii, Eq. ( 5)] we have pf 0 `f1 qpxq `pf 0 `f1 q ρ pu `ρxq

ď ´ρ 2 }x} 2 `xu `ρx, xy `ε (7)
The inequality in [START_REF] Cannarsa | Semiconcave functions[END_REF] implies that u `ρx P dom pf 0 `f1 q ρ . By applying Prop. 3, there exist two elements p 0 , p 1 P X such that u `ρx " p 0 `p1 and pf 0 `f1 q ρ pu `ρxq " f ρ0 pp 0 q `pf 1 q ρ1 pp 1 q so that ( 7) can be rewritten as pf 0 `f1 qpxq `pf 0 q ρ0 pp 0 q `pf 1 q ρ1 pp 1 q ď ´ρ 2 }x} 2 `xp 0 `p1 , xy `ε for all x P dom f 0 Xdom f 1 . We now define the following values ε 0 :" f 0 pxq `pf 0 q ρ0 pp 0 q ´xp 0 , xy `ρ0 2 }x} 2 ě 0 (8)

ε 1 :" f 1 pxq `pf 1 q ρ1 pp 1 q ´xp 1 , xy `ρ1 2 }x} 2 ě 0 (9)
which are positive in view of the definition of ρconjugate. Notice that ( 8) and ( 9) can be rewritten as

ε 0 " f 0 pxq `pf 0 q ρ0 ppp 0 ´ρ0 xq `ρ0 xq ´xpp 0 ´ρ0 xq `ρ0 x, xy `ρ0 2 }x} 2 ε 1 " f 1 pxq `pf 1 q ρ1 ppp 1 ´ρ1 xq `ρ1 xq ´xpp 1 ´ρ1 xq `ρ1 x, xy `ρ1 2 }x} 2
from which, by applying [4, Theo. 2.4.ii, Eq. ( 5)] in a similar fashion as in [START_REF] Cannarsa | Semiconcave functions[END_REF], we obtain pp 0 ´ρ0 xq P B ε0 p2,ρ0{2q f 0 pxq pp 1 ´ρ1 xq P B ε1 p2,ρ1{2q f 1 pxq which completes the proof. Remark 3: When ε " 0, the sum rule presented in the theorem above can be shown to hold for γparaconvex fuctions, γ ą 1, defined over complete metric spaces (see [START_REF] Jourani | Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions[END_REF]Theo. 5.1,Cor. 5.1]). The sum rule in [START_REF] Jourani | Subdifferentiability and subdifferential monotonicity of γ-paraconvex functions[END_REF]Theo. 5.1,Cor. 5.1] also keeps track of the modulus of subdifferentiability. Our effort in Theo. 2 is to extend this result to the more general notion of ε-subdifferentials and, in order to do so, we exploited the structure of Hilbert spaces. It is worth noticing that, as in the case of the sum rule for B pγ,Cq , also in the case of the sum rule for B ε pγ,Cq we are able to control the modulus of proximal subdifferentiability. Remark 4: Let f i pxq, i " 0, 1 be a ρ i -weakly convex function. If there exist p 0 , p 1 such that p 0 `p1 " u P B ε p2,pρ0`ρ1q{2q pf 0 `f1 qpxq and p 0 P B ε0 p2,ρ0{2q f 0 pxq, p 1 P B ε1 p2,ρ1{2q f 1 pxq then by (3), for all ε 1 i ą ε i , i " 0, 1, we have

p 0 P B ε 1 0 p2,ρ0{2q f 0 pxq, p 1 P B ε 1 1 p2,ρ1{2q f 1 pxq. In particular, we can set ε 1 1 `ε1 2 " ε in (6).
In the following theorem we show that, in presence of differentiable functions, the notion of proximal ε-subdifferentials allows to infer an inclusion which involves the gradient of the differentiable function.

Theorem 3: Let X be a Hilbert space. Let f 0 : X Ñ p´8, `8s be proper, convex and differentiable with a L 0 -Lipschitz continuous gradient on the whole space X . Let f 1 : X Ñ p´8, `8s be proper and ρ-weakly convex on X with ρ ě 0. We then have the following inclusion

B ε p2,ρ{2q pf 0 `f1 qpxq Ă ∇pf 0 qpxq `B ε p2,ρ{2`L0{2q f 1 pxq for all x P dom f 1 such that B ε p2,ρ{2q f 1 pxq ‰ H.
Proof: In view of [3, Lemma 2.64], both f 0 and ´f0 are proximally L 0 -subdifferentiable on X . Precisely, for f 0 we have that for every x, y P X p´f 0 qpyq ´p´f 0 qpxq ě (10)

x∇p´f 0 qpxq | y ´xy ´L0 2 }y ´x} 2 .
Let us choose x P dom f 1 and take v P B ε p2,ρ{2q pf 0 `f1 qpxq. For every y P X , by adding

pf 1 `f0 qpyq´pf 1 `f0 qpxq ě xv | y´xy´ρ 2 }y´x} 2 ´ε
and (10) we get

B ε p2,ρ{2q pf 1 `f0 qpxq `∇p´f 0 qpxq Ă B ε p2,ρ{2`L0{2q f 1 pxq i.e. B ε p2,ρ{2q pf 1 `f0 qpxq Ă B ε p2,ρ{2`L0{2q f 1 pxq`∇pf 0 qpxq.
Remark 5: We highlight the following facts.

' For ε ą 0, if f 1 is also lower semicontinous, Theo.

3 holds for all x P dom f 1 by Prop. 2.

' Theo. 3 still holds when f 1 is convex, i.e. ρ " 0, but not for convex subdifferential. In fact the inclusion becomes

B ε 0 pf 0 `f1 qpxq Ă ∇pf 0 qpxq `B ε p2,L0{2q f 1 pxq.

IV. INEXACT PROXIMAL MAPS

In general settings, the computation of the proximal map needs to be addressed as an independent optimisation problem. Some practical examples involves non-convex ℓ p -seminorms (i.e. p P p0, 1q) or the convex ℓ p -norms (i.e. p ě 1), unless p takes some specific values [START_REF] Chaux | A variational formulation for frame-based inverse problems[END_REF]. Another example is given by the combination of a sparsity-promoting functions with a non-orthogonal linear operator, as in the case of the popular discrete Total Variation functional [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF] (and its non-convex modifications), which has been extensively used in the context of image and signal processing. In these cases, at each point, the proximal map is defined up to a certain degree of accuracy and in the framework of proximal algorithms, it is important to carry out a convergence analysis that takes this fact into account. In order to do so, we consider the concept of ε-solution for an optimisation problem (see Def. 5) and the related notion of ε-proximal point.

Definition 8 (ε-proximal point): Let X be a normed vector space. Let function f : X Ñ p´8, `8s be proper and bounded from below. Then for all y P X and for all ε ě 0, any ε-solution to the proximal minimization problem

minimize xPX f pxq `1 2 }x ´y} 2 , (11)
is said to be a ε-proximal point for f at y. The set of all ε-proximal points of f at y is denoted as ε-prox h pyq :" tx P X | x is a ε-solution of (11)u In the following result, we provide a relationship between the ε-proximal operator and the ε-proximal subdifferentials of weakly convex function, using the sum rule from Theo. 3. Specifically, in Prop. 4, we keep track of the constant C of the p2, Cq-subdifferential of f .

Proposition 4: Let X be a Hilbert space. Let f : X Ñ p´8, `8s be a proper, lower semicontinuous ρ-weakly convex function that is bounded from below on X . Let ε ě 0, α ą 0. Then for every y P X ,

x ε P ε-prox αf pyq implies y ´xε α P B ε p2,ρ{2`1{p2αqq f px ε q. ( 12
)
If 1{α ą ρ, we obtain the equivalence py P X q x ε P ε-prox αf pyq ðñ y ´xε α P B ε p2,ρ{2`1{p2αqq f px ε q. Proof: By the definition of ε-proximal point and Remark 1 we have

x ε P ε-prox αf pyq ùñ 0 P B ε p2,ρ{2q ˆ1 2α } ¨´y} 2 `f p¨q ˙px ε q.
The assumptions in Theo. 3 are satisfied since f 0 p¨q " 1 2α } ¨´y} 2 is differentiable on the whole space and its gradient has a Lipschitz constant L 0 " 1{α, hence we also have the inclusion

0 P ∇ ˆ1 2α } ¨´y} 2 ˙px ε q `B ε p2,ρ{2`1{p2αqq f px ε q " x ε ´y α `B ε p2,ρ{2`1{p2αqq f px ε q which is equivalent to y ´xε α P B ε p2,ρ{2`1{p2αqq f px ε q.
Remark 6: We highlight that ( 12) is related to the notion of Type-2 approximation of the proximal point that is proposed in [START_REF] Rasch | Inexact first-order Primal-Dual algorithms[END_REF], [START_REF] Salzo | Inexact and accelerated proximal point algorithms[END_REF] in the convex settings. In other words, by using the ε-proximal subdifferential instead of the (convex) ε-subdifferential, we can obtain a Type 2 approximation of the proximal point directly from Def. 8.This is due to Theo. 3, at the expense of increasing the modulus of proximal subdifferentiability by 1{p2αq.

Remark 7 (ε-Subdifferential of a quadratic function ): Let X be a Hilbert space. As a consequence of [14, Formula 1.2.5], for a function of the form h 0 pxq " 1 2α }x ´y} 2 for some y P X and α ą 0, we have

B ε 0 f 0 pxq " t x ´y α `e α | 1 2α }e} 2 ď εu.
In view of Remark 7 and Theo. 2, we can provide another interpretation for the ε-proximal points of a ρ-weakly convex function in terms of proximal ε-subdifferentials, where this time we are able to keep track of the modulus of weak convexity of the function. Proposition 5: Let X be a Hilbert space. Let f : X Ñ p´8, `8s be a proper, lower semicontinuous ρ-weakly convex function that is bounded from below on X and let ε ě 0, α ą 0. If x ε P ε -prox αf pyq, then there exist ε 0 , ε 1 ě 0 with ε 0 `ε1 ď ε and there exists e P X with }e} 2 2α ď ε 0 such that y ´xε ´e α P B ε1 p2,ρ{2q f px ε q.

If 1{α ą ρ, we obtain the equivalence Proof: By definition of ε-proximal point we have

x ε P ε-prox αf pyq ùñ 0 P B ε p2,ρ{2q ˆ1 2α } ¨´y} 2 `f p¨q ˙px ε q.
We can now apply Theo. 2, according to which there exist ε 0 , ε 1 ě 0 with ε 0 `ε1 ď ε such that

0 P B ε0 0 ˆ1 2α } ¨´y} 2 ˙px ε q `B ε1 p2,ρ{2q f px ε q. ( 14
)
By applying Remark 7, we infer that there exists e P X with }e} 2 2α ď ε 0 such that x ε ´y α `e α P B ε0 0 ˆ1 2α } ¨´y} 2 ˙px ε q which implies y ´xε ´e α P B ε1 p2,ρ{2q f px ε q.

Remark 8: The interpretation provided by proposition 5 is related to the notion of Type-1 approximation of the proximal point that is proposed in [START_REF] Rasch | Inexact first-order Primal-Dual algorithms[END_REF], [START_REF] Salzo | Inexact and accelerated proximal point algorithms[END_REF] for convex functions.

Remark 9: Notice that (13) from Prop. 5 implies the inclusion [START_REF] Davis | Subgradient methods for sharp weakly convex functions[END_REF] that is equivalent to [START_REF] Davis | Subgradient methods for sharp weakly convex functions[END_REF] y ´xε α P B ε p2,ρ{2`1{p2αqq f px ε q. The inclusion in [START_REF] Hiriart-Urruty | Convex analysis and minimization algorithms II: Advanced Theory and Bundle Methods[END_REF] further leads to the following corollary which is a generalisation of [START_REF] Rasch | Inexact first-order Primal-Dual algorithms[END_REF]Lemma 2] from convex to proximal ε-subdifferentials.

Corollary 1: Let X be a Hilbert space. Let f : X Ñ p´8, `8s be a proper, lower semicontinuous ρ-weakly convex function that is bounded from below on X and let ε ě 0. If x ε P ε -prox αf pxq, then there exist e P X with }e} 2 2α ď ε such that y ´xε ´e α P B ε p2,ρ{2q f px ε q Proof: The proof is equivalent to the one from Prop. 5, with the only difference that we exploit the fact that the inclusion in ( 14) always implies the following inclusion 0 P B ε 0 ˆ1 2α } ¨´y} 2 ˙px ε q `B ε p2,ρ{2q f px ε qpx ε q.

by (3) and the fact that ε 0 and ε 1 from Theo. 2 are always smaller than ε.

V. CONCLUSIONS

We discussed inexact proximal operators (in the sense of Def. 8) for weakly convex functions defined on Hilbert spaces and their relationships to proximal ε-subdifferentials. We highlighted the main differences and similarities with the fully convex settings. An important feature of the obtained results is that in Prop. 4 and Prop. 5 we are able to control the moduli of proximal ε-subdifferentiability of f (in relation to the moduli of weak convexity). Such result could contribute to the convergence analysis of proximal algorithms for solving optimization problems containing weakly convex functions.

  from Prop. 4, which is based on Theo. 3. By definition of proximal ε-subdifferentials, (13) is equivalent to

	which stems from Cauchy-Schwarz and Young's
	inequality. It follows that
	f pxq ´f px ε q ě	B	α y ´xε	, x ´xε	F
			´ˆρ 2 `1 2α	˙}x ´xε } 2 ´ε
						px P X q		f pxq ´f px ε q ě
						B	y ´xε α	, x ´xε	F	´ρ 2	}x ´xε }	2 ´A e α	, x ´xε	E	´ε1 .
							We consider the following estimation
							A e α	, x ´xε	E	ď ε 0	`1 2α	}x ´xε } 2

Acknowledgments This work has been supported by the ITN-ETN project TraDE-OPT funded by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 861137. This work represents only the author's view and the European Commission is not responsible for any use that may be made of the information it contains.