Journal of Convex Analysis Volume 3 (1996), No. 2, 309–327

On Primal-Dual Stability in Convex Optimization

Dominique Azé, Abdelouahed Rahmouni

Département de Mathématiques, Université de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan cedex, France.

Received November 22, 1994 Revised manuscript received April 6, 1995

Conditions are given ensuring the stability of the classical duality scheme of Ekeland-Temam-Rockafellar with respect to the slice topology (a generalization of the Mosco topology to the non reflexive setting). Both primal and dual aspect are tackled allowing convergence of primal solutions along with the Lagrange multipliers. An example of application is given to optimal control of distributed parameter systems.

1. Introduction and notations

There has been recently a renewed interest in convex duality, mainly due to its connections with fully nonlinear optimal control problems, revealed by R. Vinter in [34]. On the other hand several notions of variational convergence for convex functions have been introduced and thoroughly studied by many authors ([1], [2], [3], [5], [6], [7], [9], [11], [12], [16], [17], [18], [19], [25], [26], [27], [31], [32], [33]). These variational notions of convergence lead to convergence of exact and approximate solutions of the associated minimization problems. It is quite natural to investigate the behaviour of the primal and dual functionals attached to a convex perturbation function when this perturbation function moves along some topology. This was firstly done in [13] in the finite dimensional setting. The natural topologies to be considered in this problems are those making continuous the conjugacy operation. In this work we deal with general normed spaces and we focus our attention on the slice topology introduced in [11], [12], [31], [32], [33] (see also [16], [17], [7], which specializes to the Mosco topology ([25], [26]) in the nonreflexive setting. In section 2, we study the primal stability of the duality scheme. To this end we prove a result on the slice convergence of the sum of two convex functions which might be useful in other fields of convex analysis. In section 3 we treat the case of the stability both of the primal and the dual functionals. Our approach differs from [10] where primal- dual stability was obtained by means of conditions bearing on the convex-concave Lagrangians. In our framework the stability is studied in terms of perturbations of the convex function which generates the Lagrangian. The assumptions used to prove our primal-dual stability result, namely, the equi-continuity at 0 of the value functions and the equi-coerciveness of primal functionals, are quite natural. Our result differs also from the one given in [15] and also from [30] which rely, in the finite dimensional setting, on a stronger notion of convergence for convex functions. An application is given in section 4 to a singular

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

310 D. Azé, A. Rahmouni / On primal-dual stability in convex optimization

perturbation problem in optimal control of distributed parameter systems.

Let us give some definitions. Let (X, ∥.∥) be a normed vector space and let f : X ⟶ \overline{IR} be a function. We denote by

epi f = {(x, t) ∈ X × IR : f(x) ≤ t}

its epigraph and by

epi′ f = {(x, t) ∈ X × IR : f(x) < t}

its strict epigraph. We also denote by

dom(f) = {x ∈ X : f(x) < +∞}

its effective domain and by

[f ≤ c] = {x ∈ X : f(x) ≤ c}

its sublevel sets. We say that f is proper whenever dom f ≠ ∅ and −∞ ∉ f(X). We denote by \overline{f} the lower semi continuous regularization of f characterised by

epi \overline{f} = \overline{epi f}

where \overline{epi f} stands for the closure of epi f. We denote by Conv(X) the set of proper convex functions defined on X with values in IR ∪ {+∞} and by Γ_{0}(X) the set of those f ∈ Conv(X) which are lower semicontinuous on X. Given f ∈ Conv(X) (resp. g ∈ Conv(X^{∗})), its conjugate is defined on X^{∗} (resp. on X) by

f^{∗}(y) = sup{⟨x, y⟩ − f(x) : x ∈ X} for all y ∈ X^{∗},

and

g^{∗}(x) = sup{⟨x, y⟩ − g(y) : y ∈ X^{∗}} for all x ∈ X.

The indicator function of a subset A ⊂ X is the function i_{A} defined by i_{A}(x) = 0 if x ∈ A and i_{A}(x) = +∞ if x ∉A. We denote by σ_{C} = i_{C}^{∗} the support function of a convex subset C of X, and by C^{∘} ⊂ X^{∗} its polar set defined by

C^{∘} = {y ∈ X^{∗} : σ_{C}(y) ≤ 1}.

The gauge function of a subset C ⊂ X is the function j : X ⟶ IR_{+} ∪ {+∞} defined for all x ∈ X by

j(x) = inf{t > 0 : x ∈ tC}

with the convention inf ∅ = +∞. It is known (see [24], p. 34) that

σ_{C} = j_{C^{∘}}.  (1)

whenever C ⊂ X is convex and contains 0.

Let us now review the definitions we shall need about topologies on hyperspaces. We denote by F(X) the set of nonempty closed subsets of X and by C(X) the set of nonempty closed convex subsets of X. The lower Vietoris topology τ_{V}^{−} is the topology on F(X) whose a subbase is constituted by the sets

V^{−} = {C ∈ F(X) : C ∩ V ≠ ∅}

where V ranges over the family of norm open subsets of X. A sequence (C_{n}) ⊂ F(X) converges to C ∈ F(X) with respect to τ_{V}^{−} if and only if

C ⊂ lim inf_{n→∞} C_{n}

where, given a sequence (C_{n})_{n∈IN} of subsets of X, one denotes by lim inf_{n→∞} C_{n} the set of x ∈ X such that there exists a sequence (x_{n}) which converges to x with x_{n} ∈ C_{n} eventually.

Let (f_{n})_{n∈IN} be a sequence of functions in \overline{IR}^{X}, we introduce its epigraphical upper limit f = e- lim sup_{n→∞} f_{n} (see [1]) defined by

epi f = lim inf_{n→∞}(epi f_{n}).  (2)

In [16] and [17] (see also [6], [9], [11]) was introduced the Joly topology τ_{J} in Γ_{0}(X) as the weakest topology making continuous the mappings

E : Γ_{0}(X) ⟶ (C(X × IR), τ_{V}^{−})

and

E^{∗} : Γ_{0}(X) ⟶ (C^{∗}(X^{∗} × IR), τ_{V}^{−}),

where C^{∗}(X^{∗} × IR) is the set of w^{∗}-closed convex subsets of X^{∗} × IR, defined for all f ∈ Γ_{0}(X) by

E(f) = epi f and E^{∗}(f) = (E ∘ L)(f) = epi f^{∗}.

Analogously, the dual Joly topology τ_{J}^{∗} is defined on Γ_{0}(X^{∗}) by exchanging the roles played by X and X^{∗} in the definition of τ_{J}. From the very definitions of these topologies, it follows that the conjugacy operation is continuous from (Γ_{0}(X), τ_{J}) into (Γ_{0}(X^{∗}), τ_{J}^{∗}) and from (Γ_{0}(X^{∗}), τ_{J}^{∗}) into (Γ_{0}(X), τ_{J}).

A sequence (f_{n})_{n∈IN} converges to f with respect to τ_{J} in Γ_{0}(X) if and only if

epi f ⊂ lim inf_{n→∞}(epi f_{n}) and epi f^{∗} ⊂ lim inf_{n→∞}(epi f_{n}^{∗}),  (3)

which is equivalent using (2) to

e- lim sup_{n→∞} f_{n} ≤ f and e- lim sup_{n→∞} f_{n}^{∗} ≤ f^{∗}.

If the case when X is a general normed space, this topology has been characterized only recently (see [11], [31]). Following G. Beer in [11] one defines the topology τ_{S}^{+} on C(X) as the topology whose a subbase is constituted by the family of sets

(B^{c})^{++} = {C ∈ C(X) : D(B, C) > 0}

where B ranges over the family of closed bounded convex sets of X and

D(B, C) = inf{∥b − c∥ : (b, c) ∈ B × C}.

The slice topology τ_{S} is defined by

τ_{S} = τ_{S}^{−} ∨ τ_{S}^{+}  (4)

where τ_{S}^{−} = τ_{V}^{−} with a similar definition for τ_{S}^{∗} where B is w^{∗}-compact convex. A net (C_{i})_{i∈I} converges in (C(X), τ_{S}) if and only if for all B ⊂ X closed bounded convex we have

D(B, C) ≤ lim inf_{I} D(B, C_{i}) and lim sup_{I} D(B, C_{i}) ≤ D(B, C).  (5)

In [11] it has been proved that

τ_{J} = τ_{S} and τ_{J}^{∗} = τ_{S}^{∗}.  (6)

The convergence with respect to the slice topology generalizes to non reflexive spaces the convergence in the sense of Mosco (see [25], [26], [1]). The Mosco-Beer topology τ_{M} is the topology defined on C(X) by

τ_{M} = τ_{V}^{−} ∨ τ_{M}^{+}

where τ_{M}^{+} is the topology on C(X) whose a basis is constituted by the sets

K^{c+} = {C ∈ C(X) : K ∩ C = ∅}

with K weakly compact. The dual Mosco-Beer topology τ_{M}^{∗} is defined on the set C^{∗}(X^{∗}) of w^{∗}-closed convex subsets of X^{∗} by

τ_{M}^{∗} = τ_{V}^{−} ∨ τ_{M}^{∗+}

where τ_{M}^{∗+} is the topology on C^{∗}(X^{∗}) whose a basis is constituted by the sets

K^{c+} = {C ∈ C^{∗}(X^{∗}) : K ∩ C = ∅}

with K w^{∗}-compact.

For the convenience of the reader we give the following well known lemma.

Lemma 1.1. Let X be a normed vector space and let (f_{n})_{n∈IN}, f ∈ \overline{IR}^{X}. Then e- lim sup_{n→∞} f_{n} ≤ f if and only if

for all x ∈ X there exists (x_{n}) ⟶ x such that f(x) ≥ lim sup_{n→∞} f_{n}(x_{n}).

Proof. Suppose that f ≥ e- lim sup_{n→∞} f_{n} and let x ∈ X. We can suppose that f(x) < +∞. Then given λ ∈ IR with λ > f(x), there exists a sequence (x_{n}, λ_{n}) ∈ epi f_{n} such that (x_{n}, λ_{n}) converges to (x, λ). Thus we have

lim sup_{n→∞} f_{n}(x_{n}) ≤ lim sup_{n→∞} λ_{n} = λ.

Letting λ decrease to f(x) and using a diagonalization procedure, one gets a sequence (x_{n})_{n∈IN} which converges to x and which satisfies f(x) ≥ lim sup_{n→∞} f_{n}(x_{n}). Conversely, let (x, λ) ∈ epi f and let (x_{n}) be a sequence converging to x such that

t := lim sup_{n→∞} f_{n}(x_{n}) ≤ f(x).

If t = −∞ then f_{n}(x_{n}) < λ eventually, thus (x_{n}, λ) ∈ epi f_{n} eventually and the sequence ((x_{n}, λ)) converges to (x, λ). If t ≠ −∞, let us set

λ_{n} = λ − t + max(t, f_{n}(x_{n})),

we get (x_{n}, λ_{n}) ∈ epi f_{n} and the sequence ((x_{n}, λ_{n})) converges to (x, λ), hence the result.

We shall use the following characterization of the convergence of sequence in Γ_{0}(X) with respect to the slice topology (see [3]).

Proposition 1.2. Let X be a normed vector space and let (f_{n})_{n∈IN}, f ∈ Γ_{0}(X). Then f = τ_{S} − lim_{n→∞} f_{n} if and only if

for all x ∈ X, there exists (x_{n}) ⟶ x : f(x) ≥ lim sup_{n→∞} f_{n}(x_{n})
for all y ∈ X^{∗}, there exists (y_{n}) ⟶ y : f^{∗}(y) ≥ lim sup_{n→∞} f_{n}^{∗}(y_{n}).

Proof. This is immediate from Lemma 1.1, from (3) and from the fact that τ_{J} = τ_{S}.

2. Primal stability

Let us begin by a quick review on the main features in convex duality (see [14], [20] and [29]). Let X, U be normed vector spaces whose topological dual spaces are denoted by V = X^{∗} and Y = U^{∗}. Given F ∈ Γ_{0}(X × U), the primal problem associated to the perturbation function F consists in

inf_{x∈X} F (x, 0) = inf_{x∈X} f(x).

We shall always assume that the function f(x) = F (x, 0) is proper. We set for all u ∈ U, φ(u) = inf_{x∈X} F (x, u). The value function φ is convex but not necessarily lower semicontinuous. Observe that φ^{∗}(y) = F^{∗}(0, y). The associated dual problem consists in

inf_{y∈Y} F^{∗}(0, y).

It is also useful to introduce for all v ∈ V the value function ψ(v) = inf_{y∈Y} F^{∗}(v, y). One has for all x ∈ X, ψ^{∗}(x) = F (x, 0). The convex-concave Lagrangian associated to the perturbation function is the function L : X × Y ⟶ \overline{IR} defined for all (x, y) ∈ X × Y by

L(x, y) = inf_{w∈U}(F (x, w) − ⟨w, y⟩).

It is important to observe that

F^{∗}(0, y) = sup_{x∈X}(−L(x, y)).  (7)

A pair (x̄, ȳ) ∈ X × Y solves

F (x̄, 0) = inf_{x∈X} F (x, 0) and F^{∗}(0, ȳ) = inf_{y∈Y} F^{∗}(0, y) with F (x̄, 0,) = −F^{∗}(0, ȳ)

if and only if (x̄, ȳ) is a saddle-point of the Lagrangian that is

L(x̄, y) ≤ L(x̄, ȳ) ≤ L(x, ȳ) for all (x, y) ∈ X × Y.

Moreover, assuming that L(x̄, ȳ) is finite, this is equivalent to

L′_{x}((x̄, ȳ); x − x̄) ≥ 0 and L′_{y}((x̄, ȳ); y − ȳ) ≤ 0  (8)

for all (x, y) ∈ X × Y where L(x̄, y) and L(x, ȳ) are finite with

L′_{x}((x̄, ȳ); x − x̄) = lim_{t↓0} \frac{L(x̄ + t(x − x̄), ȳ) − L(x̄, ȳ)}{t}

and

L′_{y}((x̄, ȳ); y − ȳ) = lim_{t↓0} \frac{L(x̄, ȳ + t(y − ȳ)) − L(x̄, ȳ)}{t}.

We shall use the following well-known result.

Lemma 2.1. Let X, U be normed vector spaces, let F_{n} : X × U ⟶ IR ∪ {+∞} and let F : X × U ⟶ IR ∪ {+∞} be such that epi F ⊂ lim inf_{n→∞}(epi F_{n}). Then

epi φ ⊂ lim inf_{n→∞}(epi φ_{n}).

Proof. One has epi′ φ = proj_{U×IR}( epi′ F). Let (u, λ) ∈ epi′ φ, there exists x ∈ X such that (x, u, λ) ∈ epi′ F. Thus there exists a sequence (x_{n}, u_{n}, λ_{n}) such that (x_{n}, u_{n}, λ_{n}) ∈ epi_{,}F_{n} and ((x_{n}, u_{n}, λ_{n})) converges to (x, u, λ). As (u_{n}λ_{n}) ∈ epi φ_{n} and ((u_{n}, λ_{n})) converges to (u, λ) we derive that epi′ φ ⊂ lim inf_{n→∞} ( epi φ_{n}), thus

epi φ ⊂ \overline{epi′ φ} ⊂ lim inf_{n→∞}(epi φ_{n}).

The following lemma in the spirit of [5] plays a prominent role in the sequel.

Lemma 2.2. Let X be a normed vector space and (f_{n})_{n∈IN}, (g_{n})_{n∈IN} ∈ Conv(X). Assume that for some real numbers s > 0, r ≥ 0, c ∈ IR

sB_{X} ⊂ ⋂_{n∈IN} ([f_{n} ≤ c] ∩ rB_{X} − [g_{n} ≤ c] ∩ rB_{X}).  (9)

Then if f, g ∈ Conv(X) are such that e- lim sup_{n→∞} f_{n} ≤ f and e- lim sup_{n→∞} g_{n} ≤ g, one has

e- lim sup_{n→∞}(f_{n} + g_{n}) ≤ f + g.

Proof. Let x ∈ dom(f + g). There exist sequences (x_{1n}) and (x_{2n}) converging to x such that lim sup_{n→∞} f_{n}(x_{1n}) ≤ f(x) and lim sup_{n→∞} g_{n}(x_{2n}) ≤ g(x). Let us set t_{n} = ∥x_{1n} − x∥ + ∥x_{2n} − x∥. From (9) there exist sequences (z_{1n}) and (z_{2n}) with z_{1n} ∈ [f_{n} ≤ c] ∩ rB_{X}, z_{2n} ∈ [g_{n} ≤ c] ∩ rB_{X} and

s(x_{2n} − x_{1n}) = t_{n}(z_{1n} − z_{2n}).

Let us set x_{n} = (t_{n} + s)^{−1}(t_{n}z_{1n} + sx_{1n}) = (t_{n} + s)^{−1}(t_{n}z_{2n} + sx_{2n}). One has

\begin{aligned}∥x_{n} − x∥ & ≤ & (t_{n} + s)^{−1}(t_{n}∥z_{1n} − x∥ + s∥x_{1n} − x∥) \\ & ≤ & s^{−1}t_{n}(r + ∥x∥) + ∥x_{1n} − x∥, \end{aligned}

thus (x_{n}) converges to x. Using the fact that

\begin{aligned}f_{n}(x_{n}) & ≤ & (t_{n} + s)^{−1}t_{n}f_{n}(z_{1n}) + (t_{n} + s)^{−1}sf_{n}(x_{1n}) \\ & ≤ & s^{−1}t_{n}c + f_{n}(x_{1n}) \end{aligned}

and

\begin{aligned}g_{n}(x_{n}) & ≤ & (t_{n} + s)^{−1}t_{n}g_{n}(z_{2n}) + (t_{n} + s)^{−1}sg_{n}(x_{2n}) \\ & ≤ & s^{−1}t_{n}c + g_{n}(x_{2n}), \end{aligned}

we get

lim sup_{n→∞}(f_{n} + g_{n})(x_{n}) ≤ lim sup_{n→∞} f_{n}(x_{1n}) + lim sup_{n→∞} g_{n}(x_{2n})
≤ (f + g)(x)

which ends the proof of the lemma.

Applying the preceding lemma to the indicator functions of C_{n} and C, we get the following corollary.

Corollary 2.3. Let X be a normed vector space, let (C_{n})_{n∈IN}, C, (D_{n})_{n∈IN}, D be convex subsets such that

C ⊂ lim inf_{n→∞} C_{n} and D ⊂ lim inf_{n→∞} D_{n}.

Assume that there exists s > 0, r ≥ 0, c ∈ IR such that

sB_{X} ⊂ ⋂_{n∈IN} (C_{n} ∩ rB_{X} − D_{n} ∩ rB_{X}).  (10)

Then

C ∩ D ⊂ lim inf_{n→∞}(C_{n} ∩ D_{n}).

We shall also need the following results of [4].

Theorem 2.4. Let (X, ∥.∥) be a normed vector space and let f, g ∈ Conv(X). Assume that for some real numbers s > 0, r ≥ 0 , c ∈ IR

sB_{X} ⊂ [f ≤ c] ∩ rB_{X} − [g ≤ c] ∩ rB_{X}.

Then for all y ∈ X^{∗} one has

(f + g)^{∗}(y) = min_{y′∈X^{∗}} f^{∗}(y′) + g^{∗}(y − y′).

Lemma 2.5. Let X, U be normed vector spaces, let (F_{n})_{n∈IN}, F ∈ Γ_{0}(X × U) be such that F = τ_{S}- lim_{n→∞} F_{n}. Assume that there exist s > 0, r ≥ 0, c ∈ IR such that

sB_{X×U} ⊂ ⋂_{n∈IN} (X × {0} ∩ rB_{X×U} − [F_{n} ≤ c] ∩ rB_{X×U}).  (11)

Then

F (., 0) = τ_{S}- lim_{n→∞} F_{n}(., 0).

Proof. From assumption (11) and Lemma 2.1, we get

epi (F + i_{X×{0}}) ⊂ lim inf_{n→∞} epi (F_{n} + i_{X×{0}})

which readily turns to

epi (F (., 0)) ⊂ lim inf_{n→∞} ( epi (F_{n}(., 0))).  (12)

On the other hand, we derive from Theorem 2.4 that, for all v ∈ V

(F_{n} + i_{X×{0}})^{∗}(v, 0) = min_{(v′,y′)∈V ×Y} F_{n}^{∗}(v − v′, −y′) + i_{{0}×Y} (v′, y′) = ψ_{n}(v).

It follows that ψ_{n} ∈ Γ_{0}(V). As ψ_{n}^{∗} = f_{n} = F_{n}(., 0), we get (F_{n}(., 0))^{∗} = ψ_{n}. Relying on the slice convergence of the sequence (F_{n}) to F and using Lemma 2.1, we derive that

epi (F (., 0))^{∗} ⊂ lim inf_{n→∞} epi((F_{n}(., 0))^{∗}  (13)

which combined with (12) yields the announced result.

One can apply Lemma 2.5 to the particular case of Fenchel duality. Let us consider a primal problem of the type

inf_{x∈X} (f(x) + g(A(x)))

where f and g belong respectively to Γ_{0}(X) and Γ_{0}(U) and where A : X ⟶ U is a linear continuous operator. The associated perturbation function is

F (x, u) = f(x) + g(A(x) + u).

Its conjugate is given by

F^{∗}(v, y) = f^{∗}(−A^{∗}(v) + y) + g^{∗}(v).

Theorem 2.6. Let X, U be normed vector spaces, let A ∈ L(X, U) and let (f_{n})_{n∈IN}, f ∈ Γ_{0}(X), (g_{n})_{n∈IN}, g ∈ Γ_{0}(U) be such that

τ_{S}- lim_{n→∞} f_{n} = f and τ_{S}- lim_{n→∞} g_{n} = g.

Assume there exist s > 0, r ≥ 0, c ∈ IR such that

sB_{U} ⊂ ⋂_{n∈IN} ([g_{n} ≤ c] ∩ rB_{U} − A([f_{n} ≤ c] ∩ rB_{X})).  (14)

Then

τ_{S}- lim_{n→∞} (f_{n} + (g_{n} ∘ A)) = f + (g ∘ A).

Proof. Let us set

F (x, u) = f(x) + g(A(x) + u) and F_{n}(x, u) = f_{n}(x) + g_{n}(A(x) + u).

We claim that

τ_{S}- lim_{n→∞} F_{n} = F.

Indeed, let (x, u) ∈ X × U. From the slice convergence of (f_{n}) and (g_{n}) to f and g there exist sequences (x_{n}) and (y_{n}) converging respectively to x and A(x) + u such that

lim sup_{n→∞} f_{n}(x_{n}) ≤ f(x) and lim sup_{n→∞} g_{n}(y_{n}) ≤ g(A(x) + u).

Setting u_{n} = y_{n} − Ax_{n}, we observe that u_{n} converges to u and we get

lim sup_{n→∞} F_{n}(x_{n}, u_{n}) ≤ lim sup_{n→∞} f_{n}(x_{n}) + lim sup_{n→∞} g_{n}(A(x_{n}) + u_{n})
≤ f(x) + g(Ax + u)
= F (x, u).

It follows that e- lim sup_{n→∞} F_{n} ≤ F. On the other hand as

F^{∗}(v, y) = f^{∗}(−A^{∗}(y) + v) + g^{∗}(y) and F_{n}^{∗}(v, y) = f_{n}^{∗}(−A^{∗}(y) + v) + g_{n}^{∗}(y).

we obtain by the same reasoning as above that

e- lim sup_{n→∞} F_{n}^{∗} ≤ F^{∗}.

Now let u ∈ sB_{U} and ξ ∈ sB_{X}. For all n ∈ IN, there exist u_{n} ∈ [g_{n} ≤ c] ∩ rB_{U} and x_{n} ∈ [f_{n} ≤ c] ∩ rB_{X} with u = u_{n} − A(x_{n}). Thus we get (ξ, u) = (x_{n}, u) − (x_{n} − ξ, 0) with

(x_{n}, u) ∈ [F_{n} ≤ 2c] ∩ (r + s)B_{X×U} and (x_{n} − ξ, 0) ∈ (X × {0}) ∩ (r + s)B_{X×U}

yielding

sB_{X×U} ⊂ ⋂_{n∈IN} ([F_{n} ≤ 2c] ∩ (r + s)B_{X×U} − (X × {0}) ∩ (r + s)B_{X×U}).

Hence we can apply Lemma 2.5 which ends the proof of the theorem.

Corollary 2.7. Let X be a normed vector space, let (f_{n})_{n∈IN}, f and let (g_{n})_{n∈IN}, g ∈ Γ_{0}(X) be such that

τ_{S}- lim_{n→∞} f_{n} = f and τ_{S}- lim_{n→∞} g_{n} = g.  (15)

Assume that for some real numbers s > 0, r ≥ 0, c ∈ IR

sB_{X} ⊂ ⋂_{n∈IN} ([f_{n} ≤ c] ∩ rB_{X} − [g_{n} ≤ c] ∩ rB_{X}).  (16)

Then

τ_{S}- lim_{n→∞}(f_{n} + g_{n}) = f + g.

Proof. Apply Theorem 2.6 with U = X and A = I_{X}.

One readily deduces from the above theorem a result on the continuity of the intersection of convex sets with respect to the slice topology.

Corollary 2.8. Let X be a normed vector space, let (C_{n})_{n∈IN}, (D_{n})_{n∈IN}, C, D ∈ C(X) such that

τ_{S}- lim_{n→∞} C_{n} = C and τ_{S}- lim_{n→∞} D_{n} = D.

Assume that for some real numbers s > 0, c ∈ IR

sB_{X} ⊂ ⋂_{n∈IN} (C_{n} ∩ rB_{X} − D_{n} ∩ rB_{X}).  (17)

Then

τ_{S}- lim_{n→∞}(C_{n} ∩ D_{n}) = C ∩ D.

Remark 2.9. Corollary 2.7 strictly extends Theorem 3.3 of [18] in which the continuity of the sum was obtained under the following assumption:

there exists s > 0, λ ∈ IR, x_{0} ∈ dom g such that sup_{(n,x)∈IN×B(x_{0},s)} f_{n}(x) ≤ λ.  (18)

Assume that this assumption is satisfied. From g = τ_{S}- lim_{n→∞} g_{n} and Proposition 1.2 it follows the existence of (u_{n}) → 0 such that

g(x_{0}) ≥ lim sup_{n→∞} g_{n}(x_{0} + u_{n}).

Thus there exists a constant μ ∈ IR such that for all n sufficiently large

g_{n}(x_{0} + u_{n}) ≤ μ and u_{n} ∈ \frac{s}{2}B_{X}.

Let u ∈ \frac{s}{2}B_{X}. Let us set v_{n} = u + u_{n}. One has v_{n} ∈ sB_{X}, u = (x_{0} + v_{n}) − (x_{0} + u_{n}) and f_{n}(x_{0n} + v_{n}) ≤ λ, g_{n}(x_{0n} + u_{n}) ≤ μ. Thus there exists r ≥ 0 and c ∈ IR such that

sB_{X} ⊂ ⋂_{n∈IN} ([f_{n} ≤ c] ∩ rB_{X} − [g_{n} ≤ c] ∩ rB_{X})

which is assumption (16) of Corollary 2.7. On the other hand, given a closed hyperplane H ⊂ X and a vector u ∉ H and setting f_{n} ≡ i_{H}, g_{n} ≡ i_{IRu}, we observe that (16) is in force but (18) fails to be satisfied.

Remark 2.10. Theorem 2.6 and Corollary 2.7 also extend to the infinite dimensional setting Theorem 5 and 8 of [23]. Indeed, taking into account the existence of a basis of neighborhhods of the origin in IR^{d} constituted by polyhedral sets, one easily checks as in [5], Theorem 1.1’ that assumption (16) holds whenever (15) is in force and 0 ∈ int ( dom f − dom g).

Example 2.11. Let X be a reflexive Banach space and let (f_{n}) and (g_{n}) be two sequences in Γ_{0}(X) converging respectively to f and g in the sense of Mosco. Then the sequence (f_{n} + g_{n}) converges to f + g for τ_{M}^{+} without any condition. This result fails if we replace the Mosco topology by the slice topology. This is shown by the following slight modification, brought to our attention by G. Beer in a private communication, of an example given in [11]. Let X be a non reflexive space and let y ∈ X^{∗} which is not norm achieving on the unit ball of X. Pick up x_{0} ∈ X satisfying ⟨x_{0}, y⟩ = 1 and ∥x_{0}∥ ≥ 1, and let H = y^{−1}(1). Let g ∈ Γ_{0}(X) defined by g(x) = d(x, H) + i_{B(x_{0},1)}(x) and let f ∈ Γ_{0}(X) defined by epi f = ⋃_{t≥0} t ( epi g) ∪ ({0} × IR) (observe that this set is closed from the assuptions made). Setting α = inf_{x∈X} f(x) = 0 and C = [f ≤ α] one has C ≠ ∅ and the family of closed convex C_{ϵ} = [f ≤ α + ϵ] is such that C ≠ τ_{S}^{+}- lim_{ϵ↓0} C_{ϵ}. Let us set A = epi f and D_{ϵ} = X × {α + ϵ}. One has τ_{S}- lim_{ϵ↓0} D_{ϵ} = X × {α}. Moreover A∩D_{ϵ} = [f ≤ α+ϵ]×{α+ϵ} does not converge for τ_{S}^{+}. One observes that in this example the assumption (17) is not satisfied since D_{ϵ} − A ⊂ X × [−ϵ, +∞] thus sB_{X×IR} ⊂ D_{ϵ} − A imply s ≤ ϵ. It follows that D_{ϵ} − A cannot contain a fixed ball sB_{X} for all ϵ > 0.

3. Primal-dual stability

The following result is a quantitative version of [24], Proposition 8.d.

Lemma 3.1. Let X be a normed vector space and let f ∈ Γ_{0}(X) be such that there exist λ > inf_{x∈X} f(x), ρ ≥ 0 satisfying

[f ≤ λ] ⊂ ρB_{X}.  (19)

Then for all λ_{0} ∈] inf_{x∈X} f(x), λ[ and for all v ∈ \frac{λ−λ_{0}}{2ρ} B_{X^{∗}} one has

f^{∗}(v) ≤ f^{∗}(0) + \frac{3}{2}(λ − λ_{0}).

As f ∈ Γ_{0}(X), it is bounded from below on the bounded subset [f ≤ λ] and then bounded from below on the whole X. Let us set β = inf_{x∈X} f = −f^{∗}(0) ∈ IR. Let λ_{0} ∈ IR be such that inf_{x∈X} f(x) < λ_{0} < λ. Define h = f^{∗} − ⟨x_{0}, .⟩ + λ_{0} where x_{0} ∈ [f < λ_{0}]. Observe that h = g^{∗} with g(x) = f(x + x_{0}) − λ_{0}. One has

inf_{x∈X} g = β − λ_{0} < 0.

The set C = [g ≤ λ − λ_{0}] is a closed convex subset containing 0 and

C ⊂ [f ≤ λ] − x_{0} ⊂ 2ρB_{X}.

Let us introduce the gauge function of C

j(x) = inf{t > 0 : x ∈ tC},

and let Δ = IR_{+}z be a half-line through the origin. Assuming that j(z) = +∞ we get g ≡ +∞ on Δ \ {0}. Indeed if x ∈ Δ \ {0} satisfies g(x) < +∞ then tx ∈ C for all t > 0 small enough since g(tx) ≤ tg(x) for all t ∈ [0, 1] yielding z ∈ IR_{+}^{∗}C, a contradiction with the fact that j(z) = +∞. Assuming that j is finite on Δ, we obtain that the function g − (λ − λ_{0})j is convex on Δ thus g ≤ (λ − λ_{0})j on C ∩ Δ. Observing that g ≥ (λ − λ_{0})j on (X \ C) ∩ Δ, we obtain in both cases that g ≤ (λ − λ_{0})j on C ∩ Δ and g ≥ (λ − λ_{0})j on (X \ C) ∩ Δ. It follows that

(λ − λ_{0})j + β − λ ≤ g on Δ.

Indeed assuming that x ∈ (X \ C) ∩ Δ we get g(x) ≥ (λ − λ_{0})j(x) and β − λ < 0, and assuming x ∈ C ∩ Δ one has

(λ − λ_{0})j(x) + β − λ ≤ λ − λ_{0} + β − λ ≤ β − λ_{0} ≤ g(x).

Setting W = (λ − λ_{0})C^{∘}, we get σ_{W} = (λ − λ_{0})σ_{C^{∘}} = (λ − λ_{0})j. Taking the conjugate we get h ≤ i_{W} + λ − β and \frac{1}{2ρ}B ⊂ C^{∘}. It follows that

f^{∗} ≤ λ − λ_{0} + f^{∗}(0) + ⟨x_{0}, .⟩ on \frac{λ − λ_{0}}{2ρ} B

which ends the proof of the lemma taking into account the fact that x_{0} ∈ ρB_{X}.

In the sequel, given a function F ∈ Conv(X × U) we shall use the following assumptions

sB_{X×U} ⊂ X × {0} ∩ rB_{X×U} − [F ≤ c] ∩ rB_{X×U}  (20)

for some s > 0, r ≥ 0, c ∈ IR and

s^{∗}B_{V ×Y} ⊂ {0} × Y ∩ r^{∗}B_{V ×Y} − [F^{∗} ≤ c^{∗}] ∩ r^{∗}B_{V ×Y}  (21)

for some s^{∗} > 0, r^{∗} ≥ 0, c^{∗} ∈ IR. It is easy to check that (20) is equivalent to the existence of s′ > 0, r′ ≥ 0, c′ ∈ IR such that

for all u ∈ s′B_{U}, there exists x ∈ r′B_{X} such that F (x, u) ≤ c′,  (22)

and (21) is equivalent to the existence of s^{∗}′ > 0, r^{∗}′ ≥ 0, c^{∗}′ ∈ IR such that

for all v ∈ s^{∗}′B_{V} , there exists y ∈ r^{∗}′B_{Y} such that F^{∗}(v, y) ≤ c^{∗}′.  (23)

Here stands our main result.

Theorem 3.2. Let X, U be normed vector spaces, let (F_{n})_{n∈IN}, F ∈ Γ_{0}(X × U) be such that F = τ_{S}- lim_{n→∞} F_{n}. Assume that there exist s > 0, r ≥ 0, c ∈ IR with

sB_{X×U} ⊂ ⋂_{n∈IN} (X × {0} ∩ rB_{X×U} − [F_{n} ≤ c] ∩ rB_{X×U}).  (24)

Assume also that there exist λ > lim sup_{n→∞} ( inf_{x∈X} f_{n}(x)) and ρ ≥ 0 such that

⋃_{n∈IN}[f_{n} ≤ λ] ⊂ ρB_{X}  (25)

where f_{n} = F_{n}(., 0). Then

F (., 0) = τ_{S}- lim_{n→∞} F_{n}(., 0),  (26)
F^{∗}(0, .) = τ_{S}^{∗}- lim_{n→∞} F_{n}^{∗}(0, .),  (27)
φ̄ = τ_{S}- lim_{n→∞} φ̄_{n}.  (28)

Proof. It is clear that (26) follows from Lemma 2.5. From Assumption (24), we can apply Theorem 2.4 yielding

(F_{n} + i_{X×{0}})^{∗}(v, 0) = min_{(v′,y′)∈V ×Y} F_{n}^{∗}(v − v′, −y′) + i_{{0}×Y} (v′, y′) = ψ_{n}(v).

It follows that ψ_{n} ∈ Γ_{0}(V) which implies that ψ_{n} = f_{n}^{∗}. From assumption (25) we can apply Lemma 3.1. Hence for all λ > λ_{0} > lim sup_{n→∞} ( inf_{x∈X} f_{n}(x)) we get

ψ_{n}(v) ≤ ψ_{n}(0) + \frac{3}{2}(λ − λ_{0})  (29)

for all v ∈ s^{∗}B_{V} with s^{∗} = \frac{λ−λ_{0}}{2ρ}. Let f = F (., 0) and let v ∈ dom f^{∗}. Using Lemma 2.5 and Proposition 1.2, there exists a sequence (v_{n}) in V converging to v such that f^{∗}(v) ≥ lim sup_{n→∞} f_{n}^{∗}(v_{n}). For all x ∈ X, one gets f_{n}(x) ≥ ⟨x, v_{n}⟩−f_{n}^{∗}(v_{n}) which shows the existence of m ∈ IR such that for all n large enough one has − inf_{x∈X} f_{n}(x) < m. Returning to (29) we have, for all n large enough and for all v ∈ s^{∗}B_{V}

inf_{y∈Y} F_{n}^{∗}(v, y) = ψ_{n}(v) < λ^{∗}

where λ^{∗} = m + \frac{3}{2}(λ − λ_{0}). We claim that there exists r^{∗} ≥ 0 such that for all n large enough, for all v ∈ s^{∗}B_{V} there exists y ∈ r^{∗}B_{Y} with F_{n}^{∗}(v, y) < λ^{∗}. Indeed given v ∈ s^{∗}B_{V}, let us introduce for all n ∈ IN the function ϕ̃_{n} defined for all u ∈ U by

ϕ̃_{n}(u) = inf_{x∈X}(F_{n}(x, u) − ⟨x, v⟩).

322 D. Azé, A. Rahmouni / On primal-dual stability in convex optimization

For all y ∈ Y we have ϕ̃_{n}^{∗}(y) = F_{n}^{∗}(v, y). From (24) there exists for all u ∈ sB_{U} an element x ∈ rB_{U} such that F_{n}(x, u) ≤ λ which entails ϕ̃_{n}(u) ≤ λ + rs^{∗} on sB_{X}. Thus we get ϕ̃_{n} ≤ λ + rs^{∗} + i_{sB_{U}}, which by duality implies F_{n}^{∗}(v, y) ≥ s∥y∥ − λ − rs^{∗}. Let v ∈ s^{∗}B_{V}, there exists y ∈ Y with F_{n}^{∗}(v, y) < λ^{∗} hence ∥y∥ ≤ r^{∗} with r^{∗} = \frac{λ+λ^{∗}+rs^{∗}}{s}. Thus for all n large enough

s^{∗}B_{V ×Y} ⊂ {0} × Y ∩ (r^{∗} + s^{∗})B_{V ×Y} − [F_{n}^{∗} ≤ λ^{∗}] ∩ (r^{∗} + s^{∗})B_{V ×Y} .

From Lemma 2.2 we derive

epi (F^{∗} + i_{{0}×Y}) ⊂ lim inf_{n→∞} ( epi (F_{n}^{∗} + i_{{0}×Y})),

which easily leads to

epi F^{∗}(0, .) ⊂ lim inf_{n→∞} ( epi F_{n}^{∗}(0, .)).

On the other hand Lemma 1.1 ensures that epi φ̄ ⊂ lim inf_{n→∞} epi φ̄_{n}. Hence we get

F^{∗}(0, .) = τ_{S}- lim_{n→∞} F_{n}^{∗}(0, .) and φ̄ = τ_{S}- lim_{n→∞} φ̄_{n}

and the proof of the theorem is complete.

In the particular case of the Fenchel duality, we get the following corollary. Corollary 3.3. Let X, U be normed vector spaces, let A ∈ L(X, U) and let (f_{n})_{n∈IN}, f ∈ Γ_{0}(X), (g_{n})_{n∈IN}, g ∈ Γ_{0}(U) be such that

τ_{S}- lim_{n→∞} f_{n} = f and τ_{S}- lim_{n→∞} g_{n} = g.

Assume that there exist s > 0, r ≥ 0, c ∈ IR such that

sB_{U} ⊂ ⋂_{n∈IN} ([g_{n} ≤ c] ∩ rB_{U} − A([f_{n} ≤ c] ∩ rB_{X})).  (30)

Assume also that there exist λ > lim sup_{n→∞} ( inf_{x∈X}(f_{n}(x)+g_{n}(A(x)))) and ρ ≥ 0 such that

⋃_{n∈IN} ([f_{n} + (g_{n} ∘ A) ≤ λ]) ⊂ ρB_{X}.

Then

τ_{S}- lim_{n→∞} (f_{n} + (g_{n} ∘ A)) = f + (g ∘ A)

and

τ_{S}^{∗}- lim_{n→∞} (f_{n}^{∗} ∘ (−A^{∗}) + g_{n}^{∗}) = f^{∗} ∘ (−A^{∗}) + g^{∗}.

Proof. Setting F (x, u) = f(x) + g(A(x) + u) and F_{n}(x, u) = f_{n}(x) + g_{n}(A(x) + u), we have (see the proof of Theorem 2.6)

τ_{S}- lim_{n→∞} F_{n} = F.

Moreover one has

F^{∗}(v, y) = f^{∗}(−A^{∗}(y) + v) + g^{∗}(y) and F_{n}^{∗}(v, y) = f_{n}^{∗}(−A^{∗}(y) + v) + g_{n}^{∗}(y),

thus the corollary follows from Theorem 3.2.

4. Example of application

In this section, we apply the results of the preceding one to the convergence of the adjoint space in some singularly perturbed optimal control problem described in [1], p. 336. Let Ω ⊂ IR^{d} be a bounded open subset with a smooth boundary Γ such that Ω is locally on the same side of Γ. Let us set Z = L^{2}(Γ), V = H^{1}(Ω), H = L^{2}(Ω), let ϵ > 0 and let a_{ϵ}(., .) be the continuous and elliptic bilinear form on V defined by

a_{ϵ}(y_{1}, y_{2}) = ϵ ∫_{Ω} Dy_{1}.Dy_{2} dx + ∫_{Ω} y_{1}y_{2} dx.

Let us denote by A_{ϵ} ∈ L(V, V^{∗}) the linear isomorphism defined for all y ∈ V by a_{ϵ}(y, .) = A_{ϵ}(y). Given g ∈ H and y ∈ V, one has A_{ϵ}(y) = g if and only if

\{ \begin{matrix}−ϵΔy + y = g & on Ω \\ \frac{∂y}{∂ν} = 0 & on Γ \end{matrix}  (31)

where ν is the outward normal vector to Γ at x. The optimal control problem we are dealing with is

min \frac{1}{2}∥γ_{0}y − z_{d}∥_{Z}^{2} + \frac{N}{2} ∥v∥_{H}^{2}  (32)

over v ∈ H, y ∈ V related by (31) where γ_{0}y ∈ H^{\frac{1}{2}} (Γ) denotes the trace of y on Γ and z_{d} ∈ Z is given. In order to dualize our problem, we use the general method for duality in optimal control introduced in [8]. Let us introduce the function F_{ϵ} ∈ Γ_{0}(Z × H × H) defined by

F_{ϵ}(z, v, f) = \{ \begin{matrix}\frac{1}{2}∥z − z_{d}∥_{Z}^{2} + \frac{N}{2} ∥v∥_{H}^{2} + i_{C_{ϵ}}(y, v, f) & if z = γ_{0}y \\ +∞ & otherwise \end{matrix}

where

C_{ϵ} = {(y, v, f) ∈ V × H × H : A_{ϵ}(y) = v + f}.

Problem (32) is equivalent to

min_{(z,v)∈Z×H} F_{ϵ}((z, v), 0).

Let us recall that given p ∈ H such that Δp ∈ H the trace γ_{0}p = p | Γ and γ_{1}p = \frac{∂p}{∂ν} | Γ are

defined respectively in H^{− \frac{3}{2}} (Γ) and H^{− \frac{1}{2}} (Γ) (see [22], Vol. 1, Théorème 6.5, p. 187-189) and that, for all y ∈ H^{2}(Ω) one has

− ∫_{Ω} pΔy dx = − ∫_{Ω} yΔp dx + ⟨γ_{1}p, γ_{0}y⟩_{H^{− \frac{3}{2}} (Γ),H^{\frac{3}{2}} (Γ)} − ⟨γ_{0}p, γ_{1}y⟩_{H^{− \frac{1}{2}} (Γ),H^{\frac{1}{2}} (Γ)}.

From the preceding Green’s formula the Lagrangian associated with F_{ϵ} is easily shown to be defined on Z × H × H by

L_{ϵ}(z, v, p) =

\frac{1}{2}∥z − z_{d}∥_{Z}^{2} + ⟨p, v⟩ + \frac{N}{2} ∥v∥_{H}^{2} − ⟨ϵγ_{1}p, z⟩_{H^{− \frac{3}{2}} (Γ),H^{\frac{3}{2}} (Γ)} − i_{D_{ϵ}}(p) if z ∈ γ_{0}(A^{−1}(H))

+∞  otherwise

where

D_{ϵ} = {p ∈ H : −ϵΔp + p = 0}.

It follows that a pair ((z̄_{ϵ}, v̄_{ϵ}), p̄_{ϵ}) ∈ Z × H × H is a saddle-point for L_{ϵ} if and only if there exists ȳ_{ϵ} ∈ V such that

\{ \begin{matrix}−ϵΔȳ_{ϵ} + ȳ_{ϵ} = v̄_{ϵ} & on Ω \\ \frac{∂ȳ_{ϵ}}{∂ν} = 0 & on Γ, \end{matrix}

\begin{matrix}z̄_{ϵ} = γ_{0}ȳ_{ϵ} \\ p̄_{ϵ} + Nv̄_{ϵ} = 0, \end{matrix}  (33)

\{ \begin{matrix}−ϵΔp̄_{ϵ} + p̄_{ϵ} = 0 & on Ω \\ ϵγ_{1}p̄ = z̄_{ϵ} − z_{d} & on Γ. \end{matrix}

A slight adaptation of the proof given in [1], p. 338-340, shows that the sequence (F_{ϵ}) congerges in the Mosco sense as ϵ goes to 0 on Z × H to the function F ∈ Γ_{0}(Z × H × H) defined by

F ((z, v), f) = \frac{1}{2}∥z − z_{d}∥_{Z}^{2} + \frac{N}{2} ∥v + f∥_{H}^{2}.

The Lagrangian associated to F is defined for all ((z, v), p) ∈ Z × H × H by

L((z, v), p) = ⟨p, v⟩ + \frac{1}{2}∥z − z_{d}∥_{Z}^{2} − \frac{1}{2N} ∥p∥_{H}^{2}.

The unique saddle-point of L is clearly ((z_{d}, 0), 0). The sequence of functions (F_{ϵ}(., 0)) is clearly equicoercive thus assumption (25) of Theorem 3.2 is in force. Moreover assumption (24) is also satisfied since F_{ϵ}(0, −f, f) ≤ \frac{1}{2}∥z_{d}∥_{Z}^{2} + \frac{N}{2} ∥f∥_{H}^{2} yielding

sup_{ϵ>0} φ_{ϵ}(f) ≤ \frac{1}{2}∥z_{d}∥_{Z}^{2} + \frac{N}{2} ∥f∥_{H}^{2}

where φ_{ϵ}(f) = inf_{(z,v)∈Z×H} F_{ϵ}(z, v, f) is the value function associated to F_{ϵ}; which implies (25), (see (22)). From Theorem 3.2 and from the variational properties of Mosco convergence (see [1]) we obtain that the sequence of adjoint states (p̄_{ϵ}) converges to 0 in w- L^{2}(Ω). In fact one can prove, using our duality method, that the sequence (p̄_{ϵ}) converges to 0 in s-L^{2}(Ω). Indeed some straightforward computations based on the generalized Green’s formula quoted above show that the dual functional is defined on H by

F_{ϵ}^{∗}((0, 0), p) = \frac{1}{2}∥ϵγ_{1}p∥_{Z}^{2} + ⟨ϵγ_{1}p, z_{d}⟩_{Z} + \frac{1}{2N} ∥p∥_{H}^{2} + iD̃_{ϵ}(p)

where

D̃_{ϵ} = {p ∈ H : −ϵΔp + p = 0, γ_{1}p ∈ Z}.

As the duality gap is equal to 0, we get taking into account (33)

\begin{aligned}\frac{1}{2}∥z̄_{ϵ} − z_{d}∥_{Z}^{2} + \frac{N}{2} ∥v̄_{ϵ}∥_{H}^{2} & = & −\frac{1}{2}∥ϵγ_{1}p̄_{ϵ}∥_{Z}^{2} − ⟨ϵγ_{1}p̄_{ϵ}, z_{d}⟩_{Z} − \frac{1}{2N} ∥p̄_{ϵ}∥_{H}^{2} \\ & = & −\frac{1}{2}∥z̄_{ϵ}∥_{Z}^{2} + \frac{1}{2}∥z_{d}∥_{Z}^{2} − \frac{1}{2N} ∥p̄_{ϵ}∥_{H}^{2} \end{aligned}

thus (p̄_{ϵ}) converges to 0 in s-L^{2}(Ω) relying on the fact (see [1], p. 340) that

lim_{ϵ→0} \frac{1}{2}∥z̄_{ϵ} − z_{d}∥_{Z}^{2} + \frac{N}{2} ∥v̄_{ϵ}∥_{H}^{2} = 0.

Acknowledgment. The authors would like to thank one anonymous referee for his helpful comments and suggestions which contributed to improve the presentation of this paper.

References

[1] H. Attouch: Variational convergence for functions and operators, Applicable Mathematics Series, Pitman, Boston, London, 1984.

[2] H. Attouch, D. Azé and R.J.-B. Wets: Convergence of convex- concave saddle functions: applications to convex programming and mechanics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5 (1988) 537–572.

[3] H. Attouch, G. Beer: On the convergence of subdifferential of convex functions, Arch. Math. 60 (1993) 389–400.

[4] D. Azé: Duality for the sum of convex functions in general normed spaces, Arch. Math. 6 (1994) 554–561.

[5] D. Azé, J.-P. Penot: Operations on convergent families of sets and functions, Optimization 21 (1990) 521–534.

[6] D. Azé, J.-P. Penot: Qualitative results about the convergence of convex sets and convex functions, A. Ioffe, M. Marcus and S. Reich eds, Pitman Research Notes in Mathematics Series (1992) 1–24.

[7] D. Azé, J.-P. Penot: The Joly topology and the Mosco-Beer topology revisited, Bull. Austral. Math. Soc. 48 (1993) 353–364.

[8] D. Azé: On optimality conditions for state-constrained optimal control problems governed by partial differential equations, submitted, 1994.

[9] K. Back: Continuity of the Fenchel transform of convex functions, Proc. Am. Math. Soc. 97 (1986) 661–667.

326 D. Azé, A. Rahmouni / On primal-dual stability in convex optimization

[10] K. Back: Convergence of Lagrange multipliers and dual variables for convex optimization problems, Math. Op. Res. 13 (1990) 74–79.

[11] G. Beer: The slice topology, a viable alternative to Mosco convergence in nonreflexive spaces, Nonlinear Analysis, Theory Methods Appl. 19 (1992) 271–290.

[12] G. Beer: Topologies on closed and closed convex sets, Mathematics and its applications, 268, Kluwer, London, Dordrecht, Boston, 1994.

[13] R.C. Bergström: Optimization, Convergence and Duality, Ph. D. Thesis, University of Illinois at Urbana-Champaign, 1980.

[14] I. Ekeland, R. Temam: Analyse convexe et problèmes variationnels, Dunod, Paris, 1978.

[15] E.G. Gol’štein: Theory of convex programming, Translations of mathematical monograph, Volume 36, American Mathematical Society, Providence, 1972.

[16] J.-L. Joly: Une famille de topologies et de convergences sur l’ensemble des fonctionnelles convexes, Thèse d’ état, Université de Grenoble, 1970.

[17] J.-L. Joly: Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl., IX. Sér., 52 (1973) 421–441.

[18] J. Lahrache: Topologie sur l’ensemble des convexes fermés d’un espace vectoriel normé, stabilité et analyse unilatérale, Thèse, Université de Montpellier, 1992.

[19] J. Lahrache: Stabilité et convergence dans les espaces non réflexifs, Sémin. Anal. Convexe, Univ. Sci. Tech. Languedoc, exposé n^{o} 10, 1991.

[20] P.-J. Laurent: Approximation et Optimisation, Hermann, Paris, 1972.

[21] R. Lucchetti, F. Patrone: Closure and upper semicontinuity results in mathematical programming, Nash and economic equilibria, Optimization 17 (1986) 619–628.

[22] J.-L. Lions and E. Magenes: Problèmes aux limites non homogènes et applications, Dunod, Gauthier-Villars, Paris, 1968.

[23] L. Mc Linden, R.C. Bergström: Preservation of convergence of convex sets and functions in finite dimensions, Trans. Am. Math. Soc. 268 (1981) 127–142.

[24] J.-J. Moreau, Fonctionnelles convexes: Lecture Notes Collège de France, Paris, 1966.

[25] U. Mosco: Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969) 510–585.

[26] U. Mosco: On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 45 (1974) 533–555.

[27] H. Riahi: Quelques résultats de stabilité en analyse épigraphique: approche topologique et quantitative, Thèse, University of Montpellier, France, 1989.

[28] S.M. Robinson: Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976) 130–143.

[29] R.T. Rockafellar: Conjugate duality and optimization, SIAM publication 16, Philadelphia, 1974.

[30] R. Schultz: Estimates for Kuhn-Tucker points of perturbed convex programs, Optimization 19 (1988) 29–43.

[31] Y. Sonntag, C. Zalinescu: Set convergence. An attempt of classification, International Conference on Differential Equations and Control Theory, Iaşi, Rômania, 1990.

[32] Y. Sonntag, C. Zalinescu: Set convergence. An attempt of classification, Trans. Am. Math. Soc. 340 (1993) 199–226.

[33] Y. Sonntag, C. Zalinescu: Set convergence: a survey and a classification, Set Valued Analysis 2 (1994) 339–356.

[34] R. Vinter: Convex duality and nonlinear optimal control, SIAM J. Control Opt. 31 (1993) 518–538.

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