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A general representation of delta-normal sets to sublevels of convex functions

The (delta-) normal cone to an arbitrary intersection of sublevel sets of proper, lower semicontinuous, and convex functions is characterized, using either epsilon-subdifferentials at the nominal point or exact subdifferentials at nearby points. Our tools include (epsilon-) calculus rules for sup/max functions. The framework of this work is that of a locally convex space, however, formulas using exact subdifferentials require some restriction either on the space (e.g. Banach), or on the function (e.g. epi-pointed).


Introduction
In [6] the authors prove that for a convex, proper and lower semicontinuous (lsc) function Φ : X → R ∪ {+∞} , defined on a (reflexive) Banach space X, the normal cone to the sublevel set [Φ ≤ Φ(x)] := {x ∈ X | Φ(x) ≤ Φ(x)} at a pointx in the effective domain of Φ, is characterized as the (norm) upper-limit of directions in the Fenchel subdifferential of Φ at sufficiently close points tox; that is, This relation is also valid in general Banach spaces but up to some specifications of both the lim sup and the convergence of x tox. The proof of (1) and its generalization to Banach spaces given in [6], is based on sequential calculus rules for the subdifferential of composite functions developed in [25]. In this work, assuming that X is a general locally convex (lc) space with a topological dual Y, we propose another route to approach this problem and provide different characterizations of the δ-normal set to an arbitrary intersection of sublevel sets for convex functions Φ t which are indexed in an arbitrary set T . Compared to formula (1), the present characterization only involves the reference pointx rather than nearby points, and uses ε-subdifferentials instead of exact ones. Owing to Brøndsted-Rockafellar's theorem ( [2]), these epsilon-like formulas for δnormal sets easily lead to characterizations in the line of formula (1). This passage from ε-subdifferentials to exact ones will require some natural conditions either on the underlying space or on the associated function; typically, that either X is Banach or that Φ satisfies some continuity/coercivity conditions. More generally, we shall prove that if Φ = sup t∈T Φ t , with Φ t : X → R ∪ {+∞} , t ∈ T, being convex, proper and lsc, then for every δ ≥ 0,x ∈ dom Φ and λ ∈ (−∞, +∞] we have that where the subscript ∞ refers to the recession cone, and dom Φ denotes the effective domain of Φ. The sets dom Φ andx + [Φ ≤ Φ(x)] ∞ are superfluous when Φ(x) ≤ λ < ∞, so that the left-hand side in (2) reduces to the usual δ-normal set of [Φ ≤ λ]. However, these two extra sets are necessary to make formula (2) meaningful whenx may lie outside the set [Φ ≤ λ]; for instance, when [Φ ≤ λ] = ∅. The case λ = +∞ is also meaningful since it leads to a characterization of the normal cone to the domain of Φ. It is worth recalling that when Φ t ≡ Φ and Φ(x) ≤ λ, formula (2) yields a well-known result, which was first established in [15] (see, also, [13]). Formula (2) also naturally simplifies under Slater's type conditions, giving rise to familiar results (see [24]). The passage from single to arbitrary intersections of sublevel sets will be made possible through the investigation in this work of new and general (ε-)subdifferential calculus rules for pointwise suprema, extending some previous results on this theme ( [3,7,11,12,10,18,19]). These rules are extensively used at different stages of the proof of (2).
The going back and forth between (1) and (2) will be made clear through the use of Brøndsted-Rockafellar's like-theorem, which allows rewriting (2) by means only of exact subdifferentials. However, this approach requires some extra conditions, in light of the example of proper, convex and lsc functions, defined in non-complete normed spaces, which have an empty subdifferential at every point ( [4]). Despite this limitation, we shall see that the following relation µ∂Φ(x), still hold true for a wide class of functions, which includes for example functions defined on locally convex spaces which either they or their conjugates are finite and continuous at some point. This class, referred to as the class of epi-pointed functions (e.g., [9]), also contains convex functions defined in Banach spaces up to some appropriate localization.
The need for explicit characterizations of the normal cone to sublevel sets is fundamental in optimization theory, namely in the derivation of optimality conditions for convex programming problems (e.g., [23,24]). It is also relevant in the investigation of stationarity and stability properties of different dynamic systems ( [1,5,6]).
The previous formulas are applied at the end of this work to spectral functions ( [16,17]). As it is expected, the associated normal cone will only depend on the values of the function on the range of eigenvalues vectors.
The paper is organized as follows: In section 2 we give the necessary definitions and basic notations. In section 3 we develop some approximate subdifferential calculus rules for the max function, which are needed for our analysis. In section 4 we give in Theorem 5 the main characterization of the normal cone to sublevel sets, which uses the approximate subdifferential. This characterization will be rewritten in section 5, Theorem 15, by means only of exact subdifferentials. Finally, in section 6, we apply the previous result to investigate the case of spectral functions.

Notations and Definitions
We recall in this section some definitions and notations that will be used in the sequel. We consider a dual pair (X, Y ) of real locally convex (lc, for short) spaces X and Y, defined via a bilinear form x * , x := x, x * := x * (x), x * ∈ Y , x ∈ X. By N X (x) we refer to the family of absolutely convex neighborhoods of x. The origin vectors are denoted by θ.
Given a non-empty set S ⊂ X, by S (or cl S), and aff S we denote the closure, and the affine hull of S, respectively. The relative interior of S is the interior of S relative to aff S when this set is closed, and the emptyset otherwise. The polar set, the dual cone, and the orthogonal space of S are the subsets of Y given by for all x ∈ S}, respectively. Given δ ≥ 0, the δ-normal set to S atx is the set we call N S (x) := N 0 S (x) the normal cone to S atx. We fix a function Φ : X → R ∪ {+∞} . We say that Φ is proper if its (effective) domain, dom Φ := {x ∈ X | Φ(x) < ∞} , is nonempty; convex (lower semi-continuous (lsc), resp.), if its epigraph, is convex (closed, resp.). If Φ is proper, convex and lsc we write Φ ∈ Γ 0 (X). The restriction function of Φ to S is denoted by Φ |S . The closed convex hull of Φ is defined as again, if ε = 0, we just call it directional derivative and write Φ ′ (x, v). Equivalently, for ε > 0 we have [26,Theorem 2.4.11] The asymptotic function of Φ, Φ ∞ : The conjugate of Φ is the proper, convex and lsc function defined on Y as The indicator function of S, I S : X → R ∪ {+∞} , is defined by while the support function, σ S : X → R ∪ {+∞} (when S ⊂ Y ), is defined as the conjugate of I S . We shall frequently use the following relation, which holds for every function ϕ : X → R ∪ {+∞} having a proper conjugate, We know that here, S ∞ denotes the asymptotic cone of S defined via the relation I S∞ = (I S ) ∞ .
Recall that for any set T and Φ t ∈ Γ 0 (X), t ∈ T, we have that (see [20]) Finally, given a multifunction M : U ⇒ V, defined between two topological spaces U and (V, τ ), the Painleve-Kuratowski upper limit of M atū ∈ U is defined as for some (v α ) ⊂ U converging toū. If the sets U and V are first countable, then we take sequences instead of nets. We will often omit the reference to τ and just write lim sup u→ū M (u) when the topology τ is understood.
3 ε-subdifferential calculus for pointwise suprema In this section, we develop different rules for the ε-subdifferential mapping of pointwise suprema. The setting here is that of a dual pair (X, Y ) of (real) vector spaces with an associated separating bilinear form denoted by ·, · , so that X and Y are endowed with compatible topologies.
In the following we characterize the ε-subdifferential mapping of the conjugate function.

Lemma 1 Given a function
Proof. To verify the inclusion "⊃", we fix δ > 0 and take . Then x * ∈ ∂ ε i f (x i ) and, so, Multiplying this inequality by λ i and summing up over i, we obtain (recall that f * * = (f * ) * = cof , by (3)) Since Then, by taking the supremum over u in (8), that is, x ∈ ∂ ε+δ f * (x * ). Because δ was arbitrarily chosen, it follows that x ∈ ∂ ε f * (x * ). Consequently, the inclusion "⊃" follows due to the closedness and the convexity of the set ∂ ε f * (x * ).
To establish the inclusion "⊂" we take Given a δ > 0 we choose a V ∈ N X (θ) such that σ V (x * ) ≤ δ. From the definition of cof there are elements We put Observe that from the definition of f * and the relation (9) we have Also, using (10) and the choice of V we obtain (recall (3)) Thus, since (recall (9)) Finally, the arbitrariness of V and δ > 0 leads us to the desired inclusion "⊂".
We give now a formula for the ε-subdifferential of the supremum function, which extends and improves [19,Theorem 1].
Theorem 2 Given set T and functions Φ t ∈ Γ 0 (X), t ∈ T, we put Φ := sup t∈T Φ t . Then for every x ∈ X and ε ≥ 0 we have Proof. We fix x ∈ X and δ > ε ≥ 0. If g := inf t∈T Φ * t , then writing (by (3)) according to Lemma 1 it follows that To establish the inclusion "⊂" of the current theorem we pick , and, consequently, the inclusion "⊂" follows thanks to (12).
To prove the inclusion "⊃" we take and, by multiplying by λ i and summing over i ∈ 1, k, (11)), and so the last . Thus, the desired inclusion "⊃" follows along the arbitrariness of δ > 0.
Theorem 2 becomes more explicit when the index set T is countable and the sequence (Φ n ) n is non-decreasing.
Corollary 3 With the assumptions of Theorem 2 we take T = N. If the sequence (Φ n ) n is non-decreasing, then for every x ∈ X and ε ≥ 0 we have Set m 0 := max i∈1,k n i ≥ 1. Then, from one hand, by the current assumption on the sequence (Φ n ), we obtain On the other hand, by writing the relation ξ i ∈ ∂ β i Φ n i (x) into an inequality form and, next, summing up over i ∈ 1, k, we obtain, for all u ∈ and, so, taking into account (14), by the sum rule of ε-subdifferentials (e.g., [14]) we get Hence, as V and δ were arbitrarily chosen, we deduce that This finishes the proof since the opposite inclusion "⊃" holds straightforwardly.
We recover the subdifferenial rule for the case of finitely many convex functions; see, e.g., [26,Corollary 2.8.11].
Corollary 4 With the assumptions of Theorem 2 we take T = {1, · · · , n}. Then for Proof. We may assume that x ∈ dom Φ = ∩ i∈1,n dom Φ i and ∂ ε Φ(x) = ∅. The inclusion "⊃" is straightforward. For the other inclusion we have, by Theorem 2, Then the desired inclusion follows due to the compactness of the set ∆ n .

ε-subdifferential approach
In this section, we give the desired characterization of δ-normal sets to arbitrary intersections of sublevel sets. As in the previous section, the framework here is that of a dual pair (X, Y ) of (real) lc spaces X and Y, endowed with compatible topologies with respect to a given dual pairing ·, · . We consider a family of proper, lsc, and convex where T is an arbitrary index set, together with the associated supremum function The following theorem provides the main result of this section. Its proof is based on a series of lemmas that we postpone to the Appendix.
Proof. Let us start with the proof of the inclusion "⊃". We take where ν j ↑ λ (observe that one can take ν j = λ when λ is finite). We fix an element u ∈ [Φ ≤ λ] ∩ dom Φ; hence, we may suppose that Φ(u) ≤ ν j for all j. Then from the definition of x * i,j we get Multiplying this last inequality by µ i,j and summing up over i = 1, k j give us which at the limit yields and, so, by (17), Hence, after taking the the limit on j we get x * , u ≤ 0 ≤ δ. Combining this with (18) . This proves the inclusion "⊃".
(3) Φ(x) > λ : In this case we appeal to Lemma 27, which ensures that and we proceed as in point (1) above. The proof of the theorem is complete.
Let us say some words to explain the elements involved in formula (16); namely, the appealing to the set [Φ ≤ Φ(x)] ∞ , and the consideration of the value λ = +∞ : , and so (16) gives the required explicit characterization for the normal cone to [Φ ≤ λ]. The resulting formula in this case can be compared, when all the Φ t 's are all equal, to the one given in [13] (see, also, [15]). Original characterizations for the normal cone to sublevel sets remounts to [24].
(ii) Even with the lack of the nonemptiness of the set [Φ ≤ λ], formula (16) is still meaningful, since the vectorx always belongs tox + [Φ ≤ Φ(x)] ∞ . The other interesting situation covered by (16) In this case, the presence of the termx + [Φ ≤ Φ(x)] ∞ becomes necessary, since, for otherwise, the normal cone to [Φ ≤ λ] atx can not be defined appropriately.
(iii) If λ = +∞, then [Φ ≤ λ] ∩ dom Φ = dom Φ, and the left-hand side in (16) reduces to the normal cone to the domain of Φ. In this case, the relation λ i ↑ λ is used to force the term i∈1,k µ i to go to +∞.
(iv) Due to the relation formula (16) is indeed a characterization of the normal cone to the arbitrary intersection We are going now to specify Theorem 5 to certain special cases, which lead to simpler characterizations of the δ-normal set.
Corollary 6 For everyx ∈ dom Φ and δ ≥ 0 we have that and, particularly, when Φ ≡ Φ t for all t ∈ T, Proof. This follows easily from Theorem 5 due to the relation Formula (16) takes an algebraic form in the following corollary, giving rise to a known formula ( [15]).
In the following corollary we consider the case in which the sublevel set [Φ ≤ λ] is empty.
Proof. The first equality is a direct consequence Theorem 5, while the inclusion "⊃" in the second equality holds easily. To show the converse inclusion, we fix Let t n be a minimizing sequence for this last infimum. Ift ∈ R is an accumulation point of t n , then the last relation above gives Thus we may assume that t n → +∞, so that, by the convexity of Φ, In other words, (∂ ε Φ(x)) − ⊂ (Nx +[Φ≤Φ(x)]∞ (x)) − and, so, using the bipolar Theorem, The following result puts in clear the different sets composing the right-hand side of (16).
and, in particular, Moreover, if Slater's condition holds at Φ(x), then Proof. To prove the first statement of the corollary we take x * ∈ N δ [Φ≤Φ(x)] (x) and, according to Theorem 5, we let µ i , ε i ≥ 0 and We may suppose without loss of generality that µ i converges in R + ∪ {∞} . If µ i → µ ∈ R, by taking the limit on i in the inequality we get x * , y − x ≤ (µΦ)(y) − (µΦ)(x) + δ. So, x * ∈ ∂ δ (µΦ)(x) and the inclusion "⊂" in the first equality holds. The converse inclusion follows by Theorem 5. For the second formula it suffices to observe that, when δ = 0, To prove the last statement of the corollary we only need to verify that the set lim sup µε→δ, µ→∞ µ∂ ε Φ(x) is empty whenever Slater's condition holds at Φ(x). Otherwise, if this were not the case, then there would exist µ i , ε i ≥ 0 such that µ i ε i → δ, µ i → +∞, and µ i x * i ⇀ x * . Then ε i → 0 and x * i ⇀ 0, so that θ ∈ lim sup ε→0 ∂ ε Φ(x) = ∂Φ(x), which contradicts Slater's assumption.
Before we close this section we consider now the setting of Banach spaces and we ask whether formula (16) is still valid when the lim sup is taken with respect to the norm topology. As expected, the answer is affirmative in reflexive Banach spaces as the following corollary shows: Corollary 11 With the notation of Theorem 5 we assume that (X, ) is a reflexive Banach space. Then we have where * denotes the dual norm in X * .

Subdifferential approach
As in the previous sections, here we also work with a dual pair (X, Y ) of lcs X and Y , which are endowed with compatible topologies. Our aim is to characterize the normal cone to sublevel sets by using exclusively the exact subdifferential of the nominal function.
We consider in this section some restrictions either on the underlying space, or on the nominal function. We proceed in this way because of the existence in every non-complete normed space of convex proper lsc functions (e.g., [4]), which have empty subdifferential mapping everywhere. For simplicity of the presentation, we only study the normal cone to the sublevel set In what follows, Φ : X → R ∪ {+∞} is a proper lsc convex function defined on the lcs X. First, we recall the definition of epi-pointed functions.
Definition 1 Function Φ is said to be epi-pointed if its conjugate Φ * is finite and Mackey-continuous at least at some point.
The following lemma gathers some useful properties of epi-pointed functions, which can be found in [9, Lemma 2.1.6 in and Theorem 4.2].
But U • is weak*compact, by the Banach-Alaoglu-Bourbaki theorem, and p V is weak*-continuous, and so the supremum in (24) is finite and we may assume that and we get µx * ε − ξ ∈ V . Finally, with the use of conditions (25) and (26) we can easily see that ξ belongs to the right-hand side of the desired formula. Corollary 14 With the notation of Lemma 13 we have µ∂Φ(x).
Proof. Take x * in the right-hand side, so that and ε > 0, for each i (w.l.o.g.) we obtain This entails that x * ∈ N [Φ≤Φ(x)] (x), whereas Lemma 13 gives us Up to a little modification in the way that one perturbs the nominal pointx, Lemma 13 above is still valid if the epi-pointedness condition is weakened to the continuity of the conjugate relative to its domain. We obtain then the following result, which gives both primal and dual symmetric condition ensuring the same characterization of the normal cone. One can understand that these conditions preclude the subdifferential mapping from being empty everywhere.
The notation x → Ax , A ⊂ X * , used below, refers to the convergence of the corresponding equivalence classes in the quotient space X/A ⊥ . µ∂Φ(x).
Proof. We assume first that Φ * |aff(dom Φ * ) is finite and (Mackey-)continuous in the set ri(dom Φ * ) ( = ∅). In view of Lemma 20 we may assume that θ ∈ ∂Φ(x); otherwise, we are obviously done. Hence, θ ∈ dom Φ * ⊂ aff(dom Φ * ) so that Y * := aff(dom Φ * ) is a closed subspace of X * . Denote Y := X/ aff(dom Φ * ) ⊥ (the quotient space) so that the pair (Y, Y * ) becomes a dual pair when endowed with the quotient and trace topologies of X and X * , respectively. Consider the function Φ defined on Y as Φ(x) = Φ(x) where x is in the equivalent class ofx ∈ Y. It can be easily checked that Φ ∈ Γ 0 (Y ) and that ( Φ) * (y * ) = Φ * (y * ) for all y * ∈ dom Φ * (= dom( Φ) * ); hence, Φ is epi-pointed too. Take x * ∈ N [Φ≤Φ(x)] (x) so that, by Lemma 13, x * ∈ cl(R + (range ∂Φ)) ⊂ cl(R + (dom Φ * )) ⊂ Y * and, hence, x * ∈ N [ Φ≤ Φ( x)] ( x). By applying again Lemma 13, and also Corollary 14, we get µ∂Φ(x), and we conclude. Assume now that Φ |aff(dom Φ) is finite and continuous in ri(dom Φ). Let L be a finitedimensional subspace of X which containsx and denote by Φ L the restriction of Φ to L; hence, Φ * L ∈ Γ 0 (L * ) (L * being the dual space of L) and ri(dom Φ * L ) = ∅. From the first part of the proof applied in the dual pair (L, L * ) we get We pick an . From the last relation above we find In other words, due to the arbitrariness of L, we conclude that x * ∈ lim sup x→ dom Φ * x, µ≥0 µ∂Φ(x), which yields the left inclusion "⊂" of the required statement. The other inclusions are easy and have been proved in previous opportunities (see Corollary 14).
It is worth observing that in the Banach setting any convex function can be made epi-pointed via a penalization with the indicator function of a bounded set. Then we obtain the following result, given originally in [6].

Corollary 16
Assume that X is Banach. Then for everyx ∈ dom Φ we have that where the limit is taken with respect to the weak*-topology.
Proof. We consider the dual pair (X, X * ), where X * is the topological dual of X * endowed with the weak*-topology. Fix a non-zero element ξ ∈ N [Φ≤Φ(x)] (x) and pick a θ-neighborhood V (with respect to the Mackey-topology in X * ) together with an η > 0. Next, we choose a weakly compact (and symmetric) convex set K ⊂ X such that 8K • ⊂ V . We set Φ := Φ + Ix +K (∈ Γ 0 (X)). It is easy to verify that the conjugate function Φ * is finite and bounded from above on x * 0 + K • for some x * 0 ∈ dom Φ * , and, so, is (Mackey-)continuous on x * 0 + K. It follows that Φ is epi-pointed, by definition, so that Lemma 13 applies and yields Thus, for any θ-neighborhood U ∈ N X (θ), there exist µ i > 0, in particular, we have thatx On another hand, by the sum rule in [25], for each i we find Now, for every i and z ∈ K it holds by (34), so that hence, it also follows from the seventh inequality in the table above that Consequently, using successively (27), (32), together with the choice of V in the beginning of the proof, (35) gives us Finally, since x * i ∈ ∂(µ i Φ)(x i ) (recall (30)) and we have that (30) and (29), (36), (30), and (34), together with (34), relation (37) leads us to ξ ∈ lim sup x→x, µ>0 µ∂Φ(x) + V. The desired inclusion follows then by the arbitrariness of V.
Remark 2 It is possible to obtain Corollary 16 directly from Lemma 13 by applying Borwein's version of Brøndsted-Rockafellar's theorem. Nevertheless, our approach permits to highlight the generality of Lemma 13 (and Theorem 15) in the sense that the epi-pointedness condition is not so restrictive as it may appear from a first glance. Let us, for completeness, give the direct proof: Take x * in N [Φ≤Φ(x)] (x). By Corollary 7 for each δ > 0 we have Thus, for every weak* neighborhood W of x * , there are some µ > 0 and x * 0 ∈ ∂ δ (µΦ)(x)∩ W . By Br ondsted-Rockafellar's-like Theorem ( [21]) applied to the function f := µΦ, we find x δ ∈ X and y * δ = µx * δ ∈ µ∂Φ(x δ ) = ∂(µΦ)(x δ ) such that Since W is also open in the norm topology τ · * , by taking δ small enough if necessary, the second inequality above guarantees that x * δ ∈ W . So, we get µ∂Φ(x).

Spectral functions
In this last section, we apply Theorems 5 and 15 to derive characterizations of the normal cone to sublevel sets of a (proper, convex and lsc) spectral function, by means of its restriction to the range of the eigenvalues vectors.
Here, we identify X to the Euclidean space of n × n symmetric matrices with coefficients in R, S n (R), which is endowed with the trace inner product X, Y := tr(XY ). We denote by S n − (R) the cone of semi-definite negative matrices. An extended real-valued matrix function F : S n (R) → R ∪ {+∞} is said to be spectral if it depends only on the eigenvalues of the corresponding matrix; that is, for any A ∈ S n (R) and any orthogonal matrix U (i.e., U ∈ O n (R)); the superscript T denotes the transpose matrix. We shall denote by D the set of (continuous) linear transformations A U : R n → S n (R), U ∈ O n (R), given by where diag(x) is the diagonal matrix whose main diagonal is formed by the elements of vector x.
If F ∈ Γ 0 (S n (R)), then (see [16]) there is a symmetric function f ∈ Γ 0 (R n ) such that F = f • λ, where λ(A) := (λ 1 (A), · · · , λ n (A)) is the spectra of A arranged in a non-increasing order. We know that and, consequently, for every X ∈ S n (R) and ε ≥ 0 it holds that where U ∈ O n (R) satisfies X = A U λ(X). Relation (39) also yields the following comparison between the recession functions of f and f • λ.

Lemma 17 Given a symmetric function
Proof.
Proof. By applying Theorem 5 twice we obtain which yields the first statement of the corollary. The last statement follows from Corollary 16 in a similar way.
In particular, when α = f (λ(X)) ∈ R and δ = 0, the previous characterization gives us where U ∈ O n (R) is such thatX = A U λ(X). The first equality can be easily obtained from a more general result given in [17,Corollary 5.3], by observing that the set C := [f • λ ≤ f (λ(X))] is an invariant set (see [17,Corollary 5.3]). We can proceed similarly is not necessarily invariant.
We consider in the following corollary the special and typical example of the largest eigenvalue function, λ max (X) := max i∈1,n λ i (X).

A Technical Appendix: Auxiliary Results and Proofs
In this appendix we report the auxiliary results which were needed for the proof of Theorem 5. Recall that (X, Y ) denotes a dual pair of (real) vector spaces with an associated separating bilinear form (dual pairing) denoted by ·, · , so that X and Y are endowed with compatible topologies.
The following lemmas give estimations for N δ [Φ≤λ] (x) under different conditions. The first one uses Slater's condition at λ, which means that for some x 0 ∈ X we have Φ(x 0 ) < λ.

Lemma 21
We have that Proof. The first equality is clear because To verify the inclusion we take ξ ∈ N δ dom Φ (x). We choose θ-neighborhood V ⊂ Y and n 0 ≥ 1 such thatx, 2x ∈ V • and Φ(y) − Φ(x) ≥ −n for all y ∈ V • and all n ≥ n 0 ; the existence of such a V is due to the continuity of the dual pairing, while n 0 comes from the weak lower semicontinuity of Φ and the weak compactness of V • . Fix n ≥ n 0 . Hence, for all y ∈ V • ∩ dom Φ and µ ∈ (0, δ n 2 ) we get The last inequality being also true when y ∈ V • \ dom Φ, by using the sum rule for the approximate subdifferential (see, e.g., [14]) we obtain that which proves that ξ ∈ lim sup µε→δ, ε≥0, µ↓0 µ∂ ε Φ(x).
But θ ∈ ∂Φ(x), since Slater's condition does not hold at λ, and so The conclusion follows then from the arbitrariness of ε > 0 and V.
In Lemma 27 below we analyze the case wherex does not belong to the set [Φ ≤ λ].

Lemma 26
With the notation of Lemmas 24 and 25, S is a maximal monotone operator.