Journal of Convex Analysis Volume 7 (2000), No. 1, 47–71

Epi-Distance Convergence of Parametrised Sums of Convex Functions in Non-Reflexive Spaces

A. Eberhard, R. Wenczel

Department of Mathematics, Royal Melbourne University of Technology, Melbourne 3001, Australia. e-mail: andy.eb@rmit.edu.au

Received April 24, 1998

A weakened set of conditions is established for the epi-distance convergence of a sum {f_{v} + g_{v}}_{v∈W} of parametrised closed convex functions {f_{v}}_{v∈W} and {g_{v}}_{v∈W} for v → w, on an arbitrary Banach space. They are as follows: (1) 0 ∈ sqri(dom f_{w} − dom g_{w}); and (2) X_{w} := cone(dom f_{w} − dom g_{w}) has closed algebraic complement Y_{w}; and (3) X_{v} ∩ Y_{w} = \{0\} for all v near w, (where X_{v} := \overline{span}(dom f_{v} − dom g_{v})). These are motivated by similar interiority conditions found in Fenchel duality theory. Our results are then used to investigate saddle-point convergence in Young-Fenchel duality in which both functions vary in a very general fashion.

1. Introduction

Over the last thirty years a number of authors have proposed topologies for the hyperspace of closed convex sets, the most famous being the Mosco topology. It has become evident that despite the success of this topology in characterising convergence of convex sets in reflexive Banach spaces, it fails to have many desirable properties in non-reflexive spaces (see [11]). On the other hand the Attouch-Wets topology (or epi-distance topology) has proved itself a valuable tool for obtaining quantitative estimates outside of the reflexive context (see [3],[4],[5]). Another candidate for a replacement of the Mosco topology is the slice topology of Beer [10]. A comprehensive treatment of all of these is given by Beer in [14]. Our paper concentrates on utilizing special features of Banach spaces to analyse epi-distance convergence. This has only previously been embarked upon in one instance [8]. Since the definition of the epi-distance topology only requires the existence of a metric it is usually the case that authors have only specialised to the context of a normed linear space (possibly not complete), [14]. Our considerations here are largely topological and so we do not attempt to obtain error bounds or estimates of metrics generating the Attouch-Wets topology. This endeavour has been undertaken in [8] under stronger assumptions.

Azé & Penot [8] and Beer [12] have considered the following problem: Given two families of closed proper convex functions {f_{v}}_{v∈W} and {g_{v}}_{v∈W}, mapping a normed space to the extended reals, both converging in the epi-distance sense to f_{w} and g_{w} (as v → w) respectively, when is the family {f_{v} + g_{v}}_{v∈W} epi-distance convergent to f_{w} + g_{w}? Beer [12] gives a simple condition, namely (int dom f_{w}) ∩ dom g_{w} ≠ ∅. This interiority condition transforms to the familiar Slater constraint-qualification on application to perturbed convex optimization problems. In [8] this is weakened to a more complex condition designed to extract convergence estimates. We observe here that this condition is equivalent to 0 ∈ int(dom f_{v} − dom g_{v}) for all v in a neighbourhood of w. Although the condition

ISSN 0944-6532 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

48 A. Eberhard, R. Wenczel / Epi-distance convergence of parametrised sums

(int dom f_{w}) ∩ dom g_{w} ≠ ∅ is a natural and often-used assumption in convex analysis it has been noted in recent years that it is unduly restrictive in some infinite-dimensional spaces (such as l^{1}), due to the lack of interior in the cone of positive vectors. This has motivated the introduction of weaker interiority notions such as the concept of “quasi relative interior”[15] and “strong quasi relative interior”[18]. We refer the reader to Gowda and Teboulle [16] for an insightful comparison of these constraint qualifications.

In the context of conjugate duality Robinson [20] first showed that in non-reflexive spaces the weakened condition 0 ∈ core(dom f_{w} − dom g_{w}) is sufficient to ensure strong duality. Jeyakumar [19] has shown in this context that such conditions may be used to derive other seemingly weaker constraint-qualifications 0 ∈ sqri(dom f_{w} − dom g_{w}), involving the concept of strong quasi relative interior (i.e. cone(dom f_{w} − dom g_{w}) = X_{w} is a closed subspace). In this paper we will show that for functions on a Banach space, the conditions:

(1) 0 ∈ sqri(dom f_{w} − dom g_{w}); and (2) the algebraic complement Y_{w} of X_{w} is closed and satisfies X_{v} ∩ Y_{w} = \{0\} for all v near w, (where X_{v} := \overline{span}(dom f_{v} − dom g_{v})), are sufficient for establishing epi-distance convergence of the sum {f_{v} + g_{v}}_{v∈W} (as outlined above). The condition 0 ∈ core(dom f_{w} −dom g_{w}) (and so also 0 ∈ sqri(dom f_{w} −dom g_{w})) is implied by 0 ∈ int(dom f_{w} − dom g_{w}). Surprisingly, for closed proper convex functions 0 ∈ core(dom f_{w}−dom g_{w}) implies 0 ∈ int(dom f_{w}−dom g_{w}), (and so both are equivalent). This equivalence does not seem to receive due acknowledgement in the literature (compare this with the well-known and well-documented equality of core and interior for the domains of single closed convex functions). Moreover 0 ∈ int (dom f_{w} − dom g_{w}) holding at w implies it holds locally (i.e. X_{v} = X, locally). Thus our result is a non-trivial extension of the sufficient conditions given in [8] for epi-distance convergence of a sum of parametrised closed convex functions.

These observations lead one to consider a weak set of conditions ensuring strong duality for the Young-Fenchel duality scheme which also imply stability of the perturbed problem (in the epi-distance sense). We also investigate saddle-point convergence of the associated saddle function. From within the framework of variational analysis this allows one to investigate the convergence of approximate solutions of the perturbed primal and dual optimization problems to solutions of the limiting problem. It may be shown that one can quite generally deduce the existence of an accumulation point of the approximating dual solutions. These results augment those obtained by Attouch, Wets, Brézis, Azé and Penot as well as that by other authors.

2. Preliminaries

In this section we draw together a number of results and definitions. This is done to make the development self-contained.

In the following the upper-case roman letters A, B etc, are used to denote sets in a (real) Banach space. \overline{ℝ} shall stand for the extended reals [−∞, +∞]. We will let C(X) stand for the class of all nonempty closed convex subsets of a Banach space X and CB(X) the closed bounded convex sets. Place d(a, B) = inf{ ∥a−b∥ | b ∈ B }, B(0, ρ) = { x ∈ X | ∥x∥ < ρ } and B̄(0, ρ) = {x ∈ X | ∥x∥ ≤ ρ }. Corresponding balls in the dual space X^{∗} will be denoted B^{∗}(0, ρ) and B̄^{∗}(0, ρ) respectively. The indicator function of a set A will be denoted δ_{A}.

We will use u.s.c. to denote upper-semicontinuity and l.s.c. to denote lower-semicontinuity.

Recall that a function f : X → \overline{ℝ} is called closed, proper convex on X if and only if f is convex l.s.c. is never −∞ and not is identically +∞. The class of all closed proper convex functions on X is denoted by Γ(X), and the class of all weak^{∗} closed proper convex functions on X^{∗} will be denoted by Γ^{∗}(X^{∗}). We shall use the notation cl A and \overline{A} interchangeably for the closure of a set A in a topological space (Z, τ) and, to emphasise the topology, we may write cl_{τ} A or \overline{A}^{τ}. For x ∈ Z, N_{τ}(x) denotes the collection of all τ-neighborhoods of x. For a function f : Z → \overline{ℝ}, the epigraph of f, denoted epi f, is the set {(x, α) ∈ Z × ℝ | f(x) ≤ α}, and the strict epigraph epi_{s} f is the set {(x, α) ∈ Z×ℝ | f(x) < α}. The domain, denoted dom f is the set {x ∈ Z | f(x) < +∞}. The (sub-)level set {x ∈ Z | f(x) ≤ α} (where α > inf_{Z} f) will be given the abbreviation {f ≤ α}. Any product X × Y of normed spaces will always be understood to be endowed with the box norm ∥(x, y)∥ = max{∥x∥, ∥y∥}; any balls in such product spaces will always be with respect to the box norm.

If f : (Z, τ) → \overline{ℝ}, its τ-l.s.c. hull, denoted \overline{f}^{τ} or cl_{τ} f, is defined by \overline{f}^{τ}(x) = lim inf_{x′→^{τ}x} f(x′). The (extended) lower closure \underline{cl}_{τ}f is defined to coincide with cl_{τ} f if the latter does not take the value −∞ anywhere, and to be identically −∞ otherwise.

Definition 2.1. Let F:W → 2^{X} be a multifunction from topological spaces W to X.

(1) lim sup_{v→w} F (v) = ⋂_{V ∈N (w)} \overline{⋃_{v∈V} F (v)}.

(2) lim inf_{v→w} F (v) = ⋂_{\{B⊂W |w∈\overline{B}\}} \overline{⋃_{v∈B} F (v)}.

(3) F (·) is lower-semicontinuous at w iff F (w) ⊆ lim inf_{v→w} F (v).

Remark 2.2. It is easily seen that this notion of lower-semicontinuity is equivalent to the classical formulation–namely: For any open set U intersecting F (w) there is a neighborhood V of w for which F (v) ∩ U is nonempty for every v in V.

Remark 2.3. For metrizable X, the above definitions can be shown to have the equivalent forms:

(1)

lim sup_{v→w} F (v) = {x ∈ X | ∃ nets v_{β} → w, x_{β} → x with x_{β} ∈ F (v_{β}) ∀β }
= {x ∈ X | lim inf_{v→w} d(x, F (v)) = 0}

(2)

lim inf_{v →w} F (v) = { x ∈ X | ∀ nets v_{β} → w, ∃ x_{β} → x s.t. x_{β} ∈ F (v_{β}) eventually }
= {x ∈ X | lim sup_{v→w} d(x, F (v)) = 0}

Definition 2.4. Let A be a convex set in a topological vector space and x ∈ A. Then

(1) cone A = ∪_{λ>0}λA (the smallest convex cone containing A);

(2) x ∈ qri A (quasi relative interior) iff \overline{cone(A − x)} is a subspace of X;

(3) x ∈ sqri A (strong quasi relative interior) iff cone(A − x) is a closed subspace of X;

(4) x ∈ core A (core) iff ∀y ∈ X ∃ϵ > 0 such that ∀λ ∈ [−ϵ, ϵ] we have x + λy ∈ A;

(5) For a ∈ A place aff A = a + span(A − a), the affine hull; (this is independent of the choice of a ∈ A)

(6) icr A (intrinsic core) is the core of A relative to aff A;

(7) y ∈ ri A (relative interior) iff 0 is an interior point of A − y relative to aff(A − y).

From [17] and [22] we have the following. Recall that a set A in a topological linear space X is ideally convex if for any bounded sequence {x_{n}} ⊆ A and {λ_{n}} of nonnegative numbers with ∑_{n=1}^{∞} λ_{n} = 1, the series ∑_{n=1}^{∞} λ_{n}x_{n} either converges to an element of A, or else does not converge at all. Open or closed convex sets are ideally convex, as is any finite-dimensional convex set. In particular, if X is Banach, then such series always converge, and the definition of ideal convexity only requires that ∑_{n=1}^{∞} λ_{n}x_{n} be in A. From [17, Section 17E] we have the following

Proposition 2.5. For a Banach space X,

(i) If C ⊂ X is closed convex, it is ideally convex. (ii) For ideally convex C, core C = core \overline{C} = int \overline{C} = int C. (iii) If A and B are ideally convex subsets of X, one of which is bounded, then A − B is

ideally convex.

Proof. We prove the last assertion only; the rest can be found in the cited reference. Let {a_{n} − b_{n}} ⊂ A − B be a bounded sequence, let λ_{n} ≥ 0 be such that ∑_{n=1}^{∞} λ_{n} = 1. Then {a_{n}} ⊂ A and {b_{n}} ⊂ B are both bounded, so ∑_{n=1}^{∞} λ_{n}a_{n} ∈ A and ∑_{n=1}^{∞} λ_{n}b_{n} ∈ B (both convergent). Thus ∑_{n=1}^{∞} λ_{n}(a_{n} − b_{n}) = ∑_{n=1}^{∞} λ_{n}a_{n} − ∑_{n=1}^{∞} λ_{n}b_{n} ∈ A − B.

Proposition 2.6. Let f, g ∈ Γ(X) (X a Banach space). Then

core(dom f − dom g) = int(dom f − dom g).

Proof. It suffices to show that 0 ∈ core(dom f − dom g) implies 0 ∈ int(dom f − dom g) since for any x ∈ core(dom f − dom g) we may define g̃(y) := g(y − x), a closed proper convex function, for which dom g̃ = dom g + x and so 0 ∈ core(dom f − dom g̃). If 0 ∈ core(dom f − dom g) then by a simple argument 0 ∈ core(epi f − epi g). From this it follows that whenever ρ > 0 satisfies ρ > ∥(x̄, ᾱ)∥ for some (x̄, ᾱ) ∈ epi f ∩ epi g, we have

0 ∈ core(epi f ∩ \overline{B}(0, ρ) − epi g ∩ \overline{B}(0, ρ)).  (2.1)

Indeed, if (x, α) ∈ X × ℝ, then since X × ℝ = cone(epi f − epi g) we have for some λ > 0, (x_{1}, α_{1}) ∈ epi f and (x_{2}, α_{2}) ∈ epi g that

(x, α) = λ ((x_{1}, α_{1}) − (x_{2}, α_{2}))
= tλ ([\frac{1}{t}(x_{1}, α_{1}) + (1 − \frac{1}{t})(x̄, ᾱ)] − [\frac{1}{t}(x_{2}, α_{2}) + (1 − \frac{1}{t})(x̄, ᾱ)])
∈ tλ (epi f ∩ \overline{B}(0, ρ) − epi g ∩ \overline{B}(0, ρ))

if t > 1 is selected so that \frac{1}{t}∥(x_{i}, α_{i})∥ + ∥(x̄, ᾱ)∥ < ρ for i = 1, 2. As (x, α) ∈ X × ℝ was arbitrary we have

X × ℝ = cone(epi f ∩ \overline{B}(0, ρ) − epi g ∩ \overline{B}(0, ρ))

implying (2.1). As epi f ∩\overline{B}(0, ρ) and epi g∩\overline{B}(0, ρ) are closed and convex they are ideally convex. By Proposition 2.5, epi f ∩ \overline{B}(0, ρ) − epi g ∩ \overline{B}(0, ρ) is ideally convex and so

0 ∈ int (epi f ∩ \overline{B}(0, ρ) − epi g ∩ \overline{B}(0, ρ)).  (2.2)

Using the fact we are using box norms in X × ℝ we may project onto X to obtain

0 ∈ int (\{f ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g ≤ ρ\} ∩ \overline{B}(0, ρ)) ⊆ int(dom f − dom g).

The following is a consequence of the previous proof.

Corollary 2.7. If f, g ∈ Γ(X), where X is a Banach space and ρ > inf{∥(x, α)∥ | (x, α) ∈ epi f ∩ epi g} then

0 ∈ core (dom f − dom g) implies 0 ∈ int (\{f ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g ≤ ρ\} ∩ \overline{B}(0, ρ)).

From [16] we have a characterization of sqri C.

Lemma 2.8. If C ⊆ X a linear topological space then

\{ \begin{matrix}x ∈ icr(C) \\ and aff(C − x) is a closed subspace \end{matrix} \} if and only if x ∈ sqri C

further, if x ∈ sqri C then aff(C − x) = cone(C − x) also.

We immediately have the following confirming the conjecture in [16] that strong quasi relative interior is close to being a relative interior (see [16, page 932]).

Lemma 2.9. Let f, g ∈ Γ(X) (X a Banach space). Then 0 ∈ sqri(dom f − dom g) if and only if 0 ∈ ri(dom f − dom g) and aff (dom f − dom g) is a closed subspace of X.

The infimal convolution plays a central role in our development.

Definition 2.10. Let f and g be closed convex functions on a linear topological space X into the extended reals. Then

(f□g)(x) := inf_{y∈X}(f(y) + g(x − y))

is called the inf-convolution.

It is well known that the strict epigraph of the inf-convolution is equal to the set-addition of the strict epigraphs of the individual functions:

epi_{s}(f□g) = epi_{s} f + epi_{s} g.

Also dom(f□g) = dom f + dom g and

(f□g)^{∗} = f^{∗} + g^{∗}

where f^{∗}(x^{∗}) = sup_{x∈X}(⟨x, x^{∗}⟩ − f(x)) is the Young-Fenchel conjugate of f. From [1] we have the following (which establishes a condition for f^{∗}□g^{∗} to be lower-semicontinuous and so for epi f^{∗} + epi g^{∗} to be weak^{∗} closed) giving epi(f^{∗}□g^{∗}) = epi f^{∗} + epi g^{∗}.

Proposition 2.11. Assume that f, g are proper closed convex functions from a Banach space X into the extended reals. Suppose also that 0 ∈ sqri(dom f − dom g). Then

(f + g)^{∗} = f^{∗}□ g^{∗}

and, moreover the inf-convolution is exact, that is: for each x^{∗} ∈ X^{∗} we have (f^{∗}□ g^{∗})(x^{∗}) = f^{∗}(x^{∗} − y^{∗}) + g^{∗}(y^{∗}) for some y^{∗} ∈ X^{∗}.

The proof of this theorem involved the following statement, which will be of importance in the development of our results. We state this as a lemma and for completeness we include the proof, excerpted from the indicated reference.

Lemma 2.12. Assume that f, g are proper closed convex functions from a Banach space X into the extended reals. Suppose also that 0 ∈ core(dom f − dom g). Then

H(K, r) := {(x_{1}^{∗}, x_{2}^{∗}) ∈ X^{∗} × X^{∗} | f^{∗}(x_{1}^{∗}) + g^{∗}(x_{2}^{∗}) ≤ K and ∥x_{1}^{∗} + x_{2}^{∗}∥ ≤ r}

is norm-bounded in X^{∗} × X^{∗} for any r > 0 and K > inf_{x^{∗}∈X^{∗}} (f^{∗}□ g^{∗})(x^{∗}).

Proof. In view of the Uniform Boundedness Principle, it suffices to check that for each element (x, y) of X × X there is a constant C(x, y) such that for all (x^{∗}, y^{∗}) ∈ H(K, r), we have ⟨x^{∗}, x⟩ + ⟨y^{∗}, y⟩ ≤ C(x, y). Indeed, x − y = λ(u − v) for some λ > 0, u ∈ dom f and v ∈ dom g, and then

⟨x^{∗}, x⟩ + ⟨y^{∗}, y⟩ = λ⟨x^{∗}, u⟩ + λ⟨y^{∗}, v⟩ + ⟨x^{∗} + y^{∗}, y − λv⟩
≤ λ(f^{∗}(x^{∗}) + f(u) + g^{∗}(y^{∗}) + g(v)) + ∥x^{∗} + y^{∗}∥∥y − λv∥
≤ λ(K + f(u) + g(v)) + r∥y − λv∥ := C(x, y).

Lower semi-continuity of the epi-graphical multi-function v ↦ epi_{s}(f_{v}□ g_{v}) may be deduced from that of its components using the following lemma. We include its derivation due to a lack of a solid reference.

Lemma 2.13. If F_{1}(·) and F_{2}(·) are multi-functions l.s.c. at x_{0} then F (x) = F_{1}(x)+F_{2}(x) is l.s.c. at x_{0}.

Proof. We use the classical formulation of l.s.c. (Remark 2.2). Let N be a neighborhood of a point y_{0} ∈ F (x_{0}). There exist y_{i} ∈ F_{i}(x_{0}) such that y_{0} = y_{1} + y_{2}. Taking two neighborhoods N_{i} of y_{i} such that N_{1} + N_{2} ⊆ N, there then exist neighborhoods V_{i} of x_{0} for which F_{i}(x) ∩ N_{i} ≠ ∅ for all x ∈ V_{i}. Hence for all x ∈ V_{1} ∩ V_{2},

∅ ≠ F_{1}(x) ∩ N_{1} + F_{2}(x) ∩ N_{2} ⊆ (F_{1}(x) + F_{2}(x)) ∩ (N_{1} + N_{2}) ⊆ F (x) ∩ N

giving the result.

In [24, Lemma 4.1], it is shown that the epigraphs of the closures \overline{f_{v}□ g_{v}} (in the strong topology on X) satisfy

lim inf_{v→w} epi (\overline{f_{v}□ g_{v}}) ⊇ \overline{epi} (f_{w}□ g_{w})

under the condition that dom f_{v}^{∗} ∩ dom g_{v}^{∗} ≠ ∅ for v in a neighborhood of w. This result is of limited use here as we are interested in this occurring in the dual space.

3. Convergences Used

Here we summarise the variational limit notions used in this paper. A reader conversant with variational convergences need only consult this section for definitions and notation. Let X and W be topological spaces, then for x ∈ X, w ∈ W, and {f_{v}}_{v∈W} a collection of \overline{ℝ}-valued functions, define the lower and upper epi-limits by:

(e-li_{v→w}f_{v})(x) := sup_{U∈N (x)} sup_{V ∈N (w)} inf_{v∈V} inf_{y∈U} f_{v}(y) ,
(e-ls_{v→w}f_{v})(x) := sup_{U∈N (x)} inf_{V ∈N (w)} sup_{v∈V} inf_{y∈U} f_{v}(y) .

It is well known [21] that these limits correspond to the Kuratowski limit of the epi-graph multifunction in the sense that

epi(e-ls_{v→w}f_{v}) = lim inf_{v→w} epi f_{v} and
epi(e-li_{v→w}f_{v}) = lim sup_{v→w} epi f_{v}.  (3.1)

Also, for a sequence {f_{n}}_{n=1}^{∞} of functions, on a first-countable space X, we have [2] for each x ∈ X

(e-li_{n→∞}f_{n})(x) = min_{{x_{n}}_{n=1}^{∞}→x} lim inf_{n→∞} f_{n}(x_{n}) and
(e-ls_{n→∞}f_{n})(x) = min_{{x_{n}}_{n=1}^{∞}→x} lim sup_{n→∞} f_{n}(x_{n}) .  (3.2)

Definition 3.1. Let {f_{v}}_{v∈W} be a family of functions. We say that {f_{v}}_{v∈W} is epi-u.s.c. (resp. l.s.c.) at w ∈ W if for all x ∈ X we have

(e-ls_{v→w}f_{v})(x) ≤ f_{w}(x), (resp. f_{w}(x) ≤ (e-li_{v→w}f_{v})(x)).

Equivalently for an epi-u.s.c. family the epi-graphs of f_{v} are lower Kuratowski-convergent to epi f_{w}.

Definition 3.2. A family of functions \{f_{v}\}_{v∈W} in \overline{ℝ}^{X} is epi-convergent to a function f_{w} (as v → w) if it is both epi-u.s.c. and epi-l.s.c. at w.

Since e-li_{v→w}f_{v} ≤ e-ls_{v→w}f_{v} on X, the relation defining epi-convergence is in fact an equality. Now consider the dual space and what would correspond to one half of dual slice convergence.

Definition 3.3. Let {f_{v}}_{v∈W} be a family of functions on a normed space X, and {f_{v}^{∗}}_{v∈W} the family of conjugate functions on X^{∗}. We define a bounded-weak^{∗} upper epi-limit (as v → w) of {f_{v}^{∗}}_{v∈W} by

bw^{∗}- lim sup_{v→w} epi f_{v}^{∗} := {(x^{∗}, α) ∈ X^{∗} × ℝ | ∃ nets v_{β} → w; (y∗_{β}, α_{β}) ∈ epi f_{v_{β}}^{∗}
such that α_{β} → α; y∗_{β} norm bounded; y∗_{β} →^{w^{∗}} x^{∗}}.

It is an elementary exercise to see that this set recedes to +∞ in the vertical direction and hence resembles the epigraph of some function. This prompts us to define

Definition 3.4. For x^{∗} ∈ X^{∗},

(bw^{∗}-e-li_{v→w}f_{v}^{∗})(x^{∗}) := inf{α ∈ ℝ | (x^{∗}, α) ∈ bw^{∗}- lim sup_{v→w} epi f_{v}^{∗}}.  (3.3)

It then follows that

epi_{s} (bw^{∗}-e-li_{v→w}f_{v}^{∗}) ⊆ bw^{∗}- lim sup_{v→w} epi f_{v}^{∗} ⊆ epi (bw^{∗}-e-li_{v→w}f_{v}^{∗}) .  (3.4)

Thus bw^{∗}-e-li_{v→w}f_{v}^{∗} is essentially a variational limit in the sense of [6] or [21]. The next lemma is a modification for the bounded-weak^{∗} topology of a result in [6, page 290].

Lemma 3.5. Let X be normed, W topological, and {f_{v}}_{v∈W} be a family of closed proper convex extended-real-valued functions on X. Then

e-ls_{v→w}f_{v} ≥ (bw^{∗}-e-li_{v→w}f_{v}^{∗})^{∗}.

Proof. Write U := bw^{∗}-e-li_{v→w}f_{v}^{∗}; V := e-ls_{v→w}f_{v}. We show that U^{∗} ≤ V on X. Let x ∈ X. If V (x) = +∞ or U ≡ +∞, (so U^{∗} ≡ −∞) there is nothing to prove. We then give a proof in the case where V (x) < +∞ and U not identically +∞ on X^{∗}. Let x^{∗} ∈ X^{∗} with U(x^{∗}) < +∞. Let α, β ∈ ℝ with α > V (x) , β > U(x^{∗}). Then (x, α) ∈ epi V, (x^{∗}, β) ∈ epi_{s} U ⊆ bw^{∗}– lim sup_{v→w} epi f_{v}^{∗}, so there are nets v_{γ} → w, x_{γ}^{∗} → x^{∗} (weak^{∗}), β_{γ} → β, with (x_{γ}^{∗}, β_{γ}) ∈ epi f_{v_{γ}}^{∗} for each γ and the x_{γ}^{∗} uniformly norm-bounded. Also, (x, α) ∈ lim inf_{v→w} epi f_{v}, and so there exists (x_{γ}, α_{γ})(∈ epi f_{v_{γ}}) → (x, α). For each γ, α_{γ} + β_{γ} ≥ f_{v_{γ}}^{∗}(x_{γ}^{∗}) + f_{v_{γ}}(x_{γ}) ≥ ⟨x_{γ}, x_{γ}^{∗}⟩ by the Fenchel inequality. As a result, passing to the limit, α + β ≥ ⟨x, x^{∗}⟩. (This is permissible as the x_{γ}^{∗} are norm-bounded and x_{γ} → x strongly). Rearrange to obtain α ≥ ⟨x, x^{∗}⟩ − β. Since the α > V (x) and β > U(x^{∗}) are arbitrary, we conclude that V (x) ≥ ⟨x, x^{∗}⟩ − U(x^{∗}), and since x^{∗} ∈ dom U is also arbitrary, it follows that V (x) ≥ U^{∗}(x) as claimed.

The next result provides bounds which will be used repeatedly in subsequent proofs.

Lemma 3.6. Let {f_{v}}_{v∈W} be a family of proper closed convex extended-real-valued functions on a normed space X. Suppose that {f_{v}}_{v∈W} is strongly epi-u.s.c. with respect to f_{w}. Then for each M > 0,

(∃V′ ∈ N (w))(∃μ ∈ ℝ)(∀v ∈ V′)(∀∥x^{∗}∥ ≤ M)(f_{v}^{∗}(x^{∗}) ≥ μ).  (3.5)

Proof. Let M > 0 and suppose the assertion false. Then there are nets v_{β} → w, ∥x_{β}^{∗}∥ ≤ M such that lim_{β} f_{v_{β}}^{∗}(x_{β}^{∗}) = −∞. Taking a weakly^{∗} convergent subnet x_{β_{γ}}^{∗} → x^{∗}, it follows for any real λ, that (x^{∗}, λ) ∈ bw^{∗}- lim sup_{v→w} epi f_{v}^{∗} ⊆ epi(bw^{∗}-e-li_{v→w}f_{v}^{∗}), so (bw^{∗}-e-li_{v→w}f_{v}^{∗})(x^{∗}) = −∞, which in turn implies via Lemma 3.5, that f_{w} ≥ e-ls_{v→w}f_{v} ≥ (bw^{∗}-e-li_{v→w}f_{v}^{∗})^{∗} ≡ +∞, contradicting the properness of f_{w}.

Definition 3.7. A family \{f_{v}\}_{v∈W} of \overline{ℝ}-valued functions on a normed space X converges as v → w in the epi-distance (or Attouch-Wets) topology if for each K ≥ 0 and ϵ > 0 we have a neighborhood V of w such that for all v ∈ V

epi f_{w} ∩ B(0, K) ⊆ epi f_{v} + B(0, ϵ)  (3.6)

and

epi f_{v} ∩ B(0, K) ⊆ epi f_{w} + B(0, ϵ) .  (3.7)

where the balls are in the box-norm on X × ℝ.

We can define Attouch-Wets convergence of sets in C(X) in an analogous manner. It is known to be equivalent to the convergence of the corresponding indicator functions. It is an elementary exercise to show that the epi-distance convergence f_{v} → f_{w} implies the (norm-)epi-convergence of f_{v} to \overline{f_{w}}, where the latter denotes the norm-l.s.c. hull of f_{w}.

Let X be a normed linear space with continuous dual X^{∗} and let Γ(X) (resp. Γ^{∗}(X^{∗})) denote the set of proper, lower-semicontinuous (resp. weak^{∗} lower semicontinuous) convex functions on X (resp. X^{∗}). Then it is well known [22] that the Young-Fenchel transform f ↦ f^{∗} is an order-reversing bijection of Γ(X) onto Γ^{∗}(X^{∗}). In [9] it was shown that the Young-Fenchel transform is a homeomorphism from (Γ(X), τ_{d}) to (Γ^{∗}(X^{∗}), τ_{d}) where τ_{d} is the epi-distance topology.

4. Epi-distance Convergence of Sums

Here we prove sufficient conditions for epi-distance convergence of a sum of parametrized convex functions in a general Banach space. Beer [12] has recently deduced epi-distance convergence for a sum of functions in a general normed space from the epi-distance convergence of the constituent functions, under the condition (int dom f) ∩ dom g ≠ ∅. In [8] a condition which is equivalent to 0 ∈ int (dom f − dom g) is used to deduce a similar result, in Banach spaces (although metric estimates are obtained as well, under a more technical assumption). The result proved here in Banach spaces use a strong quasi relative interiority condition more in the spirit of that found in duality theory.

Lemma 4.1. Suppose that W is topological and that {f_{v}}_{v∈W} and {g_{v}}_{v∈W} are families of proper closed convex extended-real-valued functions on a Banach space X, which are epi-distance convergent (as v → w) to f_{w} and g_{w} respectively.

(i) If 0 ∈ core(dom f_{w} − dom g_{w}), then there exists V ∈ N (w) such that for v ∈ V we have 0 ∈ core(dom f_{v} − dom g_{v}). In particular if ρ > inf{∥(x, α)∥ | (x, α) ∈ epi f_{w} ∩ epi g_{w}} we have a fixed δ > 0 such that for all v ∈ V,

\overline{B}(0, δ) ⊆ \{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) .

(ii) Suppose 0 ∈ sqri(dom f_{w} − dom g_{w}). Place X_{v} = \overline{span}(dom f_{v} − dom g_{v}) for v ∈ W. Then for each ϵ > 0, ρ > 0 there is some V′ ∈ N (w) for which

\overline{B}(0, ρ) ∩ X_{w} ⊆ X_{v} + \overline{B}(0, ϵ) for v ∈ V′.

If also for each ρ > 0 (sufficiently large) there exist V′′ ∈ N (w), η > 0 such that for all v ∈ V′′, we have

\overline{B}(0, η) ∩ X_{v} ⊆ \overline{\{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ)}  (4.1)

then {X_{v}}_{v∈W} epi-distance converges to X_{w}.

Proof. (i) By Proposition 2.6 we have 0 ∈ core(dom f_{w} − dom g_{w}) if and only if 0 ∈ int(dom f_{w} − dom g_{w}). By Corollary 2.7 it follows that

0 ∈ int (\{f_{w} ≤ ρ\} ∩ \overline{B}(0, ρ − δ) − \{g_{w} ≤ ρ\} ∩ \overline{B}(0, ρ − δ))

for δ > 0 sufficiently small that ρ−δ > inf{∥(x, α)∥ | (x, α) ∈ epi f_{w} ∩epi g_{w}}. Take δ > 0 sufficiently small so that

\overline{B}(0, 3δ) ⊆ \{f_{w} ≤ ρ\} ∩ \overline{B}(0, ρ − δ) − \{g_{w} ≤ ρ\} ∩ \overline{B}(0, ρ − δ).  (4.2)

By proposition 7.1.7 of Beer [14] we have the epi-distance convergence of the level-sets of both {f_{v}}_{v∈W} and {g_{v}}_{v∈W} and so for v ∈ V ∈ N (w) we have both

\{f_{w} ≤ ρ\} ∩ \overline{B}(0, ρ − δ) ⊆ \{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) + B̄(0, δ) and
\{g_{w} ≤ ρ\} ∩ \overline{B}(0, ρ − δ) ⊆ \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) + B̄(0, δ).  (4.3)

Combining this with (4.2) we obtain

\overline{B}(0, δ) + \overline{B}(0, 2δ) ⊆ \{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) + \overline{B}(0, 2δ),
which implies \overline{B}(0, δ) ⊆ \overline{\{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ)}

by the Rådström cancellation lemma [14]. The argument is completed by noting that the interior of \{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) coincides with that of its closure by ideal convexity (Proposition 2.5).

(ii) Since 0 ∈ core_{X_{w}}(dom f_{w} − dom g_{w}), then for ρ > 0 as defined above, an argument analogous to that in the proof of Proposition 2.6 gives that X_{w} = cone({f_{w} ≤ ρ} ∩ B(0, ρ) − {g_{w} ≤ ρ} ∩ B(0, ρ)), which implies that for some η > 0,

\overline{B}(0, η) ∩ X_{w} ⊆ \overline{\{f_{w} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{w} ≤ ρ\} ∩ \overline{B}(0, ρ)}.  (4.4)

Put K = \frac{ρ}{η} and let ϵ > 0 be arbitrary. In general we have \overline{A} ⊆ A + B(0, δ) so apply (4.3) with δ = \frac{ϵ}{3K} to get for v ∈ V′

\overline{B}(0, η) ∩ X_{w} ⊆ \overline{\{f_{v} ≤ ρ\} − \{g_{v} ≤ ρ\} + \overline{B}(0, 2δ)}
⊆ \overline{\{f_{v} ≤ ρ\} − \{g_{v} ≤ ρ\}} + \overline{B}(0, 3δ)
⊆ \overline{span (\{f_{v} ≤ ρ\} − \{g_{v} ≤ ρ\})} + \overline{B}(0, 3δ)
implying \overline{B}(0, Kη) ∩ X_{w} = K (\overline{B}(0, η) ∩ X_{w}) ⊆ K (X_{v} + \overline{B}(0, 3δ))
= X_{v} + \overline{B}(0, 3Kδ)
or \overline{B}(0, ρ) ∩ X_{w} ⊆ X_{v} + \overline{B}(0, ε) for v ∈ V′ ∈ N (w).

Using the symmetry in the definition of epi-distance convergence we have for v ∈ V′′ that \overline{B}(0, ρ) ∩ X_{v} ⊆ X_{w} + \overline{B}(0, ϵ) by a parallel argument, which utilized the uniform value of η along V′′ in (4.1) to replace (4.4). Combining both for V = V′′ ∩ V′ we get the result.

Note that in the case of 0 ∈ core (dom f_{w} − dom g_{w}) then it follows that X_{v} = X for all v ∈ V. Thus we have a trivial kind of convergence of the spaces generated in this case.

Lemma 4.2. Suppose that {f_{v}}_{v∈W} and {g_{v}}_{v∈W} are families of closed proper convex functions which epi-distance converge to f_{w} and g_{w}, respectively (as v → w). Assume that

0 ∈ sqri (dom f_{w} − dom g_{w}) .

Place

X_{v} = \overline{span} (dom f_{v} − dom g_{v})

for v ∈ W, and suppose additionally that the algebraic complement Y_{w} of X_{w} is closed, with X_{v} ∩ Y_{w} = {0} for all v near w. Then X_{v} epi-distance converges to X_{w} as v → w. Further, there is V ∈ N (w) and δ > 0 such that for all v ∈ V,

B(0, δ) ⊆ X_{v} ∩ \overline{B}(0, 1) + Y_{w} and 0 ∈ sqri(dom f_{v} − dom g_{v}).

Proof. We show that for each ρ > 0 (sufficiently large) there exist V ∈ N (w) and η > 0 such that (4.1) holds for all v ∈ V. The convergence X_{v} → X_{w} will then follow from Lemma 4.1(ii). Place

A_{v} = \overline{\{f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ)}.

where ρ > inf{∥(x, α)∥ = max{∥x∥, |α|} | (x, α) ∈ epi f_{w} ∩ epi g_{w}}. By an argument analogous to that in the proof of Corollary 2.7, we have A_{w} generating the subspace X_{w} (i.e. X_{w} = cone A_{w}) and hence 0 ∈ core_{X_{w}} A_{w} = int_{X_{w}} A_{w}, since A_{w} is closed in the Banach space X_{w}.

By the epi-distance convergence of level-sets [14, Proposition 7.1.7], and by [14, Theorem 7.4.5], both B_{v} := \{f_{v} ≤ ρ\}∩\overline{B}(0, ρ) and C_{v} := −\{g_{v} ≤ ρ\}∩\overline{B}(0, ρ) epi-distance converge to B_{w} and C_{w} as v → w and therefore, by two applications of [14, exercise 7.4.7], follow in turn the convergence of A_{v} = \overline{B_{v} + C_{v}} to A_{w} = \overline{B_{w} + C_{w}} and hence that of \overline{A_{v} + Y_{w}} to \overline{A_{w} + Y_{w}}.

Additionally, 0 ∈ int(A_{w} +Y_{w}). Indeed, there is μ > 0 such that \overline{B}(0, μ)∩X_{w} ⊆ A_{w}. Since the map from X_{w} × Y_{w} to X taking (x, y) to x + y is a continuous linear surjection, the Open Mapping Theorem provides δ′ > 0 such that \overline{B}(0, δ′) ⊆ X_{w}∩B(0, μ)+Y_{w} ⊂ A_{w}+Y_{w}.

Combining the latter interiority with the convergence \overline{A_{v} + Y_{w}} → \overline{A_{w} + Y_{w}}, the Rådström cancellation lemma [14, Chapter 7] yields, for any λ satisfying 0 < λ < δ′, a neighborhood V of w with \overline{B}(0, λ) ⊆ \overline{A_{v} + Y_{w}} for all v ∈ V, and so, from the ideal convexity of A_{v} + Y_{w} and Proposition 2.5, we have B(0, λ) ⊆ A_{v} + Y_{w}. (Note this implies that X_{v} + Y_{w} = X). Hence, choosing V sufficiently small that also X_{v} ∩ Y_{w} = {0}, we obtain for such v,

B(0, λ) ∩ X_{v} ⊆ (A_{v} + Y_{w}) ∩ X_{v} ⊆ A_{v} + X_{v} ∩ Y_{w} = A_{v}.

Taking 0 < η < λ now yields (4.1), as required. Thus follows the epi-distance convergence X_{v} → X_{w}. Also, from B(0, λ) ⊆ A_{v} + Y_{w} ⊆ X_{v} ∩ \overline{B}(0, 2ρ) + Y_{w} we obtain, on taking δ := λ/2ρ, that B(0, δ) ⊆ X_{v} ∩ \overline{B}(0, 1) + Y_{w}, which is the second assertion of the lemma. Furthermore, since \{f_{v} ≤ ρ\}∩\overline{B}(0, ρ)−\{g_{v} ≤ ρ\}∩\overline{B}(0, ρ) is by Proposition 2.5 an ideally convex subset of X, it is also likewise as a subset of the Banach space X_{v}, so its interior in X_{v} is the same as that of its (X_{v}-)closure A_{v} and we obtain that X_{v} = cone({f_{v} ≤ ρ\} ∩ \overline{B}(0, ρ) − \{g_{v} ≤ ρ\} ∩ \overline{B}(0, ρ)) so 0 ∈ sqri(dom f_{v} − dom g_{v}).

Recall from Lemma 4.2 that for some neighborhood V of w we have for all v ∈ V that X_{v} ∩ \overline{B}(0, 1) + Y_{w} contains a δ-ball at 0. If for such v we also have X_{v} ∩ Y_{w} = \{0\}, then at once X = X_{v} ⊕ Y_{w}. Thus we can form natural extensions of functionals on X_{v} to X by the following scheme:

Definition 4.3. For v ∈ W, x^{∗} ∈ X_{v}^{∗}, define an extension x̃^{∗} ∈ X^{∗} by demanding that it vanish identically on Y_{w}, and extending by linearity.

Remark 4.4. For all v ∈ V, where V is small enough that the assumptions and conclusions of Lemma 4.2 are satisfied thereon,

∥x̃^{∗}∥ = \frac{1}{δ} sup_{x∈B(0,δ)}⟨x̃^{∗}, x⟩ ≤ \frac{1}{δ} sup_{x∈\overline{B}(0,1)∩X_{v}+Y_{w}}⟨x̃^{∗}, x⟩ = \frac{1}{δ} sup_{x∈\overline{B}(0,1)∩X_{v}}⟨x^{∗}, x⟩ = \frac{1}{δ}∥x^{∗}∥_{X_{v}^{∗}}.

Note that the bound \frac{1}{δ} is uniform in v ∈ V. We may assume δ ≤ 1.

Lemma 4.5. Suppose the family {X_{v}}_{v∈W} of closed subspaces of X satisfy the following:

(i) lim sup_{v→w} X_{v} ⊆ X_{w}; (ii) X_{w} has finite codimension.

Then X_{v} ∩ Y_{w} = {0} for all v in a neighborhood of w, where Y_{w} denotes the algebraic complement of X_{w}.

Proof. Suppose the result false. Then there exist nets v_{α} → w and x_{α} ≠ 0 in X_{v_{α}} ∩ Y_{w}. Normalize so that x_{α} has unit norm for all α. By the compactness of the unit ball in Y_{w}, extracting a convergent subnet if necessary, we may assume that x_{α} has a limit x ∈ Y_{w} ∩ \overline{B}(0, 1). Then x ≠ 0, for if otherwise, x_{α} → 0, contradicting the normalization. Also, x ∈ lim sup_{v→w} X_{v} ⊆ X_{w} and we have 0 ≠ x ∈ X_{w} ∩ Y_{w}, a contradiction.

Remark 4.6. In the context of Lemma 4.2, it should be noted that the condition that X_{v} ∩Y_{w} = {0} locally in v, is in fact equivalent to the epi-distance convergence X_{v} → X_{w}, as is seen from the following lemma.

Lemma 4.7. Let X_{v} be closed subspaces of a Banach space X, with X_{v} epi-distance converging to X_{w}. Let X_{w} have closed algebraic complement Y. Then X_{v} ∩ Y = {0} for all v near w.

Proof. Since X_{w} − Y = X, [8, Corollary 2.9] implies the epi-distance convergence X_{v} ∩ Y → X_{w} ∩ Y = {0}. Hence, if 0 < ϵ < 1, then for all v in a neighborhood of w, the unit ball in the space X_{v} ∩ Y is contained in the ϵ-ball at the origin in X. This forces X_{v} ∩ Y = {0}.

We now introduce the following notation:

H_{K}(X^{∗}, v) := {(x_{1}^{∗}, x_{2}^{∗}) ∈ X^{∗} × X^{∗} | f_{v}^{∗}(x_{1}^{∗}) + g_{v}^{∗}(x_{2}^{∗}) < K , ∥x_{1}^{∗} + x_{2}^{∗}∥ < K},
H_{K}(X_{v}^{∗}, v) := {(x_{1}^{∗}, x_{2}^{∗}) ∈ X_{v}^{∗} × X_{v}^{∗} | (f_{v}|_{X_{v}})^{∗}(x_{1}^{∗}) + (g_{v}|_{X_{v}})^{∗}(x_{2}^{∗}) < K ,
∥x_{1}^{∗} + x_{2}^{∗}∥_{X_{v}^{∗}} < K} ,

where in the latter expression, the conjugation operation is with respect to the subspace X_{v}.

Also, place

H̃_{K}(X_{v}^{∗}, v) := {(x̃_{1}^{∗}, x̃_{2}^{∗}) | (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{K}(X_{v}^{∗}, v)} ⊆ X^{∗} × X^{∗},

where again the tilde denotes the extension operation of Definition 4.3.

Lemma 4.8. Assume that W is topological and that {f_{v}}_{v∈W} and {g_{v}}_{v∈W} are families of proper closed convex extended-real-valued functions on a Banach space X, which are epidistance convergent (as v → w) to f_{w} and g_{w} respectively. Assume that 0 ∈ sqri(dom f_{w} − dom g_{w}) and that for all v in a neighborhood of w, we have the following:

(i) at least one of dom f_{v}, dom g_{v} intersects the subspace X_{v} := \overline{span}(dom f_{v} − dom g_{v}); (ii) The algebraic complement Y_{w} of X_{w} is closed, and satisfies X_{v} ∩ Y_{w} = {0}.

Then for each sufficiently large K > 0, there exist V ∈ N (w) and γ > 0 such that for all v ∈ V,

H̃_{K}(X_{v}^{∗}, v) ⊆ B^{∗}(0, γ) .

Proof. Since X_{v} is a subspace for each v, it follows from (i) that both dom f_{v} and dom g_{v} are contained in X_{v}. As f_{w}^{∗}□g_{w}^{∗} = (f_{w} + g_{w})^{∗} ∈ Γ^{∗}(X^{∗}) by Proposition 2.11, and is therefore proper, there is K_{0} such that B^{∗}(0, K_{0}) ∩ epi(f_{w}^{∗}□g_{w}^{∗}) ≠ ∅. From the epidistance convergence (see [9]) f_{v}^{∗} → f_{w}^{∗} and g_{v}^{∗} → g_{w}^{∗}, follows the lower-semicontinuity (at w) of v ↦ epi f_{v}^{∗} and v ↦ epi g_{v}^{∗} (with respect to the norm on X^{∗}), so by Lemma 2.13, and the exactness of the inf-convolution f_{w}^{∗}□g_{w}^{∗} (Proposition 2.11),

epi(f_{w}^{∗}□g_{w}^{∗}) = epi f_{w}^{∗} + epi g_{w}^{∗} ⊆ lim inf_{v→w} (epi f_{v}^{∗} + epi g_{v}^{∗}) ⊆ lim inf_{v→w} epi(f_{v}^{∗}□g_{v}^{∗}) .

Hence the map v ↦ epi f_{v}^{∗}□g_{v}^{∗} is (norm-)lower-semicontinuous at w, so B^{∗}(0, K_{0}) ∩ epi f_{v}^{∗}□g_{v}^{∗} ≠ ∅ for all v in some neighborhood V_{0} of w. Let K ≥ K_{0}. Since 0 ∈ core_{X_{w}}(dom f_{w}|_{X_{w}} − dom g_{w}|_{X_{w}}) from the strong quasi relative interiority, we have by Lemma 2.12 that H_{K}(X_{w}^{∗}, w) is norm-bounded in X_{w}^{∗} × X_{w}^{∗}. (Note that H_{K}(X_{v}^{∗}, v) is nonempty for each v ∈ V_{0}, for if (x^{∗}, α) ∈ epi(f_{v}^{∗}□g_{v}^{∗}) ∩ B^{∗}(0, K) ≠ ∅, then from the definition of the inf-convolution there is y^{∗} ∈ X^{∗} such that f_{v}^{∗}(y^{∗}) + g_{v}^{∗}(x^{∗} − y^{∗}) < K and so (y^{∗}, x^{∗} −y^{∗}) ∈ H_{K}(X^{∗}, v). Taking restrictions to X_{v} then yields elements of H_{K}(X_{v}^{∗}, v), since X_{v} contains both dom f_{v} and dom g_{v}). For K ≥ K_{0}, define γ̂(K) ∈ ℝ by

γ̂(K) = sup{∥x_{i}^{∗}∥_{X_{w}^{∗}} | i = 1, 2; (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{K}(X_{w}^{∗}, w)}.  (4.5)

Place

γ := \frac{3}{δ}(γ̂(\frac{K}{δ} + 1) + 1),  (4.6)

where 0 < δ ≤ 1 is provided by Lemma 4.2 (see also Definition 4.3 and Remark 4.4).

Let M > max{γ, 1} be large enough that H_{K}(X^{∗}, w) ∩ B^{∗}(0, Mδ) ≠ ∅. By appropriate shrinkage of V_{0} we may assume, from the epi-distance convergence of X_{v} to X_{w} (Lemma 4.2) that for all v in V_{0} we have additionally:

X_{v} ∩ \overline{B}(0, 1) ⊆ X_{w} + B(0, \frac{1}{2M}), and X_{w} ∩ \overline{B}(0, 1) ⊆ X_{v} + B(0, \frac{1}{2M}) ,  (4.7)

as well as (see Lemma 3.6) for some μ ∈ ℝ,

(∀v ∈ V_{0})(∀∥x^{∗}∥ ≤ M)(f_{v}^{∗}(x^{∗}) ≥ μ and g_{v}^{∗}(x^{∗}) ≥ μ) .  (4.8)

Let β > max\{|μ|, |\frac{K}{δ} − μ|, M\}, v ∈ V_{0}, and (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{\frac{K}{δ}} (X^{∗}, v) ∩ B^{∗}(0, M). Then

f_{v}^{∗}(x_{1}^{∗})+g_{v}^{∗}(x_{2}^{∗}) < \frac{K}{δ} , ∥(x_{1}^{∗}, x_{2}^{∗})∥ < M, so from (4.8), have that μ ≤ g_{v}^{∗}(x_{2}^{∗}) < \frac{K}{δ} −f_{v}^{∗}(x_{1}^{∗}) ≤ \frac{K}{δ} − μ so |g_{v}^{∗}(x_{2}^{∗})| < β and hence

(x_{2}^{∗}, g_{v}^{∗}(x_{2}^{∗})) ∈ epi g_{v}^{∗} ∩ B^{∗}(0, β).

Similarly,

(x_{1}^{∗}, f_{v}^{∗}(x_{1}^{∗})) ∈ epi f_{v}^{∗} ∩ B^{∗}(0, β).

By the epi-distance convergence f_{v}^{∗} → f_{w}^{∗}, g_{v}^{∗} → g_{w}^{∗}, there is a neighborhood V′ of w such that for all v ∈ V′,

epi g_{v}^{∗} ∩ B^{∗}(0, β) ⊆ epi g_{w}^{∗} + B^{∗}(0, 1/3),

and similarly for the f_{v}^{∗}, so for v ∈ V_{0} ∩ V′, and (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{\frac{K}{δ}} (X^{∗}, v) ∩ B^{∗}(0, M), we obtain that (x_{1}^{∗}, f_{v}^{∗}(x_{1}^{∗})) ∈ epi f_{w}^{∗} +B^{∗}(0, 1/3) and (x_{2}^{∗}, g_{v}^{∗}(x_{2}^{∗})) ∈ epi g_{w}^{∗} +B^{∗}(0, 1/3). There then exist x̄_{1}^{∗}, x̄_{2}^{∗}, β_{1}, β_{2} with: ∥x̄_{i}^{∗} − x_{i}^{∗}∥ < 1/3 (i = 1, 2); β_{1} ≥ f_{w}^{∗}(x̄_{1}^{∗}); β_{2} ≥ g_{w}^{∗}(x̄_{2}^{∗}), with also

|β_{1} − f_{v}^{∗}(x_{1}^{∗})| < 1/3; |β_{2} − g_{v}^{∗}(x_{2}^{∗})| < 1/3,

so that

f_{w}^{∗}(x̄_{1}^{∗}) + g_{w}^{∗}(x̄_{2}^{∗}) ≤ β_{1} + β_{2} < 1/2 + f_{v}^{∗}(x_{1}^{∗}) + 1/2 + g_{v}^{∗}(x_{2}^{∗}) < \frac{K}{δ} + 1.

Also, ∥x̄_{1}^{∗} +x̄_{2}^{∗}∥ ≤ ∥x_{1}^{∗} +x_{2}^{∗}∥+1 < \frac{K}{δ} +1 and we have (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{\frac{K}{δ} +1}(X^{∗}, w)+B^{∗}(0, 1/3). It has now been proved that there exists V′ ∈ N (w) such that for all v ∈ V′

H_{\frac{K}{δ}} (X^{∗}, v) ∩ B^{∗}(0, M) ⊆ H_{\frac{K}{δ} +1}(X^{∗}, w) + B^{∗}(0, 1/3).  (4.9)

By another contraction of V_{0}, if required, we can ensure that (4.9) is also satisfied on V_{0}, as well as all the conclusions of Lemma 4.2.

Let v ∈ V_{0} and (y_{1}^{∗}, y_{2}^{∗}) ∈ H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M). Then by Remark 4.4, ∥y_{1}^{∗} + y_{2}^{∗}∥ ≤ \frac{1}{δ}∥y_{1}^{∗}|_{X_{v}} + y_{2}^{∗}|_{X_{v}}∥_{X_{v}^{∗}} < K/δ, so by (4.9),

(y_{1}^{∗}, y_{2}^{∗}) ∈ H_{\frac{K}{δ}} (X^{∗}, v) ∩ B^{∗}(0, M) ⊆ H_{\frac{K}{δ} +1}(X^{∗}, w) + B^{∗}(0, 1/3).

Taking restrictions to X_{w}, we obtain

∥(y_{1}^{∗}|_{X_{w}}, y_{2}^{∗}|_{X_{w}})∥_{X_{w}^{∗}} < γ̂(\frac{K}{δ} + 1) + 1/3 ,

since (y_{1}^{∗}|_{X_{w}}, y_{2}^{∗}|_{X_{w}}) is within distance 1/3 of H_{\frac{K}{δ} +1}(X_{w}^{∗}, w) in X_{w}^{∗} × X_{w}^{∗}, and from (4.5)

the latter set is norm-bounded by γ̂(\frac{K}{δ} + 1).

For each v ∈ V_{0}, there are x̄_{1}, x̄_{2} in B(0, 1) ∩ X_{v} such that ∥(y_{1}^{∗}, y_{2}^{∗})|_{X_{v}}∥_{X_{v}^{∗}} ≤ ⟨y_{1}^{∗}, x̄_{1}⟩ + ⟨y_{2}^{∗}, x̄_{2}⟩ + 1, and from (4.7) there are ȳ_{1}, ȳ_{2} in X_{w} within \frac{1}{2M} of x̄_{1}, x̄_{2} respectively, so that

|⟨(y_{1}^{∗}, y_{2}^{∗}), (x̄_{1}, x̄_{2}) − (ȳ_{1}, ȳ_{2})⟩| ≤ ∥(y_{1}^{∗}, y_{2}^{∗})∥ (∥x̄_{1} − ȳ_{1}∥ + ∥x̄_{2} − ȳ_{2}∥) ≤ 1.

Thus

∥(y_{1}^{∗}, y_{2}^{∗})|_{X_{v}}∥_{X_{v}^{∗}} ≤ ⟨y_{1}^{∗}, x̄_{1}⟩ + ⟨y_{2}^{∗}, x̄_{2}⟩ + 1
≤ |⟨(y_{1}^{∗}, y_{2}^{∗}), (x̄_{1}, x̄_{2}) − (ȳ_{1}, ȳ_{2})⟩| + ⟨y_{1}^{∗}, ȳ_{1}⟩ + ⟨y_{2}^{∗}, ȳ_{2}⟩ + 1
≤ 2 + 3∥(y_{1}^{∗}|_{X_{w}}, y_{2}^{∗}|_{X_{w}})∥_{X_{w}^{∗}} since ∥ȳ_{i}∥ ≤ 1 + \frac{1}{2M} ≤ 3/2
< 2 + 3(γ̂(\frac{K}{δ} + 1) + 1/3) = 3(γ̂(\frac{K}{δ} + 1) + 1)

and since y_{i}^{∗} = { \atop y_{i}^{∗}|_{X_{v}}}, we obtain (on recalling the definition (4.6) for γ)

∥(y_{1}^{∗}, y_{2}^{∗})∥ ≤ \frac{1}{δ}∥(y_{1}^{∗}, y_{2}^{∗})|_{X_{v}}∥_{X_{v}^{∗}} < \frac{3}{δ}(γ̂(\frac{K}{δ} + 1) + 1) = γ .

Hence it has been shown for v ∈ V_{0}, and γ and M > γ as above, that

H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) ⊆ B^{∗}(0, γ) .  (4.10)

This implies that

H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) = H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, γ) ,

and further, by convexity, whenever H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) is nonempty, that

H̃_{K}(X_{v}^{∗}, v) ⊆ B^{∗}(0, M) ,

for if H̃_{K}(X_{v}^{∗}, v) contains points outside B^{∗}(0, M) then we can join these with linesegments to points of H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) = H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, γ), yielding elements of H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) of norm greater than γ, contradicting (4.10).

To complete the proof of our result, then, we need to demonstrate for all v in some neighborhood V′ of w, that H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) is indeed nonempty. Once this is done, it will follow that for all v ∈ V_{0} ∩ V′,

H̃_{K}(X_{v}^{∗}, v) ⊆ B^{∗}(0, M) ,

so from (4.10),

H̃_{K}(X_{v}^{∗}, v) ⊆ B^{∗}(0, γ) .

We now attend to this final task. From the choice of M, recall that H_{K}(X^{∗}, w) ∩ B^{∗}(0, Mδ) ≠ ∅. We first show that for every v in some neighborhood V′,

H_{K}(X^{∗}, v) ∩ B^{∗}(0, Mδ) ≠ ∅ ,  (4.11)

from which the required result will follow, on taking restrictions to X_{v}. Set N := Mδ, and take (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{K}(X^{∗}, w) ∩ B^{∗}(0, N). Since the f_{v}^{∗} strongly epi-converge to f_{w}^{∗}, as do the g_{v}^{∗} to g_{w}^{∗}, there follows

K > f_{w}^{∗}(x_{1}^{∗}) + g_{w}^{∗}(x_{2}^{∗}) = (e-ls_{v→w}f_{v}^{∗})(x_{1}^{∗}) + (e-ls_{v→w}g_{v}^{∗})(x_{2}^{∗})
= sup_{U_{1}∈N_{s}(x_{1}^{∗})} lim sup_{v→w} inf_{U_{1}} f_{v}^{∗} + sup_{U_{2}∈N_{s}(x_{2}^{∗})} lim sup_{v→w} inf_{U_{2}} g_{v}^{∗}
≥ lim sup_{v→w} inf_{U_{1}} f_{v}^{∗} + lim sup_{v→w} inf_{U_{2}} g_{v}^{∗}

for any (strong) neighborhoods U_{1}, U_{2} of x_{1}^{∗} and x_{2}^{∗} respectively. Select these neighborhoods to be so small that ∥x̄_{1}^{∗} + x̄_{2}^{∗}∥ < K and ∥(x̄_{1}^{∗}, x̄_{2}^{∗})∥ < N whenever (x̄_{1}^{∗}, x̄_{2}^{∗}) ∈ U_{1} × U_{2}. Thus, since neither lim sup_{v→w} inf_{U_{1}} f_{v}^{∗} nor lim sup_{v→w} inf_{U_{2}} g_{v}^{∗} equals +∞, there is V ∈ N (w) such that for each v ∈ V, inf_{U_{1}} f_{v}^{∗} ≠ +∞, inf_{U_{2}} g_{v}^{∗} ≠ +∞, and

lim sup_{v→w} inf_{U_{1}} f_{v}^{∗} + lim sup_{v→w} inf_{U_{2}} g_{v}^{∗} ≥ lim sup_{v(∈V)→w} [inf_{U_{1}} f_{v}^{∗} + inf_{U_{2}} g_{v}^{∗}] .

Hence there is V′ ⊆ V_{0} such that K > sup_{v∈V′} [inf_{U_{1}} f_{v}^{∗} + inf_{U_{2}} g_{v}^{∗}], that is, for each v ∈ V′, there is (x̄_{1}^{∗}, x̄_{2}^{∗}) ∈ U_{1}×U_{2} such that f_{v}^{∗}(x̄_{1}^{∗})+g_{v}^{∗}(x̄_{2}^{∗}) < K, so from the choice of U_{i}, we have (x̄_{1}^{∗}, x̄_{2}^{∗}) ∈ H_{K}(X^{∗}, v)∩B^{∗}(0, N) and so (4.11) is proved. For such (x̄_{1}^{∗}, x̄_{2}^{∗}) we immediately have (x̄_{1}^{∗}|_{X_{v}}, x̄_{2}^{∗}|_{X_{v}}) ∈ H_{K}(X_{v}^{∗}, v) and ∥(x̄_{1}^{∗}, x̄_{2}^{∗})|_{X_{v}}∥_{X_{v}^{∗}} ≤ ∥(x̄_{1}^{∗}, x̄_{2}^{∗})∥ < N, so that

({ \atop x̄_{1}^{∗}|_{X_{v}}}, { \atop x̄_{2}^{∗}|_{X_{v}}}) ∈ H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, N/δ) = H̃_{K}(X_{v}^{∗}, v) ∩ B^{∗}(0, M) ,

proving nonemptiness of the latter set.

We are now in a position to prove the main result of this paper.

Theorem 4.9. Assume that W is topological and that {f_{v}}_{v∈W} and {g_{v}}_{v∈W} are families of proper closed convex extended-real-valued functions on a Banach space X, which are epi-distance convergent (as v → w) to f_{w} and g_{w} respectively. Moreover suppose that

(i) 0 ∈ sqri(dom f_{w} − dom g_{w}), (ii) the algebraic complement Y_{w} of X_{w} is closed and X_{v} ∩ Y_{w} = {0} for all v near w,

where X_{v} := \overline{span}(dom f_{v} − dom g_{v}).

Then {f_{v} + g_{v}}_{v∈W} is epi-distance convergent to f_{w} + g_{w} as v → w.

Proof. We first prove the result under the additional assumption that for all v in a neighborhood of w, at least one of dom f_{v}, dom g_{v} intersects the subspace X_{v} := \overline{span}(dom f_{v} − dom g_{v}), which in fact implies that X_{v} contains both dom f_{v} and dom g_{v}. Subsequently we will discard this assumption.

By Lemma 4.2, 0 ∈ sqri(dom f_{v} − dom g_{v}) for all v in a neighborhood of w, so by Proposition 2.11, f_{v}^{∗}□g_{v}^{∗} is exact and f_{v}^{∗}□g_{v}^{∗} = (f_{v} + g_{v})^{∗} ∈ Γ^{∗}(X^{∗}). If we assume that the family of functions {f_{v}^{∗}□g_{v}^{∗}}_{v∈W} epi-distance converge (as v → w) to f_{w}^{∗}□g_{w}^{∗} then since epi-distance convergence τ_{d} renders Young-Fenchel conjugation f ↦ f^{∗} a homeomorphism from (Γ(X), τ_{d}) to (Γ^{∗}(X^{∗}), τ_{d}) (see [9]) we may argue as follows:

τ_{d}– lim_{v→w}(f_{v} + g_{v}) = τ_{d}– lim_{v→w}(f_{v}^{∗}□g_{v}^{∗})^{∗} = (τ_{d}– lim_{v→w} f_{v}^{∗}□g_{v}^{∗})^{∗}
= (f_{w}^{∗}□g_{w}^{∗})^{∗} = f_{w} + g_{w}.

The required task is then to prove the epi-distance convergence of f_{v}^{∗}□g_{v}^{∗} to f_{w}^{∗}□g_{w}^{∗}. To establish this, we first derive (3.7) (for f_{v}^{∗}□g_{v}^{∗}), then as a corollary of that argument we shall obtain (3.6) also.

As argued in the proof of Lemma 4.8, the map v ↦ f_{v}^{∗}□g_{v}^{∗} is (norm-)lower-semicontinuous at w. Let K_{0} be such that B^{∗}(0, K_{0}) ∩ epi(f_{w}^{∗}□g_{w}^{∗}) ≠ ∅. Then the semicontinuity implies that for all v in some neighborhood V_{0} of w, B^{∗}(0, K_{0}) ∩ epi f_{v}^{∗}□g_{v}^{∗} ≠ ∅. Let K > K_{0}, ϵ > 0. We now shrink V_{0} further so that additionally, the indicated hypotheses of the theorem hold thereon, along with the conclusions of Lemmas 4.8 and 4.2.

Let v ∈ V_{0} and (x^{∗}, α) ∈ epi(f_{v}^{∗}□g_{v}^{∗})∩B^{∗}(0, K). From the exactness of the inf-convolution there exists y^{∗} ∈ X^{∗} such that (f_{v}^{∗}□g_{v}^{∗})(x^{∗}) = f_{v}^{∗}(y^{∗}) + g_{v}^{∗}(x^{∗} − y^{∗}) ≤ α < K. Thus (y^{∗}, x^{∗} − y^{∗}) ∈ H_{K}(X^{∗}, v). Let x_{1}^{∗} denote the extension (in the sense of Definition 4.3) of y^{∗}|_{X_{v}} to X. Place x_{2}^{∗} := x^{∗} − x_{1}^{∗} (so x_{1}^{∗} + x_{2}^{∗} = x^{∗}). Since X_{v} contains both dom f_{v} and dom g_{v}, there follows that

f_{v}^{∗}(x_{1}^{∗}) + g_{v}^{∗}(x_{2}^{∗}) = f_{v}^{∗}(y^{∗}) + g_{v}^{∗}(x^{∗} − y^{∗}) ≤ α .

Let z^{∗} := { \atop x_{2}^{∗}|_{X_{v}}} (the extension of x_{2}^{∗}|_{X_{v}} ∈ X_{v}^{∗}). Then since

∥x_{1}^{∗}|_{X_{v}} + z^{∗}|_{X_{v}}∥_{X_{v}^{∗}} = ∥(x_{1}^{∗} + x_{2}^{∗})|_{X_{v}}∥_{X_{v}^{∗}} ≤ ∥x_{1}^{∗} + x_{2}^{∗}∥ = ∥x^{∗}∥ < K

and

(f_{v}|_{X_{v}})^{∗}(x_{1}^{∗}|_{X_{v}}) + (g_{v}|_{X_{v}})^{∗}(x_{2}^{∗}|_{X_{v}}) = f_{v}^{∗}(x_{1}^{∗}) + g_{v}^{∗}(x_{2}^{∗}) ≤ α < K ,

we have

(x_{1}^{∗}, z^{∗}) = ({ \atop y^{∗}|_{X_{v}}}, { \atop x_{2}^{∗}|_{X_{v}}}) = ({ \atop x_{1}^{∗}|_{X_{v}}}, { \atop x_{2}^{∗}|_{X_{v}}}) ∈ H̃_{K}(X_{v}^{∗}, v) .

Hence, from Lemma 4.8 follows that ∥x_{1}^{∗}∥ ≤ ∥(x_{1}^{∗}, z^{∗})∥ ≤ γ (where γ is defined in the said Lemma, and is independent of v ∈ V_{0}, but dependent on K), and so ∥x_{2}^{∗}∥ ≤ ∥x^{∗}∥+∥x_{1}^{∗}∥ < K + γ := γ̄. Thus we have

(x_{1}^{∗}, x_{2}^{∗}) ∈ H_{K}(X^{∗}, v) ∩ B^{∗}(0, γ̄) and  (4.12)
(x^{∗}, α) = (x_{1}^{∗}, α − g_{v}^{∗}(x_{2}^{∗})) + (x_{2}^{∗}, g_{v}^{∗}(x_{2}^{∗})) ∈ epi f_{v}^{∗} ∩ (B_{X^{∗}}(0, γ̄) × ℝ)
+ epi g_{v}^{∗} ∩ (B_{X^{∗}}(0, γ̄) × ℝ), (4.13)

where B_{X^{∗}}(0, γ̄) denotes the open ball of radius γ̄ in X^{∗}. We now need to find bounds on the g_{v}^{∗}(x_{2}^{∗}). From Lemma 3.6, we may contract V_{0} if necessary so that for some real constant μ we have

(∀v ∈ V_{0})(∀∥x_{2}^{∗}∥ ≤ γ̄)(g_{v}^{∗}(x_{2}^{∗}) ≥ μ).

To obtain an upper bound, define, for v ∈ V_{0},

F_{v} := {x_{2}^{∗} ∈ X^{∗} | ∃x_{1}^{∗} ∈ X^{∗} such that (x_{1}^{∗}, x_{2}^{∗}) ∈ H_{K}(X^{∗}, v) ∩ B^{∗}(0, γ̄)}.

Note for each v ∈ V_{0}, that F_{v} ≠ ∅ since from (4.12), H_{K}(X^{∗}, v) ∩ B^{∗}(0, γ̄) is nonempty. We claim now that

(∃V′ ∈ N (w))(∃ρ ∈ ℝ)(∀v ∈ V′)(∀x_{2}^{∗} ∈ F_{v})(g_{v}^{∗}(x_{2}^{∗}) ≤ ρ).  (4.14)

To see this, suppose the contrary. Then there are nets v_{β} → w (in V_{0}), x_{2_{β}}^{∗} ∈ F_{v_{β}} with lim_{β} g_{v_{β}}^{∗}(x_{2_{β}}^{∗}) = +∞. If x_{1_{β}}^{∗} is such that (x_{1_{β}}^{∗}, x_{2_{β}}^{∗}) ∈ H_{K}(X^{∗}, v_{β}) ∩ B^{∗}(0, γ̄), then as ∥x_{1_{β}}^{∗}∥ ≤ γ̄, we may, by taking a weak^{∗}-convergent subnet if necessary, assume that it has a weak^{∗} limit, to be denoted by x_{1}^{∗}.

Let λ ∈ ℝ. Then eventually g_{v_{β}}^{∗}(x_{2_{β}}^{∗}) > K − λ so for such β, f_{v_{β}}^{∗}(x_{1_{β}}^{∗}) < K − g_{v_{β}}^{∗}(x_{2_{β}}^{∗}) < λ. Thus (x_{1}^{∗}, λ) ∈ bw^{∗}-lim sup_{v→w} epi f_{v}^{∗} ⊆ epi(bw^{∗}-e-li_{v→w}f_{v}^{∗}) and as λ ∈ ℝ is arbitrary, (bw^{∗}-e-li_{v→w}f_{v}^{∗})(x_{1}^{∗}) = −∞, so by Lemma 3.5, as f_{v} strongly epi-converges to f_{w},

f_{w} ≥ (bw^{∗}-e-li_{v→w}f_{v}^{∗})^{∗} ≡ +∞ on X ,

contradicting the properness of f_{w}. This proves (4.14).

Hence, if V_{0} ∈ N (w) is taken small enough, we have also for some ρ > 0 (dependent on K, ϵ) that

(∀v ∈ V_{0})(∀x_{2}^{∗} ∈ F_{v})(|g_{v}^{∗}(x_{2}^{∗})| ≤ ρ) .

Combining this with (4.13), it follows for all v ∈ V_{0}, that

epi(f_{v}^{∗}□g_{v}^{∗}) ∩ B^{∗}(0, K) ⊆ epi f_{v}^{∗} ∩ B^{∗}(0, K̂) + epi g_{v}^{∗} ∩ B^{∗}(0, K̂)  (4.15)

where K̂(K, ϵ) := K + max(ρ, γ̄).

Finally, by the epi-distance convergence f_{v}^{∗} → f_{w}^{∗} and g_{v}^{∗} → g_{w}^{∗} as v → w, there is V_{1} ∈ N (w) so that for all v ∈ V_{1}, epi f_{v}^{∗} ∩ B^{∗}(0, K̂) ⊆ epi f_{w}^{∗} + B^{∗}(0, ϵ/2) and similarly for the g_{v}^{∗} at w. It then follows, for all v ∈ V_{0} ∩ V_{1}, that

epi(f_{v}^{∗}□g_{v}^{∗}) ∩ B^{∗}(0, K) ⊆ epi f_{v}^{∗} ∩ B^{∗}(0, K̂) + epi g_{v}^{∗} ∩ B^{∗}(0, K̂)
⊆ epi f_{w}^{∗} + epi g_{w}^{∗} + B^{∗}(0, ϵ).

This yields (3.7). To finish, we verify (3.6). This follows immediately from (4.15) at v = w and the epi-distance convergence f_{v}^{∗} → f_{w}^{∗} and g_{v}^{∗} → g_{w}^{∗}, since for all v in a neighborhood of w,

epi(f_{w}^{∗}□g_{w}^{∗}) ∩ B^{∗}(0, K) ⊆ epi f_{w}^{∗} ∩ B^{∗}(0, K̂) + epi g_{w}^{∗} ∩ B^{∗}(0, K̂)
⊆ epi f_{v}^{∗} + epi g_{v}^{∗} + B^{∗}(0, ϵ/2) + B^{∗}(0, ϵ/2)
⊆ epi(f_{v}^{∗}□g_{v}^{∗}) + B^{∗}(0, ϵ) .

We now have proved the result, under the additional assumption that for each v, at least one of dom f_{v} or dom g_{v} intersects X_{v}. We will now remove this assumption. Take ρ > inf_{X} f_{w}. From the epi-distance convergence of level-sets [14, Proposition 7.1.7] we have v ↦ {f_{v} ≤ ρ} non-empty and lower-semicontinuous at w. Thus on taking x_{w} ∈ {f_{w} ≤ ρ} we have some x_{v} ∈ {f_{v} ≤ ρ} strongly converging to x_{w}. Place f̂_{v}(·) := f_{v}(· + x_{v}) and ĝ_{v}(·) := g_{v}(· + x_{v}). Then {f̂_{v}}_{v∈W} and {ĝ_{v}}_{v∈W} epi-distance converge to f̂_{w} and ĝ_{w}, respectively.

As 0 ∈ dom f̂_{v} and X_{v} = \overline{span}(dom f̂_{v} − dom ĝ_{v}), we may apply the previous argument to infer the epi-distance convergence of f̂_{v} + ĝ_{v} to f̂_{w} + ĝ_{w}. Translating the sum by −x_{v}, we get f_{v} + g_{v} = (f̂_{v} + ĝ_{v})(· − x_{v}) epi-distance converging to f_{w} + g_{w}.

In [8] we find a rather complicated condition ensuring epi-distance convergence of a sum (see condition (1) in Proposition 4.10 below). The form of these conditions facilitates estimation of the epi-distance metrics. Putting aside the issue of these estimates we observe that this condition is equivalent to an interiority condition. Place Δ(X) = {(x, x) | x ∈ X}. We note that the mapping P : X × X → X defined by P (x, y) = y − x is open, as is P^{−1}.

Proposition 4.10. Let f and g be proper closed convex functions on a Banach space X. Then the following are equivalent:

(i) there exists t > 0, r > 0 and s > 0 such that

(B(0, s))^{2} ⊆ Δ(X) ∩ (B(0, r))^{2} − {f ≤ r} × {g ≤ r} ∩ (B(0, t))^{2} ;  (4.16)

(ii) 0 ∈ int (dom f − dom g).

Proof. Clearly (4.16) implies 0 ∈ int (Δ(X) − dom f × dom g). Then

0 ∈ int P (Δ(X) − dom f × dom g)
= int P {(x, x) − (u, v) | x ∈ X, u ∈ dom f, v ∈ dom g}
= int{(x − v) − (x − u) | x ∈ X, u ∈ dom f, v ∈ dom g}
= int (dom f − dom g) .

For the converse we use Corollary 2.7 to deduce from 0 ∈ int (dom f − dom g) that

0 ∈ int ({f ≤ t} ∩ B(0, t) − {g ≤ t} ∩ B(0, t))

for some t > 0. Define D := Δ(X) − {f ≤ t} × {g ≤ t} ∩ (B(0, t))^{2}. Then P D = {f ≤ t} ∩ B(0, t) − {g ≤ t} ∩ B(0, t) so 0 ∈ int P D and hence 0 ∈ int P^{−1}P D. But since

P^{−1}P D = D + ker P = D + Δ(X)
= Δ(X) − {f ≤ t} × {g ≤ t} ∩ (B(0, t))^{2} + Δ(X) = D,

it follows that 0 ∈ int (Δ(X) − \{f ≤ t\} × \{g ≤ t\} ∩ (\overline{B}(0, t))^{2}). As Δ(X) and \{f ≤

t\} × \{g ≤ t\} ∩ (\overline{B}(0, t))^{2} are closed convex sets they may be viewed as the domains of their indicator functions. Thus applying Corollary 2.7 again we get for some r > 0 that

0 ∈ int (Δ(X) ∩ (B(0, r))^{2} − \{f ≤ t\} × \{g ≤ t\} ∩ (\overline{B}(0, r))^{2}).

5. Saddle-point Convergence in Fenchel Duality

Convex-concave bivariate functions are related to convex bivariate functions through partial conjugation (i.e. conjugation with respect to one of the variables). In this context we are led to the introduction of equivalence classes of saddle-functions which are uniquely associated with concave or convex parents (depending on the which variable is partially conjugated). We direct the reader to the excellent texts of Rockafellar [22] and [23] for a detailed treatment of this phenomenon. Two bivariate functions are said to belong to the same equivalence class if they have the same convex and concave parents. Such members of the same equivalence class not only have the same saddle-point but so do all linear perturbations of these two functions. Thus when discussing the variational convergence of saddle-functions one is necessarily led to the study of the convergence of the equivalence class. When referring to a saddle-function (or convex-concave bivariate function) we are by association referring to its equivalence class via the selection of a member. The following is taken from [7] from which we adapt results and proofs.

Definition 5.1. Suppose that (X, τ) and (Y, σ) are two topological spaces and {K^{n} : X × Y → \overline{ℝ}, n ∈ ℕ\} is a sequence of bi-variate functions. Define:

e_{τ}/h_{σ}-ls_{n→∞}K^{n}(x, y) = sup_{{y_{n}→^{σ}y}} inf_{{x_{n}→^{τ} x}} lim sup_{n→∞} K^{n}(x_{n}, y_{n})
h_{σ}/e_{τ}-li_{n→∞}K^{n}(x, y) = inf_{{x_{n}→^{τ} x}} sup_{{y_{n}→^{σ}y}} lim inf_{n→∞} K^{n}(x_{n}, y_{n}) .

Definition 5.2. Suppose that (X, τ) and (Y, σ) are two topological spaces and {K^{n} : X × Y → \overline{ℝ}, n ∈ ℕ\} is a sequence of bivariate functions.

(1) We say that they epi/hypo-converge in the extended sense to a function K : X×Y → \overline{ℝ} if

\underline{cl}_{x}(e_{τ}/h_{σ}-ls_{n→∞}K^{n}) ≤ K ≤ \overline{cl}^{y}(h_{σ}/e_{τ}-li_{n→∞}K^{n})

where \underline{cl}_{x} denotes the extended lower closure with respect to x (and therefore w.r.t. τ) for fixed y and \overline{cl}^{y} denotes the extended upper closure with respect to y (and therefore w.r.t. σ) for fixed x, (where generally, \overline{cl}f := −\underline{cl}(−f)).

(2) A point (x̄, ȳ) is a saddle-point of a bivariate function K : X × Y → \overline{ℝ} if for all (x, y) ∈ X × Y we have K(x̄, y) ≤ K(x̄, ȳ) ≤ K(x, ȳ).

The interest in this kind of convergence stems from the following result (see [7, Thm 2.4]).

Proposition 5.3. Let us assume that \{K^{n}, K : (X, τ) × (Y, σ) → \overline{ℝ}, n ∈ ℕ\} are such that they epi/hypo-converge in the extended sense. Assume also that (x̄_{k}, ȳ_{k}^{∗}) are saddle points of K^{n_{k}} for all k and {n_{k}} is an increasing sequence of integers, such that x̄_{n_{k}} →^{τ} x̄ and ȳ_{n_{k}}^{∗} →^{σ} ȳ^{∗}. Then (x̄, ȳ^{∗}) is a saddle point of K and

K(x̄, ȳ^{∗}) = lim_{k→∞} K^{n_{k}}(x̄_{k}, ȳ_{k}^{∗}) .

The next result from [7] uses sequential forms of the epi-limit functions, as per the following

Definition 5.4 ([7, p. 541]). Let (X, τ) be topological, f_{n} : X → \overline{ℝ}. Then

(τ-seq-e-ls_{n→∞}f_{n})(x) := inf_{{x_{n}}→^{τ}x} lim sup_{n→∞} f_{n}(x_{n})
(τ-seq-e-li_{n→∞}f_{n})(x) := inf_{{x_{n}}→^{τ}x} lim inf_{n→∞} f_{n}(x_{n})

It can be shown that these reduce to the usual (topologically defined) forms if (X, τ) is first-countable, and that the above infima are achieved. We will need these alternate forms, for generally weak topologies on normed spaces are not first-countable.

Definition 5.5. Let (X, τ) and (X^{∗}, τ^{∗}) be topological vector spaces. We shall say they are paired if there is a bilinear map ⟨·, ·⟩ : X × X^{∗} → ℝ such that the maps x^{∗} ↦ ⟨·, x^{∗}⟩ and x ↦ ⟨x, ·⟩ are (algebraic) isomorphisms such that X^{∗} ≅ (X, τ)^{∗} and X ≅ (X^{∗}, τ^{∗})^{∗} respectively.

It is readily checked that if (X, τ) and (X^{∗}, τ^{∗}) are paired, and so are (Y, σ) and (Y^{∗}, σ^{∗}), then (X × Y, τ × σ) is paired with (X^{∗} × Y^{∗}, τ^{∗} × σ^{∗}), with the pairing

⟨(x, y), (x^{∗}, y^{∗})⟩ = ⟨x, x^{∗}⟩ + ⟨y, y^{∗}⟩,

and similarly for other combinations of product spaces.

For any convex-concave saddle function K : X ×Y^{∗} → \overline{ℝ}, that is, where K is convex in the first argument and concave in the second, we may associate a convex- and concave-parent. These play a fundamental role in convex duality (see [22]) and are defined respectively as:

F (x, y) = sup_{y^{∗}∈Y^{∗}}[K(x, y^{∗}) + ⟨y, y^{∗}⟩]
G(x^{∗}, y^{∗}) = inf_{x∈X}[K(x, y^{∗}) − ⟨x, x^{∗}⟩].

(In Fenchel duality we have F (x, y) = f(x) + g(x + y)). One can show that for any closed convex function F : X × Y → \overline{ℝ}, if G := −F^{∗} relative to the natural pairing of X × Y with X^{∗} × Y^{∗}, (these yielding the primal objective F (·, 0) and dual objective G(0, −·)), we have an interval of saddle functions (all equivalent in the sense that they possess the same saddle points) given by

[\underline{K}, \overline{K}] := \{K : X × Y^{∗} → \overline{ℝ} | K convex-concave, \underline{K} ≤ K ≤ \overline{K} on X × Y^{∗}\},

where

\underline{K}(x, y^{∗}) = sup_{x^{∗}∈X^{∗}}[G(x^{∗}, y^{∗}) + ⟨x, x^{∗}⟩]
\overline{K}(x, y^{∗}) = inf_{y∈Y} [F (x, y) − ⟨y, y^{∗}⟩] .

The following result is taken from [7] and requires no additional assumption.

Proposition 5.6. Let (X, τ), (X^{∗}, τ^{∗}) and (Y, σ), (Y^{∗}, σ^{∗}) be paired topological vector spaces, with the pairings sequentially continuous; let \{F^{n}, F : X × Y → \overline{ℝ}, n ∈ ℕ\} be a family of bivariate (τ × σ)-closed convex functions. Then, if K^{n}, K are members of the corresponding equivalence classes of bivariate convex-concave saddle functions,

(i)

(τ × σ)-seq-e-ls_{n→∞}F^{n} ≤ F on X × Y
implies \underline{cl}_{x}(e_{τ}/h_{σ^{∗}}-ls \overline{K}^{n}) ≤ \underline{K} ;

(ii)

(τ^{∗} × σ^{∗})-seq-e-ls_{n→∞}(F^{n})^{∗} ≤ (F)^{∗} on X × Y
implies \overline{K} ≤ \overline{cl}^{y^{∗}}(h_{σ^{∗}}/e_{τ}-li \underline{K}^{n}) .

Proposition 5.7. Suppose that X is a Banach space and {f_{n}, f}_{n=1}^{∞} and {g_{n}, g}_{n=1}^{∞} be two families of proper closed, convex extended-real-valued functions epi-distance convergent to f and g, respectively. Then

K^{n}(x, y^{∗}) = inf_{y∈X}[f_{n}(x) + g_{n}(x + y) − ⟨y, y^{∗}⟩]

epi/hypo-converges (in the extended sense) to

K(x, y^{∗}) = inf_{y∈X}[f(x) + g(x + y) − ⟨y, y^{∗}⟩]

with respect to the strong topology on X and the weak^{∗} topology on X^{∗}.

Proof. Define proper closed convex functions on X ×X by f̂_{n}(x, y) := f_{n}(x) and ĝ_{n}(x, y) := g_{n}(x + y), and similarly define f̂ and ĝ. Then we have the epi-distance convergence f̂_{n} → f̂ and ĝ_{n} → ĝ on X × X. Note that dom f̂ = dom f × X and dom ĝ = P^{−1} dom g (and similarly for f_{n}, g_{n}), where P : X ×X → X is the addition operation. Consequently, dom f̂−dom ĝ = X ×X = dom f̂_{n} −dom ĝ_{n} for all n, so from Theorem 4.9 (or [8]) follows the epi-distance convergence

F^{n} := f̂_{n} + ĝ_{n} → F := f̂+ ĝ.

Thus also obtains [9] the epi-distance convergence (F^{n})^{∗} → F^{∗}. From the resultant strong epi-convergence on X × X and on X^{∗} × X^{∗},

F ≥ (s × s)-e-ls_{n→∞}F^{n} = (s × s)-seq-e-ls_{n→∞}F^{n}  and
F^{∗} ≥ (s^{∗} × s^{∗})-e-ls_{n→∞}(F^{n})^{∗} = (s^{∗} × s^{∗})-seq-e-ls_{n→∞}(F^{n})^{∗}
≥ (w^{∗} × w^{∗})-seq-e-ls_{n→∞}(F^{n})^{∗},

where s and s^{∗} stand for the respective norm topologies on X and X^{∗}. Now apply Proposition 5.6.

We note the following for future reference. For u ∈ X, write

φ(u) := inf_{x∈X}\{f(x) + g(x + u)\} = { \atop (f□ ĝ)}(u),  (5.1)

and similarly for φ_{n}, where for any function ψ, ψ̂(x) := ψ(−x). Note that dom φ = dom g − dom f and similarly for φ_{n}. The operation ψ ↦ ψ̂ commutes with conjugation and with epi-distance limits; the verification of this is an elementary exercise. From [22] we have the following: Calling inf_{X}(f + g) the primal problem, and inf_{X}(f_{n} + g_{n}) the approximate problems, then −φ^{∗} and −φ_{n}^{∗} are the associated dual objective functionals, and:

(x̄, ȳ^{∗})  is a saddle-point of K iff
φ(0) = (f + g)(x̄) = inf_{X}(f + g) = sup_{X^{∗}} −φ^{∗} = −φ^{∗}(ȳ^{∗}),

and similarly for φ_{n} and the saddle-points (x̄_{n}, ȳ_{n}^{∗}) of K^{n}. On taking conjugates of φ_{n} we obtain

φ_{n}^{∗} = { \atop (f_{n}□ ĝ_{n})^{∗}} = { \atop (f_{n}^{∗} + ĝ_{n}^{∗})} = f̂_{n}^{∗} + g_{n}^{∗}

and so the dual problem becomes

sup_{X^{∗}} −φ_{n}^{∗} = − (f_{n}^{∗}□g_{n}^{∗}) (0) = − inf_{y^{∗}∈X^{∗}} (f_{n}^{∗}(−y^{∗}) + g_{n}^{∗}(y^{∗})) .  (5.2)

The next result tackles the problem of finding convergent sequences of dual variables. (Note that Proposition 5.3 makes no claim about such existence). There are well-known criteria for the convergence of the primal solutions x̄_{n}– for instance, if f_{n} +g_{n} epi-distance converges to f + g, and the latter satisfies a well-posedness condition [14, Section 7.5] and [13]. From [19] we know that 0 ∈ sqri(dom f − dom g) is a sufficient condition for strong duality to hold.

Corollary 5.8. Suppose that X is a separable Banach space and {f_{n}, f}_{n=1}^{∞} and {g_{n}, g}_{n=1}^{∞} be two families of proper closed, convex extended-real-valued functions epi-distance convergent to f and g respectively. Let K^{n}, K be the associated saddle-functions as in Proposition 5.7. Assume also the following:

(i) 0 ∈ sqri(dom f − dom g); (ii) The algebraic complement Y of the subspace M = cone(dom f−dom g), is closed and

M_{n} ∩ Y = \{0\} eventually, where M_{n} denotes the subspace \overline{span}(dom f_{n} − dom g_{n}).

Then if (x̄_{n}, ȳ_{n}^{∗}) are saddle-points of K^{n} for each n and a subsequence of the x̄_{n} has a strong limit x̄, and the saddle-values along this subsequence are bounded below, then

K has a saddle-point (x̄, ȳ^{∗}) that is a (s × w^{∗}) -limit of saddlepoints (x̄_{n}, { \atop (ȳ_{n}^{∗})|_{M_{n}}}) of a subsequence of the K^{n}, with K(x̄, ȳ^{∗}) the limit of the corresponding saddle-function values.

(Here ‘s’ stands for the norm topology on X and { \atop (ȳ_{n}^{∗})|_{M_{n}}} is the extension to X^{∗} of the restriction of ȳ_{n}^{∗} to M_{n} as defined in Definition 4.3).

Proof. The proof follows from Propositions 5.7 and 5.3, on showing that the { \atop (ȳ_{n}^{∗})|_{X_{n}}} are norm-bounded in X^{∗}, so that weak^{∗}-convergent subsequences are available and are the required dual variables.

As in the proof of Theorem 4.9, there are x_{n} ∈ dom f_{n} converging to some x ∈ dom f. Place f̌_{n}(·) := f_{n}(x_{n} + ·), ǧ_{n}(·) := g_{n}(x_{n} + ·), with analogous definitions for f̌ and ǧ as translates by x. Then 0 ∈ dom f̌_{n}, dom f̌, and hence these functions satisfy the hypotheses of Lemma 4.8.

Let φ̌_{n} be the value function corresponding to f̌_{n} and ǧ_{n} via (5.1). Similarly, denote the corresponding saddle function by Ǩ^{n}. Then we immediately observe that φ̌_{n}^{∗} = φ_{n}^{∗}, from which follows that

(x̄_{n}, ȳ_{n}^{∗}) is a saddlepoint of K^{n} iff (x̄_{n} − x_{n}, ȳ_{n}^{∗}) is a saddlepoint of Ǩ^{n} ,

since (x̄_{n}, ȳ_{n}^{∗}) are an optimal pair for the primal and dual problems if and only if (x̄_{n} − x_{n}, ȳ_{n}^{∗}) are optimal for the problems based on the translated functions f̌_{n}, ǧ_{n}. Evidently the optimal values are not affected by this translation, so we also obtain that Ǩ^{n}(x̄_{n}−x_{n}, ȳ_{n}^{∗}) = K^{n}(x̄_{n}, ȳ_{n}^{∗}). Hence the saddle-values of Ǩ^{n} are also bounded below. As M_{n} contains both dom f̌_{n} and dom ǧ_{n} (recall this follows from 0 ∈ dom f̌_{n}), we obtain

Ǩ^{n}(x̄_{n} − x_{n}, ȳ_{n}^{∗}) = Ǩ^{n}(x̄_{n} − x_{n}, { \atop ȳ_{n}^{∗}|_{M_{n}}}) ,

which follows from φ̌_{n}^{∗}(ȳ_{n}^{∗}) = f̌_{n}^{∗}(−ȳ_{n}^{∗}) + ǧ_{n}^{∗}(ȳ_{n}^{∗}) = f̌_{n}^{∗}(−{ \atop ȳ_{n}^{∗}|_{M_{n}}}) + ǧ_{n}^{∗}({ \atop ȳ_{n}^{∗}|_{M_{n}}}) = φ̌_{n}^{∗}({ \atop ȳ_{n}^{∗}|_{M_{n}}}),

since M_{n} contains the domains of f̌_{n} and ǧ_{n}. Letting −α ∈ ℝ be a lower bound for the saddle-values of K^{n} (and hence of Ǩ^{n}), we have for all n large, that

(−{ \atop ȳ_{n}^{∗}|_{M_{n}}}, { \atop ȳ_{n}^{∗}|_{M_{n}}}) ∈ { \atop H_{α}(M_{n}, n)}

(where the latter set is defined relative to the translated functions f̌_{n}, ǧ_{n}), since

(f̌_{n}|_{M_{n}})^{∗}(−ȳ_{n}^{∗}|_{M_{n}}) + (ǧ_{n}|_{M_{n}})^{∗}(ȳ_{n}^{∗}|_{M_{n}}) = f̌_{n}^{∗}(−ȳ_{n}^{∗}) + ǧ_{n}^{∗}(ȳ_{n}^{∗}) = −Ǩ^{n}(x̄_{n} − x_{n}, ȳ_{n}^{∗}) < α

so that (−ȳ_{n}^{∗}|_{M_{n}}, ȳ_{n}^{∗}|_{M_{n}}) ∈ H_{α}(M_{n}, n). By Lemma 4.8, there is γ > 0 such that ∥{ \atop ȳ_{n}^{∗}|_{M_{n}}}∥ ≤

γ for all large n. Then z̄_{n}^{∗} := { \atop ȳ_{n}^{∗}|_{M_{n}}} ∈ X^{∗} is norm-bounded and hence has a weakly^{∗} convergent subsequence z̄_{n}^{∗} → z̄^{∗}. For each n, (x̄_{n} − x_{n}, z̄_{n}^{∗}) a saddlepoint for Ǩ^{n}, so (x̄_{n}, z̄_{n}^{∗}) is one for K^{n}. By Propositions 5.7 and 5.3, (x̄, z̄^{∗}) is a saddlepoint for K, with value the limit of the saddle-values along the sequence.

Remark 5.9. A case in the above result where the saddle-values are bounded below, occurs for instance when the sum f + g of the limit functions has bounded level-sets, for from the epi-distance convergence f_{n} +g_{n} → f +g (Theorem 4.9), and from [14, Corollary 7.5.3], the infima of f_{n} + g_{n} converge to that of f + g, which is finite by properness, and these correspond to the saddle-values.

References

[1] H. Attouch, H. Brézis: Duality for the sum of convex functions in general Banach spaces, Aspects of Mathematics and its Applications (J. Barroso, ed.), Elsevier Sc. Publ. (1986) 125–133.

[2] H. Attouch: Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman, London, 1984.

[3] H. Attouch, R. J-B Wets: Quantitative stability of variational systems I: The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991) 695–729.

[4] H. Attouch, R. J-B Wets: Quantitative Stability of Variational Systems II: A framework for nonlinear conditioning, SIAM J. Opt. 3(2) (1993) 359–381.

[5] H. Attouch, R. J-B Wets: Quantitative Stability of Variational Systems III: ϵ-approximate solutions, Mathematical Programming 61 (1993) 197–214.

[6] J.-P. Aubin, H. Frankowska: Set-Valued Analysis, Systems and Control: Foundations and Applications, Vol. 2, Birkhäuser, 1990.

[7] D. Azé, H. Attouch, R. J-B Wets: Convergence of convex-concave saddle functions: Applications to convex programming and mechanics, Ann. Inst. Henri Poincaré 5(6) (1988) 537–572.

[8] D. Azé, J.-P. Penot: Operations on convergent families of sets and functions, Optimization 21(4) (1990) 521–534.

[9] G. Beer: Conjugate convex functions and the epi-distance topology, Proceedings of the American Mathematical Society 108(1) (1990) 117–126.

[10] G. Beer: The slice topology: A viable alternative to Mosco convergence in non-reflexive spaces, Nonlinear Analysis: Theory, Methods and Applications 19(3) (1992) 271–290.

[11] G. Beer, M. Borwein: Mosco convergence and reflexivity, Proceedings of the American Mathematical Society, 109(2) (1990) 427–436.

[12] G. Beer, R. Lucchetti: The epi-distance topology: Continuity and stability results with applications to convex optimization problems, Mathematics of Operations Research 17(3) (1992) 715–726.

[13] G. Beer, R. Lucchetti: Convex optimization and the epi-distance topology, Transactions of the American Mathematical Society 327(2) (1991) 795–813.

[14] G. Beer: Topologies on Closed and Closed Convex Sets, Mathematics and its Applications 268, Kluwer Acad. Publ., 1993.

[15] J. M. Borwein, A. S. Lewis: Partially-finite convex programming, Mathematical Programming 57 (1992) 15–83.

[16] M. S. Gowda, M. Teboulle: A comparison of constraint qualifications in infinite-dimensional convex optimization, SIAM J. Control and Optimization 28(4) (1990) 925–935.

[17] R. B. Holmes: Geometric Functional Analysis and its Applications, Springer-Verlag, Graduate Texts in Mathematics 24 (1975).

[18] V. Jeyakumar, H. Wolkowicz: Generalizations of Slater’s constraint-qualification for infinite convex programs, Mathematical Programming 57 (1992) 85–101.

[19] V. Jeyakumar: Duality and infinite-dimensional optimization, Nonlinear Analysis: Theory, Methods and Applications 15 (1990) 1111–1122.

[20] S. M. Robinson: Regularity and stability of convex multivalued functions, Math. Oper. Res. 1 (1987) 130–143.

[21] R. T. Rockafellar, R. J-B Wets: Variational systems – an introduction, in: Multifunctions and Integrands (G. Salinetti, ed.), Springer-Verlag, Lecture Notes in Mathematics 1091 (1984) 1–54.

[22] R. T. Rockafellar: Conjugate Duality and Optimization, SIAM Publ., 1974.

[23] R. T. Rockafellar: Convex Analysis, Princeton University Press, 1970.

[24] T. Strömberg: The operation of infimal convolution, Dissertationes Math. 352 (1996) 4–58.