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∗ In the sequel, X is a real Banach space with unit ball BX , X is its topological dual, and ., . h i is the duality pairing. For x,y X, we let [x,y] := x + t(y x) : t [0, 1] ; the sets ]x,y[ ∈ { − ∈ } and [x,y[ are defined accordingly. Set-valued operators T : X ⇒ X∗ are identified with their ∗ graph T X X . For a subset A X, x X and λ> 0, we let dA(x) := infy∈A x y ⊂ × ⊂ ∈ k − k and Bλ(A) := y X : dA(y) λ . All extended-real-valued functions f : X ] , + ] { ∈ ≤ } → −∞ ∞ are assumed to be lower semicontinuous (lsc) and proper, which means that the set dom f := x X : f(x) < is non-empty. { ∈ ∞} For a lsc function f : X ] , + ], a pointx ¯ dom f and a direction u X, we → −∞ ∞ ∈ ∈ consider the following basic subderivatives (we essentially follow the terminology of Penot’s textbook [12]): - the (lower right Dini) radial subderivative:
f(¯x + tu) f(¯x) f r(¯x; u) := lim inf − , tց0 t its upper version: r f(¯x + tu) f(¯x) f+(¯x; u) := lim sup − , tց0 t and its upper strict version (the Clarke subderivative):
f(x + tu) f(x) f 0(¯x; u) := lim sup − ; tց0 t (x,f(x))→(¯x,f(¯x))
- the (lower right Dini-Hadamard) directional subderivative:
f(¯x + tu′) f(¯x) f d(¯x; u) := lim inf − , tց0 t u′→u and its upper strict version (the Clarke-Rockafellar subderivative):
f(x + tu′) f(x) f ↑(¯x; u) := sup lim sup inf − . ′ δ>0 tց0 u ∈Bδ(u) t (x,f(x))→(¯x,f(¯x))
It is immediate from these definitions that the following inequalities hold ( means ): → ≤ f r(¯x; u) f r (¯x; u) f 0(¯x; u) → + → ↑ ↑ f d(¯x; u) f ↑(¯x; u) −→ It is well known (and easily seen) that for a function f locally Lipschitz atx ¯, we have f r(¯x; u) = f d(¯x; u) and f 0(¯x; u) = f ↑(¯x; u), whereas for a lsc convex f, we have f d(¯x; u) = f ↑(¯x; u). A function f satisfying such an equality is called regular. However, in general, f d(¯x; u) < f ↑(¯x; u), and there are many other types of subderivatives f ′ which lie between f d and f ↑.
2 The inequality stated in the theorem below is (much) less elementary. It is the analytic form of Treiman’s theorem [14] on the inclusion of the lower limit of Boulingand contin- gent cones at neighbouring points ofx ¯ into the Clarke tangent cone atx ¯ in the context of a Banach space (in finite dimensional spaces, equality holds between these objects, as was shown earlier by Cornet [3] and Penot [11]). A proof of this inequality (or equality in fi- nite dimensional spaces) based on this geometrical approach was given by Ioffe [6] (see also Rockafellar-Wets [13, Theorem 8.18]). For a proof (in the general context of Banach spaces) using a multidirectional mean value inequality rather than the above geometric approach, see Correa-Gajardo-Thibault [4].
Theorem 1 (Link between subderivatives). Let X be a Banach space, f : X ] , + ] → −∞ ∞ be lsc, x¯ dom f and u X. Then: ∈ ∈ f ↑(¯x; u) sup lim sup inf f d(x; u′). ≤ ε>0 x→x¯ u′∈B(u,ε)
3 Subdifferentials
Given a lsc function f : X ] , + ] and a pointx ¯ dom f, we consider the following → −∞ ∞ ∈ two basic subsets of the dual space X∗: - the Moreau-Rockafellar subdifferential (the subdifferential of convex analysis):
∗ ∗ ∗ ∂MRf(¯x) := x X : x ,y x¯ + f(¯x) f(y), y X ; { ∈ h − i ≤ ∀ ∈ } - the Clarke subdifferential, associated to the Clarke-Rockafellar subderivative:
∗ ∗ ∗ ↑ ∂C f(¯x) := x X : x , u f (¯x; u), u X . { ∈ h i≤ ∀ ∈ } All the classical subdifferentials (proximal, Fr´echet, Hadamard, Ioffe, Michel-Penot, . . . ) lie between these two subsets. It is well known that for a lsc convex f, ∂MRf = ∂C f, so all the classical subdifferentials coincide in this case. In the sequel, we call subdifferential any operator ∂ that associates a set-valued mapping ∂f : X ⇒ X∗ to each function f on X so that
∂MRf ∂f ∂C f ⊂ ⊂ and the following Separation Principle is satisfied in X: (SP) For any lsc f, ϕ with ϕ convex Lipschitz near x¯ dom f, if f + ϕ admits a local ∈ minimum at x¯, then 0 ∂f(¯x)+ ∂ϕ(¯x), where ∈ ∗ ∗ b ∗ ∂f(¯x) := x¯ X : there is a net ((xν ,x ))ν ∂f with { ∈ ν ⊂ ∗ w∗ ∗ ∗ b (xν,f(xν )) (¯x,f(¯x)), xν x¯ , lim sup xν ,xν x¯ 0 . (1) → −→ ν h − i≤ }
Remark 1.1. (a) In our paper [9], the set ∂f(¯x) defined in (1) is called the weak*-controlled closure of the set-valued map ∂f at pointx ¯. The reason to consider such a closure is that, b even for a convex lsc function f, the a priori simpler strong weak∗-closure of the graph of ×
3 ∂f = ∂MRf is too big for the Separation Principle to be meaningful. The graph of ∂MRf is not strong weak∗-closed in general: see, e.g., [8] for a discussion on what would be sufficient × to add to the strong weak∗ topology on X X∗ to guarantee the closure of such graphs. × × More precisely, the graph of the convex subdifferential is strong weak∗-closed for each lsc × convex function if and only if X is finite dimensional (see [1]). It is worth noting (and easily seen) that, as expected, always ∂MRf = ∂MRf. ∗ ∗ (b) If we require the net ((xν,xν ))ν b ∂f in (1) to be actually a sequence ((xn,xn))n, ⊂ ∗ n N (in which case the control assertion lim sup x ,xn x¯ 0 is automatically satisfied), ∈ n h n − i≤ we obtain the so-called ‘limiting subdifferentials’. A widely used such limiting subdifferential is the weak∗ sequential closure of the Fr´echet subdifferential, known as the Mordukhovich subdifferential. (c) The Separation Principle (SP) is a very simple property expected to be satisfied by a subdifferential ∂ in a Banach space X. This property is actually equivalent to various other properties of the subdifferential ∂ in the Banach space X: see [9]. We recall that the Clarke subdifferential, the Michel-Penot subdifferential and the Ioffe subdifferential satisfy the Separation Principle in any Banach space. The elementary subdif- ferentials (proximal, Fr´echet, Hadamard, . . . ), as well as their viscosity and limiting versions, satisfy the Separation Principle in appropriate Banach spaces: the Fr´echet subdifferential in Asplund spaces, the Hadamard subdifferential in separable spaces, the proximal subdifferen- tial in Hilbert spaces. The Moreau-Rockafellar subdifferential does not satisfy the Separation Principle for the whole class of lsc (non necessarily convex) functions: it is not a subdiffer- ential for this wide class. See, e.g. [7, 9, 12] and the references therein. The following link between the radial subderivative and arbitrary subdifferentials was established in [10, Theorem 2.1] (see also [9, Theorem 3.2]):
Theorem 2 (Link between radial subderivative and subdifferentials). Let X be a Banach space, f : X ] , + ] be lsc, x¯ dom f and u X. Then, there is a sequence ∗ → −∞ ∞ ∈ ∈ ((xn,x )) ∂f such that xn x¯, f(xn) f(¯x), n ⊂ → → r ∗ ∗ f (¯x; u) lim inf xn, u and lim sup xn,xn x¯ 0. ≤ n h i n h − i≤
4 Subderivative-subdifferential duality formula
A sequence (xn) X is said to be directionally convergent to x¯ from the direction v X, ⊂ ∈ written xn v x¯, if there are two sequences tn 0 (that is, tn 0 with tn > 0) and → ց → vn v such that xn =x ¯ + tnvn for all n; equivalently: for every ε > 0 the sequence → ′ ′ (xn x¯) eventually lies in the open drop ]0, εB(v, ε)[ := tv : 0 < t < ε, v B(v, ε) . − 2 { ∈ } Observe that for v = 0, ]0, εB(v, ε)[ = B(0, ε ) so xn v x¯ simply means xn x¯. We let → → D(¯x,v,ε) :=x ¯ + ]0, εB(v, ε)[. We call subderivative associated to a subdifferential ∂f at a point (¯x, u) dom f X, the ∈ × support function of the set ∂f(¯x) in the direction u, which we denote by
f ∂(¯x; u) := sup x¯∗, u :x ¯∗ ∂f(¯x) . {h i ∈ }
4 In the theorem below, given a function f : X ] , + ], we denote by f ′ : dom f X R → −∞ ∞ × → any function lying between the subderivatives f d and f ↑, that is:
f d f ′ f ↑. ≤ ≤ Subderivatives and subdifferentials are linked by the following formula: Theorem 3 (Subderivative-subdifferential duality formula). Let X be a Banach space, f : X ] , + ] be lsc, x¯ dom f and u X. Then, for any direction v X and any real → −∞ ∞ ∈ ∈ ∈ number α 0, one has ≥ lim sup f r(x; u + α(¯x x)) = lim sup f ′(x; u + α(¯x x)) (2a) x→vx¯ − x→vx¯ − = limsup f ∂(x; u + α(¯x x)). (2b) x→vx¯ − Proof. First step. We claim that
lim sup f ↑(x; u + α(¯x x)) lim sup f d(x; u + α(¯x x)). (3) x→vx¯ − ≤ x→vx¯ − To prove this inequality, we take λ R such that ∈ lim sup f d(x; u + α(¯x x)) < λ (4) x→vx¯ − and show that λ is greater than or equal to the left-hand side of (3). From (4) we can find δ > 0 such that
x D(¯x,v,δ) f d(x; u + α(¯x x)) < λ. (5) ∈ ⇒ − Let z =x ¯ + tv′ D(¯x,v,δ/2) and let µ 5 Second step. We claim that lim sup f r(x; u + α(¯x x)) lim sup f ∂(x; u + α(¯x x)). (9) x→vx¯ − ≤ x→vx¯ − As in the first step, to prove this inequality we take λ R such that ∈ lim sup f ∂(x; u + α(¯x x)) < λ (10) x→vx¯ − and show that λ is greater than or equal to the left-hand side of (9). From (10) we can find δ > 0 such that x D(¯x,v,δ) sup x∗, u + α(¯x x) :x ¯∗ ∂f(x) < λ. (11) ∈ ⇒ {h − i ∈ } Let z =x ¯ + tv′ D(¯x,v,δ/2). By Theorem 2, for any µ z D(¯x,v,δ/2) f r(z; u + α(¯x z)) < λ, ∈ ⇒ − showing that (9) holds. ∂ ′ ∂ ′ ↑ ′ ′ Third step. Since ∂f ∂C f, we have f (z; u ) f C (z; u ) f (z; u ) for every u X. ⊂ ≤ ≤ ∈ Hence, the right-hand side of (9) is less than or equal to the left-hand side of (3). On the other hand, f d f r. So all the expressions in formulas (3) and (9) are equal. The desired ≤ set of equalities (2a)–(2b) follows because f d f ′ f ↑. ≤ ≤ Remark 3.1. (a) In the special case v = 0 and α = 0, the formula (2a) was proved by Borwein-Str´ojwas [2, Theorem 2.1 and Corollary 2.3] (b) For f locally Lipschitz atx ¯, the formulas (2) do not depend on α 0 since ≥ lim sup f r(x; u + α(¯x x)) = lim sup f r(x; u). x→vx¯ − x→vx¯ But they may depend on the direction v X: for f : x R f(x) := x and u = 0, one ∈ ∈ 7→ −| | 6 has lim sup f r(x; u)= u < lim sup f r(x; u)= u . x→u0 −| | x→0 | | (c) For arbitrary lsc f, the value of the expressions in (2) depends on α 0 even for ≥ convex f. Indeed, as was recalled in Remark 1.1 (a), the graph of the subdifferential ∗ ∗ ∗ r ∂MRf(¯x)= x X : x , u f (¯x; u), u X { ∈ h i≤ ∀ ∈ } 6 is generally not strong weak∗-closed. Therefore, for an arbitrary lsc convex f the function × x f r(x; u) is generally not upper semicontinuous, that is 7→ f r(¯x; u) < lim sup f r(x; u), x→x¯ while always (see Proposition 4 below) f r(¯x; u)= inf lim sup f r(x; u + α(¯x x)). α≥0 x→x¯ − Proposition 4 (Radial subderivative for convex lsc functions). Let X be a Banach space, f : X ] , + ] be convex lsc, x¯ dom f and u X. Then, → −∞ ∞ ∈ ∈ f r(¯x; u)= inf lim sup f r(x; u + α(¯x x)). (12) α≥0 x→x¯ − Proof. Of course, f r(¯x; u) is always not greater than the expression of the right-hand side of (12). It is not smaller either since, for every t> 0, lim sup f r(x; tu +x ¯ x) lim sup (f(¯x + tu) f(x)) = f(¯x + tu) f(¯x), x→x¯ − ≤ x→x¯ − − hence, writing α = 1/t for t> 0, 1 inf lim sup f r(x; u + α(¯x x)) inf lim sup f r(x; tu +x ¯ x) α≥0 x→x¯ − ≤ t>0 x→x¯ t − f(¯x + tu) f(¯x) inf − = f r(¯x; u). ≤ t>0 t 5 Radially accessible functions A lsc function f : X ] , + ] is said to be radially accessible atx ¯ dom f from a → −∞ ∞ ∈ direction u X provided ∈ f(¯x) = lim inf f(¯x + tu), tց0 or equivalently, provided there exists a sequence tn 0 such that f(¯x + tnu) f(¯x). (The ց → case u = 0 is a tautology.) Examples. 1. Every lsc function f : X ] , + ] which is radially upper semicontinuous → −∞ ∞ atx ¯ dom f from u is evidently radially accessible atx ¯ from u. This is the case of convex ∈ lsc functions f for any u X such thatx ¯ + u dom f. ∈ ∈ 2. If f r(¯x; u) < , then f is radially accessible atx ¯ from u. Indeed, let γ R such that r ∞ ∈ f (¯x; u) < γ. Then, there exists tn 0 such that f(¯x + tnu) f(¯x)+ γtn, and consequently ց r ≤ lim sup f(¯x + tnu) f(¯x). The condition f (¯x; u) < however is not necessary: the n ≤ ∞ continuous function f : R R given by f(x) := x has f r(0; u)= for any u = 0. → | | ∞ 6 3. The function f : R R given by p → 0 if x =0 or x = 1/n, for n = 1, 2,... f(x) := 1 otherwise. is lsc on R, not upper semicontinuous at 0 along the ray R+u for u> 0 but radially accessible at 0 from such u> 0. 7 4. The function f : R R given by → 1 if x> 0 f(x) := 0 if x 0 ≤ is lsc on R but not radially accessible at 0 from u = 1. We notice that f r(0; 1) = + , while ∞ f r(x; 1) = 0 for any x> 0. Radial accessibility is a mild regularity property. Yet this property leads to a more consistent behaviour of subdifferentials and subderivatives. We give two illustrations. Assume the lsc function f is radially accessible atx ¯ dom f from a direction u. Then first, dom ∂f ∈ contains a sequence graphically and directionally convergent tox ¯ (Theorem 5), and second, r the upper radial subderivative f+(¯x; u) is stable with respect to radially convergent sequences (Proposition 7). From the latter statement we derive that the upper radial subderivative is a lower bound for the expressions in (2) with v = u (Theorem 8). We recall the statement of Ekeland’s variational principle [5]: Variational Principle. For any lsc function f defined on a closed subset S of a Banach space,x ¯ dom f and ε > 0 such that f(¯x) inf f(S)+ ε, and for any λ > 0, there exists ∈ ≤ xλ S such that xλ x¯ λ, f(xλ) f(¯x) and the function x f(x) + (ε/λ) x xλ ∈ k − k ≤ ≤ 7→ k − k attains its minimum on S at xλ. Theorem 5 (Directional density of subdifferentials). Let X be a Banach space, f : X → ] , + ] be lsc, x¯ dom f and u X such that f is radially accessible at x¯ from u. −∞ ∞ ∈ ∗∈ Then, there exists a sequence ((xn,xn))n ∂f such that xn u x¯, f(xn) f(¯x) and ∗ ⊂ → → lim sup x ,xn x¯ 0. nh n − i≤ Proof. Let ε> 0. Since f is lsc atx ¯, there exists δ ]0, ε2[ such that ∈ 2 f(¯x) inf f(Bδ(¯x)) + ε , (13) ≤ and since f(¯x) = lim inftց f(¯x + tu), there exists µ > 0 such that µ(ε + u ) < δ and 0 k k f(¯x + µu) f(¯x)+ ε2. Summarizing, we can find real numbers δ and µ satisfying ≤ 0 <µ(ε + u ) <δ<ε2, and (14a) k k 2 f(¯x + µu) inf f(Bδ(¯x)) + ε . (14b) ≤ Now we apply Ekeland’s variational principle to f on the set Bδ(¯x) at pointx ¯ + µu with λ = µε. Observe that the ball Bλ(¯x + µu) is contained in the ball Bδ(¯x) by (14a). We therefore obtain a point xε X such that ∈ xε (¯x + µu) <µε, f(xε) f(¯x + µu), and (15a) k − k ≤ y f(y) + (ε/µ) y xε admits a local minimum at xε. (15b) 7→ k − k In view of (15b), we may apply the Separation Principle at point xε with the convex Lipschitz ∗ function ϕ : y (ε/µ) y xε to obtain a subgradient x ∂f(xε) such that 7→ k − k ε ∈ x∗ ε/µ. b (16) k εk≤ 8 ∗ Now, take (¯xε, x¯ ) ∂f such that ε ∈ 2 x¯ε xε < µε, f(¯xε) f(xε) < ε , and (17a) k ∗ − ∗k | − ∗ | x¯ x , x¯ xε > ε, x¯ , x¯ε xε < ε. (17b) h ε − ε − i − h ε − i It follows from the first parts of (15a) and (17a) that x¯ε (¯x + µu) < 2µε, (18) k − k from the second parts of (15a) and (17a) combined with (13) and (14b) that 2 f(¯xε) f(¯x) 2ε , (19) | − |≤ and from (16) and (17b) that ∗ x¯ , x¯ x¯ε > ε(ε + u + 2), (20) h ε − i − k k since ∗ ∗ ∗ ∗ ∗ x¯ε, x¯ x¯ε = xε, x¯ xε + x¯ε xε, x¯ xε + x¯ε,xε x¯ε h − i h ∗ − i h − − i h − i > x xε x¯ 2ε −k εkk − k− > (ε/µ)µε + µ u 2ε = ε(ε + u + 2). − k k− − k k Therefore, if for every n N, we let ε = 1/n and choose µn = µ satisfying (14a), so that 0 < ∈ ∗ ∗ ∗ µn < 1/n, we obtain a sequence ((xn,xn))n in ∂f by setting xn :=x ¯ε and xn :=x ¯ε. It follows from (18), (19) and (20) that this sequence satisfies the requirements of the theorem. Remark 5.1. (a) The case u = 0 in Theorem 5 is known, see, e.g. [9, 10]. (b) The case u = 0 is new even for convex lsc functions (recall that such functions are 6 radially accessible at any pointx ¯ dom f from any u such thatx ¯ + u dom f). ∈ ∈ (c) For u = 0, the conclusion of Theorem 5 can be false at points where the function is 6 not radially accessible. Let f : R ] , + ] given by → −∞ ∞ 0 if x =0 or x = 1/n, for n = 1, 2,... f(x) := + otherwise. ∞ Then, f is lsc on R but not radially accessible at any pointx ¯ = 1/n from u = 0. We observe 6 that all the points x =x ¯ close tox ¯ are not in dom f, hence ∂f(x)= . 6 ∅ (d) For u = 0, we cannot claim in the conclusion of Theorem 5 to find a radially convergent 6 2 sequence (xn) instead of a directionally convergent one. Consider the function f : R → ] , + ] given, for x = (ξ ,ξ ), by −∞ ∞ 1 2 √ξ if (ξ ,ξ ) R R f(x) := − 1 1 2 ∈ + × + otherwise. ∞ Then, f is convex lsc on R2 and f(0,t) = 0 for every t R, so f is radially continuous at ∈ x¯ = (0, 0) in the direction u = (0, 1). But, for every t R we have ∂f(0,t) = , so there is ∈ ∅ no sequence (xn) radially convergent tox ¯ from the direction u = (0, 1) with ∂f(xn) = . 6 ∅ 9 We recall the statement of the mean value inequality using the radial subderivative [9, 10]: Lemma 6 (Basic mean value inequality). Let X be a Hausdorff locally convex space, f : X ] , + ] be lsc, x¯ X and x dom f. Then, for every real number λ f(¯x) f(x), → −∞ ∞ ∈ ∈ ≤ − there exist t [0, 1[ and x := x + t (¯x x) [x, x¯[ such that f(x ) f(x)+ t λ and 0 ∈ 0 0 − ∈ 0 ≤ 0 λ f r(x ;x ¯ x). ≤ 0 − Proposition 7 (Radial stability of the upper radial subderivative). Let X be a Hausdorff locally convex space, f : X ] , + ] be lsc, x¯ dom f and u X such that f is radially → −∞ ∞ ∈ ∈ accessible at x¯ from u. Then, there is a sequence µn 0 such that f(¯x + µnu) f(¯x) and ց → r r f (¯x; u) lim inf f (¯x + µnu; u). (21) + ≤ n→+∞ In particular, r r f+(¯x; u) inf lim sup f (x; u + α(¯x x)). (22) ≤ α≥0 x→ux¯ − r Proof. If f+(¯x; u) = , there is nothing to prove. Otherwise, it suffices to show that for r −∞ every λ λτn λτn Applying Lemma 6 to (24a) we obtain t [0, 1[ and x :=x ¯ + tk u + t (τn tk )u =x ¯ + µnu 0 ∈ 0 n 0 − n with µn := tk + t (τn tk ) [tk ,τn[ such that n 0 − n ∈ n f(¯x + µnu) f(¯x + tkn u)+ t0λτn, (25a) ≤ r λτn/(τn tk ) Observe that in view of (24b) one has τn/(τn tk ) 1, so, letting n + we obtain − n → → ∞ r µn 0, lim sup f(¯x + µnu) f(¯x), λ lim inf f (¯x + µnu; u). ց n→+∞ ≤ ≤ n→+∞ This completes the proof of the first statement since we also have f(¯x) lim inf f(¯x + µnu) ≤ n→+∞ by the lower semicontinuity of f atx ¯. 10 To show the second statement, let xn :=x ¯ + µnu with µn 0 such that ց r r f (¯x; u) lim inf f (xn; u). (26) + ≤ n→+∞ Since u + α(¯x xn) = (1 αµn)u, it follows that, for any α 0, − − ≥ r r r lim inf f (xn; u + α(¯x xn)) = lim inf (1 αµn)f (xn; u) = lim inf f (xn; u). (27) n→+∞ − n→+∞ − n→+∞ Since xn u x¯, we derive from (26) and (27) that, for every α 0, → ≥ r r r f+(¯x; u) lim inf f (xn; u + α(¯x xn)) lim sup f (x; u + α(¯x x)). ≤ n→+∞ − ≤ x→ux¯ − Plugging the formula (2b) of Theorem 3 into the inequality (22) of Proposition 7, we r immediately obtain that the upper radial subderivative f+(¯x; u) is a lower bound for the directional limit superior of the support of any subdifferential: Theorem 8 (Link between upper radial subderivative and subdifferentials). Let X be a Banach space, f : X ] , + ] be lsc, x¯ dom f and u X such that f is radially → −∞ ∞ ∈ ∈ accessible at x¯ from u. Then, r ∂ f+(¯x; u) inf lim sup f (x; u + α(¯x x)). (28) ≤ α≥0 x→ux¯ − 6 Appendix: a direct proof of Theorem 8 For the sake of completeness, we provide a direct proof Theorem 8. In fact, we shall establish an inequality more accurate than (28), in the same vein as Theorem 2 and Theorem 5. Theorem 9 (Refined link between upper radial subderivative and subdifferentials). Let X be a Banach space, f : X ] , + ] be lsc, x¯ dom f and u X such that f is radially → −∞ ∞ ∈ ∗ ∈ accessible at x¯ from u. Then, there is a sequence ((xn,x )) ∂f such that xn u x¯, n ⊂ → f(xn) f(¯x) and → r ∗ f (¯x; u) lim inf x , u + α(¯x xn) , α 0. (29) + ≤ n h n − i ∀ ≥ Proof. The pattern of the proof is similar to that of [10, Theorem 2.1] but the argument has to be refined in order to obtain a directionally convergent sequence (xn). r First step. If u = 0 or if f+(¯x; u)= , the result follows from Theorem 5. Otherwise, r −∞ N assume u = 0, let γ f(¯x + µnu) f(¯x) < 1/n, (31a) | r − | γ 11 We may assume 0 <µn < √δ. By (31b), there exists tn ]0, 1 µn] such that ∈ − f(¯x + µnu) f(¯x + µnu + tu) γt, t [0,tn]. (32) ≤ − ∀ ∈ Let Kn := [¯x + µnu, x¯ + (µn + tn)u] K. Then, (32) can be rewritten as ⊂ g(¯x + µnu) g(x), x Kn. (33) ≤ ∀ ∈ Take r> 0 such that 3 g(¯x + µnu) < inf g + µntn, (34) Br(Kn) and, observing that both infBr(Kn) g and infBδ(Kn) g are finite, choose αn > 0 such that 2 inf g inf g + αnr . Br(Kn) ≤ Bδ (Kn) Then 2 inf g (g + αndKn )(x), x Bδ(Kn), Br(Kn) ≤ ∀ ∈ and therefore, by (34), 2 3 g(¯x + µnu) (g + αnd )(x)+ µ tn, x Bδ(Kn). (35) ≤ Kn n ∀ ∈ 2 Now, apply Ekeland’s variational principle to the function g + αndKn on the set Bδ(Kn) at 3 2 pointx ¯ + µnu Kn with ε = µ tn and λ = µ tn. Observe that the ball Bλ(¯x + µnu) is ∈ n n contained in Bδ(Kn) since for every x Bλ(¯x + µnu), we have dKn (x) x (¯x + µnu) 2 ∈ ≤ k − k≤ λ = µ tn < δ. We then obtain a point xn X satisfying n ∈ 2 2 xn (¯x + µnu) <µ tn, g(xn)+ αnd (xn) g(¯x + µnu) (36a) k − k n Kn ≤ ∗ 2 y f(y)+ z ,y + αnd (y)+ µn y xn admits a local minimum at xn. (36b) 7→ h i Kn k − k It follows from the first half of (36a) that 2 xn x¯ B(µnu, µ tn)= µnB(u, µntn), − ∈ n showing that xn D(¯x, u, ε) for n sufficiently large. On the other hand, the second half of ∈ (36a) and (31a) entail ∗ ∗ 2 f(xn) f(¯x + µnu)+ z x¯ + µnu xn f(¯x)+ z µ tn + 1/n, ≤ k kk − k≤ k k n showing that f(xn) In view of (36b), we may apply the Separation Principle at point xn with the convex ∗ 2 ∗ Lipschitz function ϕ : y z ,y + αndKn (y)+ µn y xn to obtain points xn ∂f(xn), ∗ 2 ∗ 7→∗ h i k − k ∈ ζ ∂d (xn) and β B with n ∈ Kn n ∈ b ∗ ∗ ∗ ∗ 0= xn + z + αnζn + µnβn. (37) ∗ We claim that the pair (xn,x ) ∂f satisfies (30b) for large n N. Assume it can be shown n ∈ ∈ that for all α 0 and for large n N, ≥ ∈b ∗ ζ , u + α(¯x xn) 0. (38) h n − i≤ 12 Then, it follows that for all α 0 and for large n N, ≥ ∈ ∗ ∗ ∗ ∗ xn, u + α(¯x xn) = z , u + α(¯x xn) αn ζn, u + α(¯x xn) 2µn βn, u + α(¯x xn) h − i h− ∗ − i− h − i− h − i γ α z x¯ xn 2µn u + α(¯x xn) , ≥ − k kk − k− k − k ∗ which implies that x , u + α(¯x xn) > γ (α + 1)ε for n sufficiently large, as claimed. h n − i − Second step. To complete the proof of (30b) it remains to prove (38). We first consider the case α = 0, that is, we show ζ∗, u 0, n N. (39) h n i≤ ∀ ∈ Let PK xn Kn be any point such that xn PK xn = dK (xn). We have n ∈ k − n k n 2 x¯ + µnu PK xn x¯ + µnu xn + xn PK xn < 2µ tn. k − n k ≤ k − k k − n k n 2 So, PK xn =x ¯ + µnu + τnu with τn u < 2µ tn. Hence, tn τn > 0 for large n, and n k k n − (tn τn)u =x ¯ + (µn + tn)u PK xn. − − n ∗ ∗ ∗ Notice that ζ = 2dK (xn)ξ where ξ ∂dK (xn). Then: n n n n ∈ n ∗ ∗ (tn τn) ζn, u = ζn, x¯ + (µn + tn)u PKn xn − h i h ∗ − i ∗ = ζn, x¯ + (µn + tn)u xn + ζn,xn PKn xn h ∗ − i h − ∗ i = 2dK (xn) ( ξ , x¯ + (µn + tn)u xn + ξ ,xn PK xn ) n h n − i h n − n i 2dK (xn)( dK (xn)+ xn PK xn ) = 0. ≤ n − n k − n k This proves (39). Now consider the case α> 0. Write α = 1/t. We must show that for large n N, ∈ ∗ ∗ ζ , u + α(¯x xn) = (1/t) ζ , x¯ + tu xn 0. (40) h n − i h n − i≤ But, for n so large that µn Fourth step. The theorem is derived from (30a)–(30b) as follows. 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