Journal of Convex Analysis Volume 3 (1996), No. 2, 207–219

Subdifferential Calculus Without Qualification Assumptions

Jean-Paul Penot

Université de Pau et des Pays de l’Adour, Lab. de Math. Appliquées, URA – CNRS 1204, Avenue de l’Université, 64000 Pau, France. e-mail: jean-paul.penot@univ-pau.fr

Received October 10, 1995 Revised manuscript received January 9, 1996

Some classical formulas of convex subdifferential calculus in a general Banach space are presented without any qualification condition, but in a limiting form. Classical formulas are recovered when a qualification condition such as the Attouch-Brézis condition is assumed.

Several decades after the blossoming of convex analysis, the subject of subdifferential calculus rule is still very active (see [1]–[4], [15], [16], [23], [30]–[32] for instance). There are several reasons for that; among them is the attention devoted to calculus rules. Simple formulas such as

∂ (f_{1} + f_{2}) (x) = ∂f_{1} (x) + ∂f_{2} (x)  (0.1)
∂(g ∘ A)(x) = ∂g(A(x)) ∘ A,  (0.2)

in which A is linear continuous, f_{1}, f_{2} and g are closed proper convex functions finite at x and A(x) respectively, are not valid without additional assumptions, a well known and disappointing fact, especially for beginners in convex analysis. A general formula for ∂ (f_{1} + f_{2}) (x) which does not require any qualification condition has been proposed in [15] (see also [16]). But it involves the approximate subdifferential ∂_{ε}f of convex analysis. Thus this condition is specific to the convex situation. It is our purpose here to present a formula which is valid without any additional condition and which uses only the ordinary subdifferential of convex analysis. Such an aim has already been pursued in [2], [3], [14] and [29]; moreover, in [2], [3] a connection with a notion of variational sum of operators is pointed out. However the conditions we impose on the sequences involved in our formula differ from the conditions of Attouch-Baillon-Théra and Thibault: they are more precise and their interpretation is easier and more amenable to the general nonconvex, nonsmooth case. Since our approach aims at a link with the general situation of nonsmooth analysis, an interest for such conditions seems to be justified. It may have consequences on the evolution of nonsmooth analysis. Moreover, our conditions bear on the coupling functional and it is well known in nonlinear functional analysis and in the study of partial differential equations that delicate limiting procedures are to be found for such terms (see [7], [10],

[19] for instance). Thus the analysis of convex functionals which are obtained in terms of integrals or integro-differential terms as in [5], [8], [13], [25], [26] may benefit from a closer analysis of calculus rules.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

208 J-P Penot / Subdifferential calculus without qualification assumptions

1. Preliminaries

The following result will be our basic tool. It consists in a slight supplement to the famous Brønsted-Rockafellar’s theorem [9] and to an extension due to J. Borwein [6], [23]. It has its own interest. It involves the ε-approximate subdifferential of f given by

∂_{ε}f(u) := {u^{∗} ∈ X^{∗} : ∀x ∈ X f(x) − f(u) ≥ ⟨u^{∗}, x − u⟩ − ε} .

This set is a nonempty weak^{∗} closed convex set whose support function h(·, ∂_{ε}f(x_{0})) := sup {⟨·, x_{0}^{∗}⟩ : x_{0}^{∗} ∈ ∂_{ε}f(x_{0})} satisfies

h(v, ∂_{ε}f(x_{0})) = inf_{t>0} \frac{1}{t} (f(x_{0} + tv) − f(x_{0}) + ε)  (1.1)

for each v ∈ X. Moreover one has

∂f(x_{0}) = ⋂_{ε>0} ∂_{ε}f(x_{0}).  (1.2)

Proposition 1.1. Given a closed proper convex function f on a Banach space X, α > 0, ε > 0, u_{0} ∈ dom f, u_{0}^{∗} ∈ ∂_{ε}f (u_{0}) one can find (x_{ε}, x_{ε}^{∗}) ∈ ∂f and γ ∈ [−1, 1] such that

\begin{aligned}∥x_{ε} − u_{0}∥ + α | ⟨u_{0}^{∗}, x_{ε} − u_{0}⟩ | & ≤ \sqrt{ε}, \\ ∥x_{ε}^{∗} − (1 + αγ\sqrt{ε})u_{0}^{∗}∥ & ≤ \sqrt{ε}, \\ |⟨x_{ε}^{∗} − u_{0}^{∗}, x_{ε} − u_{0}⟩| & ≤ ε, \\ |⟨x_{ε}^{∗}, x_{ε} − u_{0}⟩| & ≤ ε + α^{−1}\sqrt{ε}, \\ |f (x_{ε}) − f(u_{0})| & ≤ ε + α^{−1}\sqrt{ε}. \end{aligned}

Further on we will take α = 1, but it may be useful to dispose of such a parameter: the choice α = ε^{− \frac{1}{2}} is also an interesting one.

Proof. We follow the general line of the proof of [6]. Here we endow X with the equivalent norm ∥ · ∥_{α} given by ∥x∥_{α} := ∥x∥ + α | ⟨u_{0}^{∗}, x⟩ | for x ∈ X. Since u_{0}^{∗} ∈ ∂_{ε}f (u_{0}) , u_{0} is an ε-approximate minimizer of the function g := f − ⟨u_{0}^{∗}, .⟩. The Ekeland’s variational principle (see for instance [12]) applied to g yields x_{ε} ∈ X such that

g(x_{ε}) + \sqrt{ε}∥x_{ε} − u_{0}∥_{α} ≤ g(u_{0}),
0 ∈ ∂ (g + \sqrt{ε} ∥· − x_{ε}∥_{α}) (x_{ε}) .

Since g(u_{0}) ≤ g(x_{ε}) + ε, the first relation implies that

∥x_{ε} − u_{0}∥ + α | ⟨u_{0}^{∗}, x_{ε} − u_{0}⟩ |≤ \sqrt{ε}.

Using familiar subdifferential calculus rules, and denoting by B^{∗} the closed unit ball of X^{∗}, the second relation yields

u_{0}^{∗} ∈ ∂ (f + \sqrt{ε} ∥· − x_{ε}∥ + α\sqrt{ε} | ⟨u_{0}^{∗}, · − x_{ε}⟩ |) (x_{ε}) ⊂ ∂f (x_{ε}) + \sqrt{ε}B^{∗} + α\sqrt{ε} [−1, 1] u_{0}^{∗}.

In other terms, there exists x_{ε}^{∗} ∈ ∂f (x_{ε}) , γ ∈ [−1, 1] and v^{∗} ∈ B^{∗} such that

x_{ε}^{∗} = (1 + αγ\sqrt{ε})u_{0}^{∗} + \sqrt{ε}v^{∗}.

It follows that

|⟨x_{ε}^{∗} − u_{0}^{∗}, x_{ε} − u_{0}⟩| ≤| αγ\sqrt{ε}⟨u_{0}^{∗}, x_{ε} − u_{0}⟩ | + | \sqrt{ε}⟨v^{∗}, x_{ε} − u_{0}⟩ |≤ ε

and

|⟨x_{ε}^{∗}, x_{ε} − u_{0}⟩| ≤ ε+ | ⟨u_{0}^{∗}, x_{ε} − u_{0}⟩ |≤ ε + α^{−1}\sqrt{ε}.

Since x_{ε}^{∗} ∈ ∂f (x_{ε}) one has

f (u_{0}) − f (x_{ε}) ≥ ⟨x_{ε}^{∗}, u_{0} − x_{ε}⟩ ≥ −ε − α^{−1}\sqrt{ε}.

Since u_{0}^{∗} ∈ ∂_{ε}f (u_{0}) one has

f (x_{ε}) − f (u_{0}) ≥ ⟨u_{0}^{∗}, x_{ε} − u_{0}⟩ − ε ≥ −α^{−1}\sqrt{ε} − ε

and the last inequality of the statement holds.

The last assertion of the following corollary could not be obtained without the additional information we got. Thus, it shows the importance of this information. It suggests a new definition of stabilized or limiting normals in the nonconvex case; such a proposal will not be considered here.

Corollary 1.2. Given x_{0} in the domain dom f of a closed proper convex function f on X one can find a sequence (x_{n}, x_{n}^{∗}) in the graph of ∂f such that (x_{n}) → x_{0}, (f (x_{n})) → f (x_{0}) , (⟨x_{n}^{∗}, x_{n} − x_{0}⟩) → 0. Moreover, each possible weak^{∗} cluster point of a sequence or a net (x_{n}^{∗}) satisfying these properties belongs to ∂f(x_{0}).

Proof. For each n ∈ IN \ {0} we can find y_{n}^{∗} ∈ ∂_{1/n^{2}}f (x_{0}) . Setting u_{0} = x_{0}, u_{0}^{∗} = y_{n}^{∗}, α = 1, ε = 1/n^{2} in the preceding proposition, we get (x_{n}, x_{n}^{∗}) ∈ ∂f such that

\begin{aligned}∥x_{n} − x_{0}∥ & ≤ 1/n, \\ |f (x_{n}) − f (x_{0})| & ≤ 2/n, \\ |⟨x_{n}^{∗}, x_{n} − x_{0}⟩| & ≤ 2/n. \end{aligned}

If x_{0}^{∗} is a weak^{∗} cluster point of (x_{n}^{∗}), for each x ∈ X one has

\begin{aligned}f(x) & ≥ lim sup_{n}(f(x_{n}) + ⟨x_{n}^{∗}, x − x_{0}⟩ + ⟨x_{n}^{∗}, x_{0} − x_{n}⟩) \\ & ≥ f(x_{0}) + ⟨x_{0}^{∗}, x − x_{0}⟩. \end{aligned}

so that x_{0}^{∗} ∈ ∂f(x_{0}).

We will make use of the following recent results whose proofs are elementary.

Proposition 1.3. ([15], [16]) Let A be a linear continuous operator between two Banach spaces X, Y, and let g : Y → IR^{•} = IR ∪ {∞} be a closed convex function finite at some y_{0} ∈ Y. Then, denoting by A^{T} the transpose of A and by cl^{∗}C the weak^{∗}-closure of a subset C of X^{∗} one has, for any x_{0} ∈ A^{−1}(y_{0}),

∂(g ∘ A)(x_{0}) = ⋂_{ε>0} cl^{∗}A^{T} (∂_{ε}g(y_{0})).  (1.3)

When X is reflexive, the weak^{∗} closure can be replaced by the norm closure.

This result is a consequence of relations ((1.1), (1.2)) and of the relation

∂_{ε}f(x_{0}) = cl^{∗}(A^{T} (∂_{ε}g(y_{0}))),  (1.4)

where f := g ∘ A; this relation amounts to the obvious equalities

\begin{aligned}h(v, cl^{∗}(A^{T} (∂_{ε}g(y_{0})))) & = h(v, A^{T} (∂_{ε}g(y_{0}))) = h(Av, ∂_{ε}g(y_{0})), \\ inf_{t>0} \frac{1}{t} (f(x_{0} + tv) − f(x_{0}) + ε) & = inf_{t>0} \frac{1}{t} (g(y_{0} + tAv) − g(y_{0}) + ε). \end{aligned}

Proposition 1.4. ([15], [16]) Given two closed proper convex functions f_{1}, f_{2} on the Banach space X and x_{0} ∈ dom f_{1} ∩ dom f_{2} one has

∂ (f_{1} + f_{2}) (x_{0}) = ⋂_{ε>0} cl^{∗} (∂_{ε}f_{1} (x_{0}) + ∂_{ε}f_{2} (x_{0})) .

2. The main results

The main results we have in view are the following ones. Let us start with composition, the fundamental operation from the point of view of category theory. We first consider a simple case.

Proposition 2.1. Let A be a linear continuous operator between two Banach spaces X, Y, let b ∈ Y and let g : Y → IR^{•} := IR ∪ {∞} be a closed proper convex function. Let f := g ∘ (A + b), x_{0} ∈ dom f. Suppose that for each element ε of a subset E of (0, ∞) containing 0 in its closure the set A^{T} (∂_{ε}g(y_{0})) is weak^{∗}-closed. Then for each x_{0}^{∗} ∈ ∂f(x_{0}) there exist sequences (y_{n})_{n∈IN}, (y_{n}^{∗})_{n∈IN} such that y_{n}^{∗} ∈ ∂g(y_{n}) for each n ∈ IN, (y_{n})_{n∈IN} → y_{0} := A(x_{0}) + b, (g(y_{n}))_{n∈IN} → g(y_{0}), (A^{T} (y_{n}^{∗}))_{n∈IN} → x_{0}^{∗}, and (⟨y_{n}^{∗}, y_{n} − y_{0}⟩)_{n∈IN} → 0.

Proof. Let (ε_{n}) be a sequence of E converging to 0. Proposition 1.3 enables one to find some u_{n}^{∗} ∈ ∂_{ε_{n}}g(y_{0}) such that x_{0}^{∗} = A^{T} (u_{n}^{∗}). Then Proposition 1.1 provides some y_{n} ∈ B(y_{0}, ε_{n}), y_{n}^{∗} ∈ ∂g(y_{n}), γ_{n} ∈ [−1, 1] such that ∥y_{n}^{∗} − (1 + γ_{n}\sqrt{ε_{n}})u_{n}^{∗}∥ ≤ \sqrt{ε_{n}}, | ⟨y_{n}^{∗}, y_{n} − y_{0}⟩ |≤ ε_{n} + \sqrt{ε_{n}}, | g(y_{n}) − g(y_{0}) |≤ ε_{n} + \sqrt{ε_{n}}. As (1 + γ_{n}\sqrt{ε_{n}})A^{T} (u_{n}^{∗}) = (1 + γ_{n}\sqrt{ε_{n}})x_{0}^{∗} → x_{0}^{∗}, we get the result.

When X is reflexive, the condition we obtained characterizes the elements of ∂f(x_{0}), as shown at the end of the proof of the following statement which deals with the general case.

Theorem 2.2. Let A, b, f, g be as above. Then, for any x_{0} ∈ dom f the subdifferential of f at x_{0} is the set of weak^{∗}-limit points x_{0}^{∗} of nets (x_{n}^{∗})_{n∈N} such that there exists a net (y_{n})_{n∈N} converging to y_{0} := A(x_{0}) + b and for each n ∈ N, some y_{n}^{∗} ∈ ∂g(y_{n}) with x_{n}^{∗} = A^{T} (y_{n}^{∗}), (g(y_{n}))_{n∈N} → g(y_{0}) and (⟨y_{n}^{∗}, y_{n} − y_{0}⟩)_{n∈N} → 0.

If y_{0}^{∗} is any weak^{∗}-limit point of such a net (y_{n}^{∗})_{n∈N} then x_{0}^{∗} = A^{T} (y_{0}^{∗}) and y_{0}^{∗} ∈ ∂g(y_{0}). If X is reflexive one can take sequences instead of nets and impose convergence of (x_{n}^{∗})_{n∈N} to x_{0}^{∗} in norm.

Proof. Without loss of generality we may suppose b = 0. Let P be a family of seminorms on X^{∗} whose unit balls form a base of neighborhoods of 0 for the weak^{∗} topology. Given x_{0}^{∗} ∈ ∂f(x_{0}), using Proposition 1.3, for any k ∈ K := IN\{0} and for any p ∈ P one can find z_{k,p}^{∗} ∈ ∂_{1/k^{2}}g(y_{0}) such that p(A^{T} (z_{k,p}^{∗})−x_{0}^{∗}) < 1/k. Then, by Proposition 1.1 and Corollary 1.2, one can pick (y_{k,p}, y_{k,p}^{∗}) ∈ ∂g and γ_{k,p} ∈ [−1, 1] such that ∥y_{k,p} −y_{0}∥ < 1/k, | g(y_{k,p})−g(y_{0}) |< 2/k, ⟨y_{k,p}^{∗}, y_{k,p} −y_{0}⟩ ≤ 2/k and ∥z_{k,p}^{∗} −(1+k^{−1}γ_{k,p})y_{k,p}^{∗}∥ < 1/k. Setting

N := K × P with the product order and x_{n}^{∗} := A^{T} y_{k,p}^{∗} for n := (k, p), and observing that

(1 + k^{−1}γ_{k,p})^{−1}A^{T} (z_{k,p}^{∗}) → x_{0}^{∗}

for the weak^{∗} topology, so that

A^{T} (y_{k,p}^{∗}) → x_{0}^{∗}

by the continuity of A^{T} , we get the announced nets. The assertion about limit points of the net (y_{k,p}^{∗}) follows from Corollary 1.2 and the weak^{∗} continuity of A^{T} . Moreover, we observe that when X is reflexive we can replace the weak^{∗}-closure by the norm closure and take P := {k^{−1}∥ · ∥ : k ∈ K} , so that we get a sequence rather than a net and strong convergence instead of weak^{∗} convergence. Conversely, if x_{0}^{∗}, (y_{n}^{∗})_{n∈N} , (y_{n})_{n∈N} are as in the statement, for each x ∈ X we have

⟨x_{0}^{∗}, x − x_{0}⟩ = lim_{n}⟨y_{n}^{∗}, A(x) − A(x_{0})⟩ = lim_{n}⟨y_{n}^{∗}, A(x) − y_{n}⟩

≤ lim sup_{n}(g(A(x)) − g(y_{n})) ≤ g(A(x)) − g(A(x_{0}))

by the lower semicontinuity of g, and thus x_{0}^{∗} belongs to ∂f(x_{0}).

Now let us turn to the sum.

Theorem 2.3. Let X be a Banach space, let f = f_{1} + f_{2} with f_{i} closed proper convex on X and let x_{0} ∈ dom f. Then ∂f (x_{0}) is the set of weak^{∗}- limits of nets of the form (x_{1,n}^{∗} + x_{2,n}^{∗})_{n∈N} with x_{i,n}^{∗} ∈ ∂f_{i} (x_{i,n}), (x_{i,n})_{n∈N} → x_{0}, (f_{i} (x_{i,n}))_{n∈N} →

f_{i} (x_{0}) , (⟨x_{i,n}^{∗}, x_{i,n} − x_{0}⟩)_{n∈N} → 0 for i = 1, 2. If X is reflexive the preceding nets

can be replaced by sequences and the convergence of x_{1,n}^{∗} + x_{2,n}^{∗} can be taken in the strong topology.

Proof. Let us show how this result can be deduced from the composition result. Given f = f_{1} + f_{2} , we set Y := X × X and we define A and g by A(x) := (x, x), g(y) :=

212 J-P Penot / Subdifferential calculus without qualification assumptions

f_{1}(x_{1}) + f_{2}(x_{2}) for y = (x_{1}, x_{2}); then f = g ∘ A. Given x_{0}^{∗} ∈ ∂f(x_{0}), let (y_{n}^{∗})_{n∈N} , (y_{n})_{n∈N} be as in Theorem 2.1, with y_{n}^{∗} := (x_{1,n}^{∗}, x_{2,n}^{∗}), y_{n} := (x_{1,n}, x_{2,n}), y_{0} = (x_{0}, x_{0}). Since x_{i,n}^{∗} ∈ ∂f_{i}(x_{i,n}) as easily seen, we have

f_{1}(x_{1,n}) − f_{1}(x_{0}) ≤ ⟨x_{1,n}^{∗}, x_{1,n} − x_{0}⟩ ≤ ⟨y_{n}^{∗}, y_{n} − y_{0}⟩ − (f_{2}(x_{2,n}) − f_{2}(x_{0})).

Since (g(y_{n})) → g(y_{0}), the lower semicontinuity of f_{1} and f_{2} entails that (f_{i}(x_{i,n}) − f_{i}(x_{0})) → 0 and we get (⟨x_{1,n}^{∗}, x_{1,n} − x_{0}⟩) → 0 from the convergence (⟨y_{n}^{∗}, y_{n} − y_{0}⟩) → 0; similarly (⟨x_{2,n}^{∗}, x_{2,n} − x_{0}⟩) → 0. Moreover

A^{T} (x_{1,n}^{∗}, x_{2,n}^{∗}) = x_{1,n}^{∗} + x_{2,n}^{∗} → x_{0}^{∗}

for the weak^{∗} convergence. Conversely, let us suppose

x_{0}^{∗} = lim_{n} (x_{1,n}^{∗} + x_{2,n}^{∗})

with

x_{i,n}^{∗} ∈ ∂f_{i} (x_{i,n}) , (x_{i,n}) → x_{0}, (α_{i,n}) := (f_{i} (x_{i,n}) − f_{i} (x_{0})) → 0,
(γ_{i,n}) := (⟨x_{i,n}^{∗}, x_{i,n} − x_{0}⟩) → 0 for i = 1, 2.

Then for each x ∈ dom f we have

\begin{aligned}& f (x) − f (x_{0}) − ⟨x_{0}^{∗}, x − x_{0}⟩ \\ = & ∑_{i=1}^{2}(f_{i} (x) − f_{i} (x_{i,n}) + α_{i,n} − γ_{i,n} − ⟨x_{i,n}^{∗}, x − x_{i,n}⟩) + ⟨x_{1,n}^{∗} + x_{2,n}^{∗} − x_{0}^{∗}, x − x_{0}⟩. \end{aligned}

Passing to the limit and using the relations

f_{i} (x) − f_{i} (x_{i,n}) − ⟨x_{i,n}^{∗}, x − x_{i,n}⟩ ≥ 0

we get

f (x) − f (x_{0}) − ⟨x_{0}^{∗}, x − x_{0}⟩ ≥ 0,

and x_{0}^{∗} ∈ ∂f(x_{0}).

Remark 2.4. The preceding statement shows that the familiar conditions

(x_{i,n}) → x_{0}, (f_{i} (x_{i,n})) → f_{i} (x_{0})

which are of common use in nonsmooth analysis have to be supplemented with the third condition

(⟨x_{i,n}^{∗}, x_{i,n} − x_{0}⟩) → 0

which represents a coupling condition on the sequence (x_{i,n}, x_{i,n}^{∗}). Obviously, the three conditions imply the condition

(f_{i} (x_{i,n}) − ⟨x_{i,n}^{∗}, x_{i,n} − x_{0}⟩) → f_{i} (x_{0})

imposed in [29]. The latter condition seems to be more difficult to interpret. Moreover, our conditions being more stringent, given an element x_{0}^{∗} of ∂f(x_{0}) we get more information about its limiting decomposition. This additional information may be precious, for instance for writing optimality conditions in terms of subdifferentials.

Remark 2.5. When receiving the first version of this paper, L. Thibault has kindly informed us that a related condition appears in [2] and [3] in the Hilbertian case, where it is obtained with quite different methods. Again, our three conditions are more precise when X is reflexive, so that one uses sequences : the conditions (x_{1,n}^{∗} + x_{2,n}^{∗}) → x_{0}^{∗},

(⟨x_{i,n}^{∗}, x_{i,n} − x_{0}⟩) → 0 for i = 1, 2 imply the conditions

(⟨x_{i,n}^{∗}, x_{1,n} − x_{2,n}⟩) → 0 for i = 1, 2.

However, the variational interpretation obtained in [3] is no more present here and there is no connection with sums of monotone operators as in [3].

Remark 2.6. The preceding two theorems are equivalent statements. In order to show that the sum theorem implies the composition theorem let f, g, A, b be given as in Theorem 2.2, and let us set f_{1}(x, y) = g(y), f_{2}(x, y) = ι_{G}(x, y), where G = {(x, Ax + b) : x ∈ X} and ι_{G} is the indicator function of G, so that f = f_{1} + f_{2}. The derivation is then straightforward in view of the obvious relationship between ∂f_{1} and ∂g.

Remark 2.7. Let us observe that for any nets (y_{n})_{n∈N}, (y_{n}^{∗})_{n∈N} as in Theorem 2.2 and for each ε > 0 one has y_{n}^{∗} ∈ ∂_{ε}g (y_{0}) for n ∈ N, n ≥ n(ε) where n(ε) is chosen in such a way that | ⟨y_{n}^{∗}, y_{n} − y_{0}⟩ | + | g(y_{n}) − g(y_{0}) |≤ ε for n ≥ n(ε) since for each y ∈ Y one has

g(y) − g(y_{0}) ≥ ⟨y_{n}^{∗}, y − y_{0}⟩ + ⟨y_{n}^{∗}, y_{0} − y_{n}⟩ + g(y_{n}) − g(y_{0}).

Let us derive a sufficient condition for a composition formula.

Corollary 2.8. Suppose with the notations of Proposition 2.1 that for some η > 0 and each r > 0 there exists some s > 0 such that

y^{∗} ∈ ∂_{η}g(y_{0}), ∥A^{T} (y^{∗})∥ ≤ r ⇒ ∥y^{∗}∥ ≤ s.

Then

∂(g ∘ A)(x_{0}) = A^{T} (∂g(y_{0})).

Proof. The Krein-Smulian Theorem shows that for ε < η the convex set C := A^{T} (∂_{ε}g(y_{0})) is weak^{∗}-closed since for each r > 0

C ∩ rB_{X^{∗}} = A^{T} (∂_{ε}g(y_{0}) ∩ sB_{Y^{∗}}) ∩ rB_{X^{∗}}

and the image by A^{T} of the weak^{∗} compact set ∂_{ε}g(y_{0}) ∩ sB_{Y^{∗}} is weak^{∗} compact. Then if we take sequences (u_{n}^{∗})_{n∈N}, (y_{n})_{n∈N}, (y_{n}^{∗})_{n∈N} as in Proposition 2.1, the assumption shows that (u_{n}^{∗})_{n∈N} is bounded, hence has a converging subnet whose limit y_{0}^{∗} is also the limit of (y_{n}^{∗})_{n∈N}, and belongs to ∂g(y_{0}) by Corollary 1.2 and obviously x_{0}^{∗} = A^{T} (y_{0}^{∗}).

In order to derive sufficient conditions for the validity of formula (0.1) let us introduce the following notion which differs slightly from analogous concepts of nonsmooth analysis (see [20], [21], [27], [17], [28] for instance).

Definition 2.9. Given a closed proper convex function f on X, x_{0} ∈ dom f, the asymptotic subdifferential of f at x_{0} is the set ∂_{∞}f (x_{0}) of u^{∗} ∈ X^{∗} such that there exist nets (t_{n})_{n∈N} → 0_{+}, ((x_{n}, x_{n}^{∗}))_{n∈N} in ∂f with (x_{n})_{n∈N} → x_{0}, (f (x_{n}))_{n∈N} → f (x_{0}) , (⟨x_{n}^{∗}, x_{n} − x_{0}⟩)_{n∈N} → 0 and (t_{n}x_{n}^{∗})_{n∈N} → u^{∗} in X^{∗} weak^{∗}. When ∂f(x_{0}) is nonempty, taking x_{n} = x_{0} for each n we see that this set is larger than the asymptotic cone (∂f(x_{0}))_{∞} of ∂f(x_{0}) and is larger than the corresponding sets in [17], [20], [21], [27], [28] given by

∂^{∞}f (x_{0}) = {u^{∗} ∈ X^{∗} : (u^{∗}, 0) ∈ N (E_{f}, (x_{0}, r_{0}))}  (2.1)

where r_{0} = f (x_{0}) , E_{f} is the epigraph of f and N (C, a) denotes the normal cone to C at a :

N (C, a) = {x^{∗} : ⟨x^{∗}, x − a⟩ ≤ 0 ∀x ∈ C} .

Here we recall that ∂f (x_{0}) × {−1} = N (E_{f}, (x_{0}, r_{0})) ∩ (X^{∗} × {−1}). In fact, given y_{0}^{∗} ∈ ∂f(x_{0}), for any u^{∗} ∈ ∂^{∞}f(x_{0}) and any (t_{n}) → 0_{+} we have x_{n}^{∗} := t_{n}^{−1}u^{∗} +y_{0}^{∗} ∈ ∂f(x_{0}) as N (E_{f}, (x_{0}, r_{0})) is a convex cone, and (t_{n}x_{n}^{∗}) → u^{∗}. Let us note that the coupling condition (⟨x_{n}^{∗}, x_{n} − x_{0}⟩) → 0 is useful in order to shrink ∂_{∞}f(x_{0}) as most as possible, so that the qualification condition presented below will not be too exacting. Let us first give a more precise identification of ∂_{∞}f(x_{0}).

Proposition 2.10. For any closed proper convex function f on X and any x_{0} ∈ D_{f} := dom f one has ∂_{∞}f (x_{0}) ⊂ ∂^{∞}f (x_{0}) = N(D_{f}, x_{0}). Equality holds if ∂f(x_{0}) is nonempty.

Proof. The second equality is a direct consequence of the definitions. Since we already observed the inclusion ∂_{∞}f (x_{0}) ⊃ ∂^{∞}f (x_{0}) when ∂f(x_{0}) is nonempty, and since for u^{∗}, t_{n}, x_{n}, x_{n}^{∗} as in Definition 2.9 we have ((x_{n}, r_{n})) → (x_{0}, r_{0}) for r_{0} = f (x_{0}) , r_{n} = f (x_{n}) , (t_{n} (x_{n}^{∗}, −1)) → (u^{∗}, 0) weak^{∗} in (X × IR)^{∗},

⟨t_{n} (x_{n}^{∗}, −1) , (x_{n}, r_{n}) − (x_{0}, r_{0})⟩ = t_{n}⟨x_{n}^{∗}, x_{n} − x_{0}⟩ − t_{n} (r_{n} − r_{0}) → 0,

the first inclusion and the proposition are consequences of the following result of independent interest.

Lemma 2.11. Let C be a convex subset of a normed vector space Z, let (a_{n})_{n∈N} be a net in C with limit a, and let a_{n}^{∗} ∈ N (C, a_{n}) be such that (⟨a_{n}^{∗}, a_{n} − a⟩) → 0 and (a_{n}^{∗}) weak^{∗} converges to some a^{∗}. Then a^{∗} ∈ N (C, a).

Let us observe that if we deal with a sequence rather than a net the condition (⟨a_{n}^{∗}, a_{n} − a⟩) → 0 is automatically satisfied.

Proof. The result is a special case of the last assertion of Corollary 1.2 for the indicator function of C.

We are ready to present a qualification condition which is new in the context of convex analysis. We need to introduce some terminology.

Definition 2.12. A convex subset C of a n.v.s. Z is said to be normally compact at z ∈ C if for any net (z_{n})_{n∈N} of C with limit z, any net (z_{n}^{∗})_{n∈N} such that z_{n}^{∗} ∈ N(C, z_{n}), ∥z_{n}^{∗}∥ = 1 for each n ∈ N has a non null cluster weak^{∗} point z^{∗}. A convex function f : X → IR^{•} on a n.v.s. X is said to be subdifferentially compact at x_{0} ∈ dom f if, for any net ((x_{n}, x_{n}^{∗}))_{n∈N} in ∂f with (x_{n}) → x_{0}, (∥x_{n}^{∗}∥) → ∞, the net (∥x_{n}^{∗}∥^{−1}x_{n}^{∗}) has a non null weak^{∗} cluster point. Clearly, in a finite dimensional n.v.s., any convex set is normally compact and any function is subdifferentially compact. Moreover one has the following criteria. Note that assertion (a) below is valid for the more general class of epi-Lipschitzian sets, but here we do not wish to leave the framework of convex analysis.

Lemma 2.13.

a) If the interior of a convex set C is nonempty, then C is normally compact at each z ∈ C;

b) If the epigraph E_{f} of a convex function f is normally compact at z_{0} := (x_{0}, f(x_{0})), then f is subdifferentially compact at x_{0};

c) If the convex function f is continuous at some point of its domain, then f is subdifferentially compact at each point of its domain.

Proof. a) Let a ∈ C and r > 0 be such that the ball B(a, r) is contained in C. Then, for any net (z_{n})_{n∈N} of C with limit z ∈ C, any net (z_{n}^{∗})_{n∈N} such that z_{n}^{∗} ∈ N(C, z_{n}), ∥z_{n}^{∗}∥ = 1 for each n ∈ N, one has for each u in the unit ball of Z

⟨z_{n}^{∗}, a + ru − z_{n}⟩ ≤ 0

hence

⟨z_{n}^{∗}, a − z_{n}⟩ ≤ −r∥z_{n}^{∗}∥ = −r,

so that each cluster weak^{∗} point z^{∗} of (z_{n}^{∗})_{n∈N} satisfies

⟨z^{∗}, a − z⟩ ≤ −r

and is nonzero.

b) Let ((x_{n}, x_{n}^{∗}))_{n∈N} be a net in ∂f with (x_{n}) → x_{0}, (∥x_{n}^{∗}∥) → ∞, for z_{n} := (x_{n}, f(x_{n})). For n large enough (u_{n}^{∗}, r_{n}) := (∥x_{n}^{∗}∥^{−1}x_{n}^{∗}, −∥x_{n}^{∗}∥^{−1}) ∈ N(E_{f}, z_{n}) and is a unit vector; then, if (u^{∗}, r) is a non null weak^{∗} cluster point of ((u_{n}^{∗}, r_{n})) one has r = 0, u^{∗} ≠ 0 and and the net (∥x_{n}^{∗}∥^{−1}x_{n}^{∗}) has a non null weak^{∗} cluster point u^{∗}. Assertion c) follows from the preceding two assertions.

Corollary 2.14. Let X be a Banach space and let f_{1}, f_{2} be two closed proper convex functions on X. Let x_{0} ∈ dom f_{1} ∩ dom f_{2} be such that

∂_{∞}f_{1} (x_{0}) ∩ (−∂_{∞}f_{2} (x_{0})) = {0}  (2.2)

and such that f_{1} is subdifferentially compact at x_{0}. Then

∂ (f_{1} + f_{2}) (x_{0}) = ∂f_{1} (x_{0}) + ∂f_{2} (x_{0}) .

Proof. Given x_{0}^{∗} ∈ ∂ (f_{1} + f_{2}) (x_{0}), Theorem 2.1 ensures that for i = 1, 2 there exists

nets (x_{i,n})_{n∈N} → x_{0}, (x_{i,n}^{∗})_{n∈N} such that

x_{i,n}^{∗} ∈ ∂f_{i}(x_{i,n}), (f_{i} (x_{i,n})) → f_{i} (x_{0}) , (⟨x_{i,n}^{∗}, x_{i,n} − x_{0}⟩) → 0 with (x_{1,n}^{∗} + x_{2,n}^{∗}) →^{w^{∗}}x_{0}^{∗}.

Let r_{n} = ∥x_{1,n}^{∗}∥. If (r_{n}) has a bounded subnet we may assume that (x_{1,n}^{∗}) w^{∗}−converges

and then (x_{2,n}^{∗}) w^{∗}−converges too. In view of Corollary 1.2, the limit (x_{1}^{∗}, x_{2}^{∗}) of

(x_{1,n}^{∗}, x_{2,n}^{∗}) belongs to ∂f_{1} (x_{0}) × ∂f_{2} (x_{0}), and x_{1}^{∗} + x_{2}^{∗} = x_{0}^{∗}. If (r_{n}) → ∞, setting

u_{1,n}^{∗} = r_{n}^{−1}x_{1,n}^{∗} we use our assumption to get that (u_{1,n}^{∗}) has a converging subnet whose

limit u_{1}^{∗} is non null and belongs to ∂_{∞}f_{1} (x_{0}). Since

lim r_{n}^{−1}x_{2,n}^{∗} = lim r_{n}^{−1}x_{0}^{∗} − lim r_{n}^{−1}x_{1,n}^{∗} = −u_{1}^{∗}

belongs to ∂_{∞}f_{2} (x_{0}) we reach a contradiction with our assumption (2.2). The opposite inclusion being obvious, we get the announced equality.

Remark 2.15. The qualification condition (2.2) seems to have been used for the first time (and in a nonconvex framework, but in finite dimensions) in [20], [21].The familiar qualification condition: f_{1} is finite and continuous at some point a of dom f_{2} implies assumption (2.2). In fact, if B(a, r) ⊂ dom f_{1} and if u^{∗} ∈ ∂_{∞}f_{1} (x_{0}) ∩ (−∂_{∞}f_{2} (x_{0})), in view of Proposition 2.10 we have ⟨u^{∗}, a + ru − x_{0}⟩ ≤ 0 for each u in the unit ball B_{X}, and ⟨−u^{∗}, a − x_{0}⟩ ≤ 0, hence −r⟨u^{∗}, u⟩ ≤ 0 for each u ∈ B_{X} and u^{∗} = 0.

Let us conclude by observing that the classical qualification condition of [27]

IR_{+}(dom f_{1} − dom f_{2}) = X  (2.3)

entails condition (2.2) since it can be equivalently written

IR_{+}(dom f_{1} − x_{0}) − IR_{+}(dom f_{2} − x_{0}) = X

so that for any x ∈ X and any u^{∗} ∈ N(dom f_{1}, x_{0})∩(−N(dom f_{2}, x_{0})) one has ⟨u^{∗}, x⟩ ≤ 0, hence u^{∗} = 0. It is shown in [29] that condition (2.3) is a qualification condition in order to get a sum rule. In fact it can be shown that the more general condition

X_{0} := IR_{+}(dom f_{1} − x_{0}) − IR_{+}(dom f_{2} − x_{0})) is a closed vector subspace  (2.4)

of Attouch and Brézis ([1], see also [18] Th 3.6.1 for a similar condition in an algebraic framework) can be deduced from Theorem 2.3. Similarly, the following qualification condition

Y_{0} := IR_{+}(dom g − y_{0}) − A(X) is a closed vector subspace  (2.5)

can be derived from Theorem 2.2.

Corollary 2.16. Given two closed proper convex functions f_{1}, f_{2} on the Banach space X such that condition (2.4) is satisfied, then for any x_{0} ∈ dom f_{1} ∩ dom f_{2} one has

∂ (f_{1} + f_{2}) (x_{0}) = ∂f_{1} (x_{0}) + ∂f_{2} (x_{0}) .

Given f, g, X, Y, A, b, x_{0,} y_{0} as in Theorem 2.2 such that condition (2.5) is satisfied one has

∂f(x_{0}) = A^{T} (∂g(y_{0})).

Proof. For the sake of brievity, we prove only the second assertion, both being related, as seen in a previous remark. Here one has to observe that if C_{1} and C_{2} are two convex cones of X such that X_{0} := C_{1} − C_{2} is a closed vector subspace of X, then Y_{0} := C_{1} × C_{2} − D is a closed vector subspace of Y := X × X, D being the diagonal of Y. Let us first suppose Y_{0} = Y. Let us show that the assumption of Corollary 2.8 is satisfied. Given r > 0 let us show that the set D := ∂_{ε}g(y_{0}) ∩ (A^{T})^{−1}(rB_{X^{∗}}) is bounded. This follows from the Banach-Steinhaus Theorem and the fact that for each y ∈ Y one can find t > 0, z ∈ dom g, x ∈ X such that y = t(z − y_{0}) − A(x) hence for each y^{∗} ∈ D

\begin{aligned}⟨y^{∗}, y⟩ & = t⟨y^{∗}, z − y_{0}⟩ − ⟨A^{T} (y^{∗}), x⟩ \\ & ≤ t(g(z) − g(y_{0}) + ε) + r∥x∥, \end{aligned}

a number independent of y^{∗}. Now, as is well-known ( see [1]), the general case can be deduced from the special case by setting g_{0}(y) := g(y+y_{0}) for y ∈ Y_{0} and by observing that any continuous linear extension y^{∗} ∈ Y^{∗} of some y_{0}^{∗} ∈ ∂g_{0}(0) belongs to ∂g(y_{0}) as Y_{0} contains A(X) and dom g − y_{0}, (so that

g′(y_{0}, v) = ∞

for any v ∈ Y \ Y_{0}) and is such that A^{T} (y^{∗}) = x_{0}^{∗} if A^{T} (y_{0}^{∗}) = x_{0}^{∗}.

Acknowledgment. The author is grateful to L. Barbet for a careful reading, to A.D. Ioffe, M. Théra and an anonymous referee for some references and to L. Thibault for helpful discussions. He also acknowledges useful comments and criticisms by another anonymous referee.

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