Necessary and sufficient conditions for stable conjugate duality

Nonlinear Analysis 64 (2006) 1998 – 2006 www.elsevier.com/locate/na

Necessary and sufficient conditions for stable conjugate duality R.S. Burachika, b,∗ , V. Jeyakumarc , Z.-Y. Wud, c a School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat 3353,

Victoria, Australia b School of Mathematics and Statistics, University of South Australia, Mawson Lakes SA 5095, Australia c School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia d Department of Mathematics and Computer Science, Chongqing Normal University,

Chongqing 400047, PR China Received 3 May 2005; accepted 26 July 2005

Abstract The conjugate duality, which states that inf x∈X
(x, 0)=maxv∈Y  −
∗ (0, v), whenever a regularity condition on
is satisfied, is a key result in convex analysis and optimization, where
: X × Y → R ∪ {+∞} is a convex function, X and Y are Banach spaces, Y  is the continuous dual space of Y and
∗ is the Fenchel–Moreau conjugate of
. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, inf {
(x, 0) + x ∗ (x)} = max {−
∗ (−x ∗ , v)},

x∈X

v∈Y 

∀x ∗ ∈ X 

and then obtain a new epigraph regularity condition for the conjugate duality. The regularity condition is shown to be much more general than the popularly known interior-point type conditions. As an easy consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel–Rockafellar duality theorem. In the case where one of the functions involved is a polyhedral convex function, we provide generalized interior-point conditions for the epigraph regularity

∗ Corresponding author at. School of Information Technology and Mathematical Sciences, University of Ballarat, Ballarat 3353,Victoria, Australia. E-mail addresses: [email protected] (R.S. Burachik), [email protected] (V. Jeyakumar), [email protected] (Z.-Y. Wu).

0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.07.034

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condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: 90C25; 49J52; 49J53; 49J35; 65K10 Keywords: Conjugate duality; Constraint qualifications; Convex programming; Polyhedral functions; Sublinear functions

1. Introduction A fundamental duality scheme for studying the convex minimization problem min f (x),

x ∈ X,

(1.1)

where f : X → R ∪ {+∞}, is by a representation function
: X × Y → R ∪ {+∞} such that
(x, 0) = f (x). In which case the dual problem associated to (1.1) is given by max −
∗ (0, v),

v ∈ Y ,

(1.2) ∗

where X and Y are Banach spaces with the duals X  and Y  respectively, and
is the Fenchel–Moreau conjugate of
. A conjugate duality states that inf
(x, 0) = max −
∗ (0, v),

x∈X

(1.3)

v∈Y

whenever a regularity condition on
is satisfied. In other words, under the regularity condition, there is no duality gap and the dual problem has a solution. The conjugate duality, which is a key to the study of convex optimization, constrained best approximation and interpolation (see [6]), enables, for instance, one to find the optimal solution of the original problem by solving the corresponding dual optimization problem. A central question in convex analysis and optimization is to find a general regularity condition for the conjugate duality. From the point of view of applications, it is vital to find conditions on
, which characterize the stable conjugate duality: inf {
(x, 0) + x ∗ (x)} = max {−
∗ (−x ∗ , v)},

x∈X

v∈Y

∀x ∗ ∈ X .

Various sufficient conditions for the duality have been given in the literature (see [18] and other reference therein). However, these regularity conditions are either (global) interiorpoint type conditions [1,13,16,17] which frequently restrict applications or are based on local conditions. In recent years, it has been shown in [3–5] that Fenchel’s duality holds under a dual epigraph condition, which is strictly weaker than the usual interior-point conditions [1,9,10], and the bounded linear regularity condition [2]. The purpose of this paper is to establish a necessary and sufficient condition for the stable conjugate duality and derive new epigraph regularity conditions for conjugate duality. These conditions are much more general than the popularly known interior-point type conditions, employed in the literature [1,13,16,17]. We also present an epigraph closure condition that is necessary and sufficient for a stable Fenchel–Rockafellar duality theorem involving

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R.S. Burachik et al. / Nonlinear Analysis 64 (2006) 1998 – 2006

composite functions. As a consequence, we show that a stable Fenchel duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. In the case where one of the functions involved is a polyhedral convex function, we provide generalized interior-point conditions for the epigraph closure condition. As an application, we give general sufficient conditions for duality results of convex programming problems involving polyhedral constraints.

2. Preliminaries: epigraphs of conjugate functions We begin by fixing some definitions and notations. We assume throughout that X and Y are Banach spaces. The continuous dual space of X will be denoted by X  and will be endowed with the weak* topology. For the set D ⊂ X the closure of D will be denoted cl D. If a set A ⊂ X , the expression cl A will stand for the weak* closure. The indicator function / D. The support function D D is defined as D (x) = 0 if x ∈ D and D (x) = +∞ if x ∈ is defined by D (u) = supx∈D u(x). The normal cone of D is given by ND (x) := {v ∈ X : D (v) = v(x)} = {v ∈ X  : v(y − x)
0, ∀y ∈ D} when x ∈ D, and ND (x) := ∅ when x∈ / D. Let f : X → R ∪ {+∞} be a proper lower semi-continuous convex function. Then, the conjugate function of f , f ∗ : X → R ∪ {+∞}, is defined by f ∗ (v) = sup{v(x) − f (x) | x ∈ dom f }, where the domain of f, dom f , is given by dom f = {x ∈ X|f (x) < + ∞}. The epigraph of f, Epi f , is defined by Epi f = {(x, r) ∈ X × R | x ∈ dom f, f (x)
r}. Recall that the lower semicontinuous hull of a function h : X → R ∪ {+∞}, denoted by lsc h, is defined as the function whose epigraph is the topological closure of the epigraph of h, and that h is lower semicontinuous if and only if lsc h = h. It is also convenient to deal with the following concept of closure of h, denoted by cl h, and defined as
lsc h(·) if lsc h(·) > − ∞, cl h(·) = −∞ if lsc h(x) = −∞ for at least one x ∈ X. We say that h is closed when cl h = h. It follows from these definitions that a proper convex function h is closed if and only if it is lower semicontinuous. In the latter situation, we have that Epi(cl h) = cl(Epi h). The subdifferential of f, jf : X ⇒ X is defined as jf (x) = {v ∈ X |f (y) f (x) + v(y − x), ∀y ∈ X}. Note also that jD =ND . If f : X → R∪{+∞} is a proper lower semi-continuous sublinear function, i.e., f is convex and positively homogeneous (f (0)=0, and f (x)=f (x), ∀x ∈ X, ∀ ∈ (0, ∞)), then jf (0) is non-empty and for each x ∈ dom f , jf (x) = {v ∈ jf (0)|v(x) = f (x)}.

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Let Z be a real Banach space. Given two proper lower semi-continuous convex functions h1 , h2 : Z → R ∪ {+∞}, their infimal convolution h1 ⊕ h2 is the function h1 ⊕ h2 : Z → R ∪ {+∞} defined by h1 ⊕ h2 (z) :=

inf {h1 (z1 ) + h2 (z2 )}.

z1 +z2 =z

The infimal convolution of h1 with h2 is said to be exact provided the infimum above is achieved at each z ∈ Z. We also need the following lemma, which is well-known, see for instance Precupanu [14]. For related results, see [7,8]. Let U, V be two topological ¯ Then the marginal function h : V → R ¯ is defined by spaces and let  : U × V → R. h(v) := inf u∈U (u, v) and the projection operator, P r V ×R : U × V × R → V × R, is given P r V ×R (u, v, r) = (v, r). ¯ be a proper function. Lemma 2.1 (Theorem 2.1 Precupanu [14]). Let  : U × V → R ¯ defined by Then, P r V ×R (Epi ) is closed if and only if the marginal function h : V → R, ¯ is attained whenever h(v) = inf u∈U (u, v) is lower semicontinuous and inf u∈U (u, v) h(v) ¯ > − ∞, for v¯ ∈ V .

3. Stable conjugate duality In this section we establish characterizations of conjugate duality properties and derive a dual epigraph condition for conjugate duality. Our main result, Theorem 3.1, establishes a necessary and sufficient condition for the stable conjugate duality. We begin by establishing a basic result on the lower semicontinuity of a marginal function and a duality property ¯ The involving conjugate functions. Let X, Y be Banach spaces and let
: X × Y → R. ∗ ∗ ∗ ∗  ∗ ∗ ¯ ∗  marginal function  : X → R of
is given by (x ) = inf y ∈Y
(x , y ), for x ∈ X . ¯ be a proper lower semicontinuous convex function. Lemma 3.1. Let
: X × Y → R Suppose that inf x∈X {
(x, 0)} < + ∞. Then the following statements are equivalent. (i) The marginal function  of
∗ is weak* lower semicontinuous. (ii) inf x∈X {
(x, 0) + x, x ∗ } = supy ∗ ∈Y  {−
∗ (−x ∗ , y ∗ )}, ∀x ∗ ∈ X . Proof. Note first that ∗ (x) = (
∗ )∗ (x, 0) =
(x, 0),

∀x ∈ X

so that ∗ is proper. Hence  is also proper and lsc  = ∗∗ . Therefore, for each x ∗ ∈ X , sup (−
∗ (−x ∗ , y ∗ )) = −(−x ∗ )
− lsc (−x ∗ ) = inf {
(x, 0) + x, x ∗ }. (3.1)

y ∗ ∈Y 

x∈X

Now, it follows from (3.1) that (i) holds if and only if (ii) holds.



We now establish our main result as an easy application of this Lemma.

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Theorem 3.1 (Stable conjugate duality). Let
: X × Y → R ∪ {+∞} be a proper and lower semi-continuous convex function. Suppose that  := inf x∈X {
(x, 0)} < + ∞. Then the following statements are equivalent: (i) inf x∈X {
(x, 0) + x ∗ (x)} = maxv∈Y  {−
∗ (−x ∗ , v)}, ∀x ∗ ∈ X . (ii) P r X ×R (Epi
∗ ) is weak* closed. Proof. Since  < ∞, we have that inf x∈X {
(x, 0) + x ∗ (x)} < ∞ for all x ∗ ∈ X . This fact, together with (3.1) yields (x ∗ ) > − ∞ for all x ∗ ∈ X  . By Lemma 3.1, (i) is equivalent to the fact that  is weak* lower semicontinuous and all values (x ∗ ) are attained. Now, it follows from Lemma 2.1 that the latter situation is equivalent to the closedness of P r X ×R (Epi
∗ ).  Remark 3.1. It is worth noting that Theorem 3.1 gives a necessary and sufficient condition for the equality
(·, 0)∗ (−x)∗ = min
∗ (x ∗ , v), v∈Y

∀x ∗ ∈ X ,

which is equivalent to the equality (i) of Theorem 3.1. For numerous sufficient conditions for this equality, see Theorem 2.7.1 and Corollary 2.7.3 of [18] and see also [17]. Theorem 3.1 shows that these sufficient conditions also ensure that the set P r X ×R (Epi
∗ ) is weak* closed. Now we establish a necessary and sufficient condition for the conjugate duality as a consequence of Theorem 3.1. Proposition 3.1. Let
: X × Y → R ∪ {+∞} be a proper and lower semi-continuous convex function. Let  := inf x∈X
(x, 0) be finite. Then the following statements are equivalent: (i) inf x∈X
(x, 0) = maxv∈Y  −
∗ (0, v). (ii) (0, −) ∈ P r X ×R (Epi
∗ ). Proof. From (3.1) with x ∗ = 0 we have −

∗ (0, y ∗ ) for every y ∗ ∈ Y  . If (i) holds, then there exist y ∗ such that
∗ (0, y ∗ ) = −, and so (0, y ∗ , −) ∈ Epi
∗ . Hence, (0, −) ∈ P r X ×R (Epi
∗ ). Conversely, if (ii) holds, then there exists y ∗ ∈ Y  such that
∗ (0, y ∗ )
− . But −

∗ (0, y ∗ ). Hence, (i) holds.  Corollary 3.1 (Generalized conjugate duality). Let
: X × Y → R ∪ {+∞} be a proper and lower semi-continuous convex function such that  := inf x∈X
(x, 0) < + ∞. If P r X ×R (Epi
∗ ) is weak* closed then inf
(x, 0) = max −
∗ (0, v).

x∈X

v∈Y

Proof. The conclusion follows from Theorem 3.1 by taking x ∗ = 0.



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Let us see how the epigraph closure condition in the conjugate duality results can be expressed in the particular case where
is described in terms of the sum of two convex functions. Recall that for a continuous linear map A : X → Y the adjoint operator of A, denoted by A
, is the unique linear application from Y  to X  with the property (A
v)(x) = v(Ax) for all v ∈ Y  , x ∈ X. Associated to the linear mapping A, we consider the mapping A
× I : Y  × R → X × R defined as (A
× I )(w, ) = (A
w, ). Note that the image of Epi g ∗ through this application is (A
× I )(Epi g ∗ ) = {(A
w, )|(w, ) ∈ Epi g ∗ } = {(A
w, )|g ∗ (w)
}. Let (Epi g ∗ )A := (A
× I )(Epi g ∗ ). Theorem 3.2 (Stable Fenchel–Rockafellar duality). Let A : X → Y be a continuous linear mapping. Let f : X → R ∪ {+∞} and g : Y → R ∪ {+∞} be proper and lower semi-continuous convex functions such that A(dom f ) ∩ dom g  = ∅. Then the following statements are equivalent: (i) inf x∈X {f (x) + g(Ax) + x ∗ (x)} = maxv∈X {−f ∗ (A∗ v − x ∗ ) − g ∗ (−v)}, ∀x ∗ ∈ X . (ii) Epi f ∗ + (Epi g ∗ )A is weak* closed. Proof. Define
: X × Y → R ∪ {+∞} by
(x, y) = f (x) + g(Ax + y) for each (x, y) ∈ X × Y . Then
∗ (u, v) = f ∗ (u − A
v) + g ∗ (v). Then, condition (i) becomes inf
(x, 0) + x ∗ (x) = max {−
∗ (−x ∗ , −v)},

x∈X

v∈Y

∀x ∗ ∈ X .

(3.2)

The equivalence will follow from Theorem 3.1. Note that the assumptions of Theorem 3.1 hold because A(dom f ) ∩ dom g = ∅, so there exists y0 ∈ dom g such that y0 = Ax 0 with x0 ∈ dom f . Hence, inf x∈X
(x, 0)
f (x0 ) + g(Ax 0 ) < + ∞. By Theorem 3.1, (3.2) is equivalent to the weak* closedness of the set P r X ×R (Epi
∗ ). The latter fact is equivalent to (ii) since P r X ×R (Epi
∗ ) = {(x ∗ , t)|∃(y ∗ , s) ∈ Y  × R : (x ∗ − A∗ y ∗ , s) ∈ Epi f ∗ , (y ∗ , t − s) ∈ Epi g ∗ } = Epi f ∗ + (A∗ × I )(Epi g ∗ ) = Epi f ∗ + (Epi g ∗ )A .  For various (primal) sufficient conditions for Stable Fenchel–Rockafellar duality, see [17,18]. Note that for a closed convex cone K ⊂ Y , the polar cone of K is given by K 0 := {v ∈ Y  |v(y)
0, ∀y ∈ K}. Corollary 3.2 (Stable duality for convex programs). Let A : X → Y be a continuous linear mapping; let f : X → R ∪ {+∞} be a proper and lower semi-continuous convex function. Let C ⊂ dom f be a closed convex set, b ∈ Y and let K ⊂ Y be a closed convex cone. Assume that A(C) ∩ (K + b)  = ∅. Then the following statements are equivalent: (i) inf

x∈C Ax−b∈K

{f (x) + x ∗ (x)} = maxv∈K 0 − (f + C )∗ (A∗ v − x ∗ ) + v(b), ∀x ∗ ∈ X ,

(ii) Epi(f + C )∗ + (Epi ∗b+K )A is weak* closed. Proof. By Theorem 3.2, applied to the functions f + C and g := b+K , we readily see the equivalence between (i) and (ii). 

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When the functions f and g are sublinear, we see that stable Fenchel duality is equivalent to the subdifferential sum formula. Corollary 3.3 (Subdifferential sum formula). Let f, g : X → R ∪ {+∞} be proper lower semi-continuous sublinear functions with dom f ∩ dom g  = ∅. Then the following statements are equivalent: (i) inf x∈X {f (x) + g(x) + x ∗ (x)} = maxv∈X {−f ∗ (v − x ∗ ) − g ∗ (−v)}, ∀x ∗ ∈ X . (ii) j(f + g)(x) = jf (x) + jg(x), ∀x ∈ dom f ∩ dom g. (iii) Epi f ∗ + Epi g ∗ is weak* closed. Proof. The equivalence of (i) and (iii) follows from Theorem 3.2 by taking X = Y and A = I , the identity mapping. The equivalence of (ii) and (iii) follows from Corollary 3.1 of [5].  As an application of our stable conjugate duality results, one can easily obtain subdifferential sum formulas and minimax Theorems involving a linear operator. The details are left to the reader and similar results can be found in [18].

4. Duality and polyhedral convex functions This section studies the case in which one of the convex functions is a polyhedral convex function, i.e., a function which has as epigraph a polyhedral convex set. Recall that, for f, g : Rn → R∪{+∞} convex and lower-semicontinuous functions, the classical Fenchel’s Duality Theorem [15, Theorem 31.1] states that one has inf {f (x) + g(x)} = max {−f ∗ (v) − g ∗ (−v)},

x∈X

v∈X

(4.1)

when ri(dom f ) ∩ ri(dom g)  = ∅. However, if one of these functions, say g, is polyhedral, the latter condition can be weakened to ri(dom f ) ∩ dom g  = ∅. Moreover, when both functions are polyhedral, then the relative interiors can be replaced by the domains of the functions. This fundamental result has been recently extended to arbitrary Banach spaces in [13, Theorem 3.3], where the concept of relative interior has been replaced by the strong quasi relative interior (see [11,12,18]), which is denoted by sqri(C), and defined as sqri(C) = {x ∈ C| cone(C − x) is a closed subspace}. Note that if the set C is contained in a finite dimensional space, then ri(C)=sqri(C). In the case in which both functions f and g are lower semi-continuous polyhedral convex functions on Rn , then it is well known that both sets Epi f ∗ and Epi g ∗ are closed polyhedrons, and their sum Epi f ∗ + Epi g ∗ , being also a polyhedral, is also closed. We now show that, when g is polyhedral, our closure condition holds whenever sqri (dom f ) ∩ dom g  = ∅. This condition is used in [13, Theorem 3.3], which is stated below.

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Theorem 4.1. Let f : X → R ∪ {+∞} and g : Y → R ∪ {+∞} be convex proper and lower semicontinuous functions, and let A : X → Y be a linear and continuous operator. Suppose that sqri(A(dom f )) ∩ dom g = ∅ and that g is polyhedral. Then inf {f (x) + g(Ax)} = max {−f ∗ (A
v) − g ∗ (−v)}.

x∈X

v∈X

(4.2)

Theorem 4.2. Let f : X → R ∪ {+∞} and g : Y → R ∪ {+∞} be convex proper and lower semicontinuous functions, and let A : X → Y be a linear and continuous operator. Suppose that g is polyhedral and that dom g∩sqri(A(dom f ))  = ∅. Then Epi f ∗ +(Epi g ∗ )A is weak* closed. Proof. Let x ∗ ∈ X  and define fˆ(x) := f (x) + x ∗ (x). Then, fˆ is also a proper convex function and dom fˆ =dom f , so that we still have dom g ∩sqri(A(dom fˆ))  = ∅. Moreover, for each v ∈ X , fˆ∗ (v) = f ∗ (v − x ∗ ). By Theorem 4.1, we know that if g is polyhedral and dom g ∩ sqri(A(dom fˆ))  = ∅, then inf (fˆ(x) + g(Ax)) = max∗ (−fˆ∗ (A
v) − g ∗ (−v)).

x∈X

v∈X

Re-writing the expression above we get inf (f (x) + g(Ax) + x ∗ (x)) = max∗ (−f ∗ (A
v − x ∗ ) − g ∗ (−v)).

x∈X

v∈X

Having shown the above equality for arbitrary x ∗ ∈ X  , Theorem 3.2 applies and we can conclude that Epi f ∗ + (Epi g ∗ )A is weak* closed.  A corollary of the above result is an application to convex programs with polyhedral constraints. Corollary 4.1 (Duality for convex programs with polyhedral constraints). Let A : X → Y be a continuous linear mapping; let f : X → R ∪ {+∞} be a proper and lower semicontinuous convex function. Let C be a closed convex subset of dom f , b ∈ Y and let K ⊂ Y be a polyhedral cone. Assume that A(C) ∩ (K + b)  = ∅. Under one of the following conditions: (i) Y = Rn and ri(A(C)) ∩ (b + K)  = ∅, (ii) sqri(A(C)) ∩ (b + K)  = ∅, (iii) Epi(f + C )∗ + (Epi ∗b+K )A is weak* closed, one has inf{f (x)|x ∈ C, Ax − b ∈ K} = max{−(f + C )∗ (A∗ v) + v(b)|v ∈ K 0 }.

(4.3)

Proof. By Corollary 3.2, we have that condition (iii) above implies (4.3), which is condition (i) of Corollary 3.2 for the choice x ∗ = 0. Using now Theorem 4.2 applied to the functions f + C and g := b+K , we have that both (ii) and its finite dimensional version (i) are stronger than (iii). 

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Acknowledgements The authors are grateful to the referee for his/her constructive comments on the paper and also thankful to Professor Wen Song for his comments on the preliminary version of the paper. References [1] H. Attouch, H. Brézis, Duality for the sum of convex functions in general Banach spaces, in: J.A. Barroso (Ed.), Aspects of Mathematics and its Applications, North Holland, Amsterdam, 1986, pp. 125–133. [2] H.H. Bauschke, J.M. Borwein, W. Li, Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization, Math. Progr. 86 (1999) 135–160. [3] R.S. Burachik, V. Jeyakumar, A simple closure condition for the normal cone intersection formula, Proc. Amer. Math. Soc. 133 (6) (2004) 1741–1748. [4] R.S. Burachik, V. Jeyakumar, A new geometric condition for Fenchel’s duality in infinite dimensions, Math. Progr. Ser. B, published on-line July 14, 2005. [5] R.S. Burachik, V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal. 12 (2) (2005) 279–290. [6] F. Deutsch, Best Approximation in Inner Product Spaces, Springer, New York, 2000. [7] J. Gwinner, V. Jeyakumar, Stable minimax on noncompact sets, Fixed Point Theory and Applications (Marseille, 1989), Pitman Research Notes in Mathematics Series, vol. 252, Longman Science and Techniques, Harlow, 1991, pp. 215–220. [8] J. Gwinner, J.-C. Pomerol, On weak* closedness, coerciveness, and inf-sup the theorems, Arch. Math. 52 (2) (1989) 159–167. [9] J.-B. Hiriart-Urruty, -subdifferential calculus, in: Convex Analysis and Optimization, Pitman, Boston, 1982, pp. 43–92. [10] J.-B. Hiriart-Urruty, R.R. Phelps, Subdifferential calculus using -subdifferentials, J. Funct. Anal. 118 (1993) 154–166. [11] V. Jeyakumar, Duality and infinite dimensional optimization, Nonlinear Anal. 15 (1990) 1111–1122. [12] V. Jeyakumar, H. Wolkowicz, Generalizations of Slater’s constraint qualification for infinite convex programs, Math. Progr. 57 (1) (1992) 85–102. [13] K.F. Ng, W. Song, Fenchel duality in infinite-dimensional setting and its applications, Nonlinear Anal. 55 (7–8) (2003) 845–858. [14] T. Precupanu, Closedness conditions for the optimality of a family of non-convex optimization problems, Math. Operationsforsch. Statist. Ser. Optim. 15 (1984) 339–346. [15] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [16] T. Strömberg, The operation of infimal convolution, Diss. Math. 352 (1996) 1–61. [17] C. Zalinescu, A comparison of constraint qualifications in infinite dimensional convex programming revisited, J. Austral. Math. Soc. Ser. B 40 (3) (1999) 353–378. [18] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific, London, 2002.