Nonlinear Analysis 75 (2012) 3672–3681
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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
Asymptotic closure condition and Fenchel duality for DC optimization
problems in locally convex spaces
D.H. Fang a , C. Li b,∗ , X.Q. Yang c
a
College of Mathematics and Statistics, Jishou University, Jishou 416000, China
b
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
c
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
article
info
Article history:
Received 2 September 2011
Accepted 20 January 2012
Communicated by S. Carl
MSC:
Primary 90C26
90C46
Keywords:
Fenchel duality
DC programming
Locally convex space
abstract
We consider the DC optimization problem
(P )
inf {(f1 (x) − f2 (x)) + (g1 (Ax) − g2 (Ax))},
x∈ X
where f1 , f2 , g1 and g2 are proper convex functions defined on locally convex Hausdorff
topological vector spaces X and Y respectively, and A is a linear continuous operator
from X to Y . Adopting the standard convexification technique, a Fenchel dual problem of
(P ) is given. By using properties of the epigraph of conjugate functions, some sufficient
and necessary conditions for the Fenchel duality and for the stable Fenchel duality of
(P ) are provided.
© 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Let X and Y be real locally convex Hausdorff topological vector spaces, whose respective dual spaces, X ∗ and Y ∗ , are
endowed with the weak∗ -topologies w ∗ (X ∗ , X ) and w ∗ (Y ∗ , Y ), respectively. Let f : X → R := R ∪ {+∞} and g : Y → R
be proper functions, and let A : X → Y be a linear continuous operator satisfying A(dom f ) ∩ dom g ̸= ∅. Consider the
primal problem
(P )
inf {f (x) + g (Ax)}
x∈X
and its associated Fenchel dual problem
(D )
sup {−f ∗ (−A∗ y∗ ) − g ∗ (y∗ )},
y∗ ∈Y ∗
where f ∗ and g ∗ are the Fenchel conjugates of f and g, respectively, and A∗ : Y ∗ → X ∗ stands for the adjoint operator.
It is well-known that the optimal values of these problems, v(P ) and v(D ) respectively, satisfy the so-called weak
duality, i.e., v(P ) ≥ v(D ), but a duality gap may occur, i.e. we may have v(P ) > v(D ). A challenge in convex analysis
has been to give sufficient conditions which guarantee the strong duality, i.e. the situation when there is no duality gap
and the dual problem has at least an optimal solution. In the case when f and g are proper convex functions, several
interiority-type conditions were given in order to preclude the existence of such a duality gap in different settings (see,
for instance, [1–3] and [4, Theorem 2.8.3]). Taking inspiration from Burachik and Jeyakumar [5,6], the epigraph technique
∗
Corresponding author. Tel.: +86 571 87952025; fax: +86 571 87953794.
E-mail address: cli@zju.edu.cn (C. Li).
0362-546X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2012.01.023
D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
3673
has been used extensively in many problems in optimization such as the linear semi-infinite optimization problem and the
conical programming problem and in particular in their dual problems; see for example, [1,5–23]. Especially, Li et al. [20]
considered the case when f and g are not necessarily convex and they gave some conditions ensuing the strong duality
between (P ) and (D ).
Recently, the DC (difference of two convex functions) programming problem has received much attention. As mentioned
in [11], not only from the viewpoint of optimization theory but also from the viewpoint of applications, problems of DC
programming are very important, and so they have been extensively studied in the literature; see [11–14,24–29] and
the references therein. Here we specially mention the works on duality defined via convexification techniques due to Boţ
et al. [26], Dinh et al. [13,14], and Horst and Thoai [27]. More precisely, for the optimization problem with a DC objective
function and finitely many DC inequality constraints and by using the interiority condition, Boţ et al. [26] provided a
Fenchel–Lagrange duality result and an extended Farkas lemma, and Martinez-Legaz and Volle [28] established a Lagrange
duality; while, for a conical optimization problem with a DC objective function, Dinh et al. [13,14] gave a Fenchel–Lagrange
duality result and an extended Farkas lemma in terms of the epigraph closure condition. Very recently, inspired by the
works mentioned above, we considered in [15] for the first time the optimization problem (i.e., the problem (P ) but with
f := f1 − f2 and g := g1 − g2 being two DC functions) and introduced its dual problem defined via the convexification
technique. More precisely, the primal problem is defined by
(P )
inf {f1 (x) − f2 (x) + g1 (Ax) − g2 (Ax)},
(1.1)
x∈X
while its dual problem is defined by
(D)
sup {−f1∗ (u∗ − A∗ y∗ ) + f2∗ (u∗ ) − g1∗ (y∗ + v ∗ ) + g2∗ (v ∗ )},
inf
u∗ ∈dom f ∗ y∗ ∈Y ∗
2
v ∗ ∈dom g2∗
(1.2)
where f1 , f2 : X → R, g1 , g2 : Y → R are proper convex functions. Here and throughout the whole paper, following [4, page
39] and [15], we adapt the convention that (+∞) + (−∞) = (+∞) − (+∞) = +∞, (+∞) · 0 = +∞ and (−∞) · 0 = 0.
Then, for any two proper convex functions h1 , h2 : X → R, we have that

∈R
h1 (x) − h2 (x) = −∞
= +∞
x ∈ dom h1 ∩ dom h2 ,
x ∈ dom h1 \ dom h2 ,
x ̸∈ dom h1 .
(1.3)
Hence,
h1 − h2 is proper ⇐⇒ dom h1 ⊆ dom h2 .
(1.4)
Clearly, the dual problem (D) is different from (D ). As explained in [15], the primal problem (P ) and its dual problem (D)
are strongly related to the following convex subproblems:
(P(u∗ ,v∗ ) )
inf {f1 (x) + f2∗ (u∗ ) − ⟨u∗ , x⟩ + g1 (Ax) + g2∗ (v ∗ ) − ⟨Ax, v ∗ ⟩}
x∈X
and the corresponding dual subproblems:
(D(u∗ ,v∗ ) )
sup {−(f1 − u∗ + f2∗ (u∗ ))∗ (−A∗ y∗ ) − (g1 − v ∗ + g2∗ (v ∗ ))∗ (y∗ )},
(1.5)
y∗ ∈Y ∗
where u∗ ∈ dom f2∗ and v ∗ ∈ dom g2∗ . In the special case when f2 and g2 are lower semicontinuous (lsc in brief), problems
(P ) and (D) can be reformulated equivalently as
inf
P(u∗ ,v ∗ )
(u∗ ,v ∗ )∈dom f2∗ ×dom g2∗
and
inf
(u∗ ,v ∗ )∈dom f2∗ ×dom g2∗
D(u∗ ,v ∗ ) ,
(1.6)
respectively; hence the weak C -duality holds between (P ) and (D), that is, v(P ) ≥ v(D), where v(P ) and v(D) denote
the optimal values of problems (P ) and (D), respectively. However, in the general case, the equivalent expressions about
problems (P ) and (D) are no longer true, and indeed the weak C -duality between (P ) and (D) does not necessarily hold as
shown in [15, Example 3.1].
The weak C -duality and/or the strong C -duality (in the sense that v(P ) = v(D) and the dual problem (D(u∗ ,v ∗ ) ) has an
optimal solution for any (u∗ , v ∗ ) ∈ dom f2∗ × dom g2∗ satisfying v(D(u∗ ,v ∗ ) ) = v(D)) have been studied in [15], where
sufficient and/or necessary conditions are established respectively for characterizing the weak C -duality, the strong C duality, the stable strong C -duality, and the stable total duality.
The purpose of the present paper is to study the problem of the C -duality, that is, only the situation when v(P ) = v(D).
Our main contribution in this paper is to provide complete characterizations for the C -duality and for the stable C -duality
of (P ) via the newly introduced asymptotic closure conditions.
In general, we only assume that f1 , f2 and g1 , g2 are convex functions (not necessarily lsc) and that, to avoid the triviality
in our study for (1.1),
Ω := A(dom (f1 − f2 )) ∩ dom (g1 − g2 ) ̸= ∅.
The paper is organized as follows. The next section contains some necessary notations and preliminary results. The C duality and the stable C -duality are considered in Section 3.
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2. Notations and preliminaries
The notation used in the present paper is standard (cf. [4]). In particular, we assume throughout the whole paper that
X and Y are real locally convex Hausdorff topological vector spaces, and let X ∗ denote the dual space, endowed with the
weak∗ -topology w ∗ (X ∗ , X ). By ⟨x∗ , x⟩ we shall denote the value of the functional x∗ ∈ X ∗ at x ∈ X , i.e. ⟨x∗ , x⟩ = x∗ (x). Let Z
be a set in X . The interior (resp. closure, convex cone hull) of Z is denoted by intZ (resp. cl Z , cone Z ). If W ⊆ X ∗ , then cl W
denotes the weak∗ -closure of W . For the whole paper, we endow X ∗ × R with the product topology of w ∗ (X ∗ , X ) and the
usual Euclidean topology.
Let f be a proper function defined on X . The effective domain, the conjugate function and the epigraph of f are denoted
by dom f , f ∗ and epi f respectively; they are defined by
dom f := {x ∈ X : f (x) < +∞},
f ∗ (x∗ ) := sup{⟨x∗ , x⟩ − f (x) : x ∈ X }
for each x∗ ∈ X ∗ ,
and
epi f := {(x, r ) ∈ X × R : f (x) ≤ r }.
It is well known and easy to verify that epi f ∗ is weak∗ -closed. The lsc hull of f is denoted by cl f : X → R and defined by
(cl f )(x) := sup inf h(y) for each x ∈ X ,
V ∈N (x) y∈V
where N (x) denotes the set of the neighborhoods of x. By definition, one checks that
epi(cl f ) = cl(epi f )
(cf. [30, page 7]), and by [4, Theorem 2.3.1], we have that
f ∗ = (cl f )∗ .
(2.1)
By [4, Theorem 2.3.4], if cl f is proper and convex, then the following equality holds
f ∗∗ = cl f .
(2.2)
If g , h are proper, then
epi g ∗ + epi h∗ ⊆ epi (g + h)∗
(2.3)
g ≤ h ⇒ g ∗ ≥ h∗ ⇔ epi g ∗ ⊆ epi h∗ .
(2.4)
and
Moreover, let x0 ∈ X , if g (x) = h(x + x0 ) for x ∈ X ; then by [4, Theorem 2.3.1 (vi)],
g ∗ (x∗ ) = h∗ (x∗ ) − ⟨x0 , x∗ ⟩
for each x∗ ∈ X ∗ .
(2.5)
∗
We end this section with a remark that an element p ∈ X can be naturally regarded as a function on X in such a way
that
p(x) := ⟨p, x⟩ for each x ∈ X .
(2.6)
Thus the following facts are clear for any a ∈ R and any function h : X → R:
(h + p + a)∗ (x∗ ) = h∗ (x∗ − p) − a for each x∗ ∈ X ∗ ,
(2.7)
epi(h + p + a) = epi h + (p, −a).
(2.8)
∗
∗
3. Asymptotic closure conditions and Fenchel dualities
Let X and Y be real locally convex Hausdorff topological vector spaces, and let A : X → Y be a linear continuous operator.
Let f1 , f2 : X → R and g1 , g2 : Y → R be proper convex functions such that f1 − f2 and g1 − g2 are proper functions and that
Ω = A(dom (f1 − f2 )) ∩ dom (g1 − g2 ) ̸= ∅.
Then, by (1.4), we have that
∅ ̸= dom f1 ⊆ dom f2 and ∅ ̸= dom g1 ⊆ dom g2 .
For simplicity, we denote
H ∗ := dom f2∗ × dom g2∗ .
(3.1)
D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
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To make the dual problem considered here well-defined, we further assume that cl f2 and cl g2 are proper. Then H ∗ ̸= ∅. Let
p ∈ X ∗ . Consider the following DC optimization problem with a linear perturbation:
(Pp )
inf {f1 (x) − f2 (x) + g1 (Ax) − g2 (Ax) − ⟨p, x⟩},
(3.2)
x∈X
and its dual problem
(Dp )
sup {−f1∗ (p + u∗ − A∗ y∗ ) + f2∗ (u∗ ) − g1∗ (y∗ + v ∗ ) + g2∗ (v ∗ )}.
inf
(u∗ ,v ∗ )∈H ∗ y∗ ∈Y ∗
(3.3)
In the case when p = 0, problem (Pp ) and its dual problem (Dp ) are reduced to problem (P ) (see (1.1)) and its dual problem
(D) (see (1.2)). Let v(Pp ) and v(Dp ) denote the optimal values of (Pp ) and (Dp ), respectively. Following [15], we say that the
weak C -duality holds (between (P ) and (D)) if v(D) ≤ v(P ), and that the stable weak C -duality holds (between (P ) and (D))
if the weak C -duality between (Pp ) and (Dp ) holds for each p ∈ X ∗ . This section is devoted to the study of the C -duality and
the stable C -duality between (P ) and (D), which are defined as follows.
Definition 3.1. We say that
(a) the C -duality holds (between (P ) and (D)) if v(D) = v(P );
(b) the stable C -duality holds (between (P ) and (D)) if for each p ∈ X ∗ , the C -duality holds between (Pp ) and (Dp ).
As showed in [15, Example 3.1], the weak C -duality does not necessarily hold in general. In order to characterize
completely the C -duality and the stable C -duality between (P ) and (D), we below introduce some new regularity conditions.
To this aim, we shall consider the identity operator IdR on R, and the image set (A∗ × IdR )(Z ) of a set Z ⊆ Y ∗ × R through
the map A∗ × IdR : Y ∗ × R → X ∗ × R defined by
(x∗ , r ) ∈ (A∗ × IdR )(Z ) ⇔
 ∗
∃y ∈ Y ∗ such that (y∗ , r ) ∈ Z
and A∗ y∗ = x∗ ,
(where the map A∗ × IdR was for the first time introduced in [9]). Moreover, we will make use of the following characteristic
set K defined by
K :=

epi f1∗ + (A∗ × IdR )(epi g1∗ ) − (u∗ , f2∗ (u∗ )) − (A∗ v ∗ , g2∗ (v ∗ )) .


(3.4)
(u∗ ,v ∗ )∈H ∗
Definition 3.2. The family (f1 , f2 , g1 , g2 ; A) is said to satisfy
(a) the semi-(FRC ) ((SFRC )) if
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R) ⊇ K ∩ ({0} × R);
(3.5)
(b) the semi-(CC ) ((SCC )) if
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ⊇ K ;
(3.6)
(c) the asymptotic (FRC ) ((AFRC )) if
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R) = cl[K ∩ ({0} × R)];
(3.7)
(d) the asymptotic (CC ) ((ACC )) if
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ = cl K .
(3.8)
Remark 3.1. The notions of the (SFRC ) and the (SCC ) were introduced in [15] for the weak C -duality and the stable weak
C -duality. Moreover, in order to characterize the strong C -duality (that is, v(P ) = v(D) and for each (u∗ , v ∗ ) ∈ H ∗ satisfying
v(D(u∗ ,v∗ ) ) = v(D), the dual problem (D(u∗ ,v∗ ) ) has an optimal solution, where D(u∗ ,v∗ ) is defined by (1.5)), Fang et al. [15,
Definitions 3.3 and 4.1] introduced the notions of the (FRC ) and the (CC ) which are defined as follows:
(FRC ) epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R) = K ∩ ({0} × R)
and
(CC ) epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ = K .
By definition, it is easy to see that
the (FRC ) H⇒ the (AFRC ) H⇒ the (SFRC )
and
the (CC ) H⇒ the (ACC ) H⇒ the (SCC ).
(3.9)
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D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
The following two examples show that each of the converse implications in (3.9) does not hold in general. This means
that the notions of the (CC ), the (ACC ) and the (SCC ) are mutually different.
Example 3.1. Let X = Y =: R2 and let A be the identity. Consider the functions f1 , f2 , g1 , g2 : R2 → R as defined in
[15, Example 3.5], that is, f1 := δA and g1 := δB and f2 = g2 := 0, where
A := {(x − 1, y) ∈ R2 : x2 + y2 ≤ 1}
and
B := {(1 − x, −y) ∈ R2 : x2 + y2 ≤ 1}.
Then, by [15, Example 3.5],
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ = R2 × [0, +∞)
and
K = {(r1 , r2 , α) : r2 = 0, α ≥ 0} ∪ {(r1 , r2 , α) : r2 ̸= 0, α > 0}.
Hence,
cl K = R2 × [0, +∞) = epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ̸= K .
This means that the (ACC ) and the (SCC ) hold but the (CC ) does not hold. Therefore, the implications ‘‘the (SCC ) H⇒ the
(CC )’’ and ‘‘the (ACC ) H⇒ the (CC )’’ do not hold.
Example 3.2. Let X = Y =: R and let A be the identity. Define the functions f1 , f2 , g1 , g2 : R → R respectively by
0
1

f1 (x) :=
+∞
x < 0,
x = 0,
x > 0,

+∞
g1 (x) :=
1
0
x < 0,
x = 0,
x>0
and f2 = g2 := 0.
Then f1 − f2 + g1 ◦ A − g2 ◦ A = 2 + δ{0} and
epi(f1 − f2 + g1 ◦ A − g2 ◦ A)∗ = R × [−2, +∞).
(3.10)
Furthermore, one checks by the definition of K that K = R × [0, +∞). This together with (3.10) implies that
K ⊆ epi(f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ̸= cl K .
Hence, the (SCC ) holds but the (ACC ) does not hold.
Proposition 3.1 was proved in [15, Theorems 3.7 and 4.3] and provides the sufficient and necessary conditions for the
weak C -duality and the stable weak C -duality to hold.
Proposition 3.1. (i) The weak C -duality holds if and only if the family (f1 , f2 , g1 , g2 ; A) satisfies the (SFRC ).
(ii) The stable weak C -duality holds if and only if the family (f1 , f2 , g1 , g2 ; A) satisfies the (SCC ).
The following lemma is a slight extension of [15, Lemma 3.4], the proof of which is a direct application (to f1 (·) − ⟨p, ·⟩
in place of f1 (·)) of [15, Lemma 3.4], together with (2.8).
Lemma 3.1. Let p ∈ X ∗ and r ∈ R. Then the following assertions hold.
(i) (p, r ) ∈ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ if and only if v(Pp ) ≥ −r.
(ii) (p, r ) ∈ K if and only if v(Dp ) ≥ −r and for each (u∗ , v ∗ ) ∈ H ∗ there is y∗ ∈ Y ∗ satisfying
− f1∗ (p + u∗ − A∗ y∗ ) + f2∗ (u∗ ) − g1∗ (y∗ + v ∗ ) + g2∗ (v ∗ ) ≥ −r .
(3.11)
Our first theorem provides a complete characterization in terms of the condition (AFRC ) for the C -duality to hold.
Theorem 3.1. The C -duality holds between (P ) and (D) if and only if the family (f1 , f2 , g1 , g2 ; A) satisfies the (AFRC ).
Proof. Suppose that the C -duality holds between (P ) and (D). Then v(P ) = v(D) and, by Proposition 3.1(i), the family
(f1 , f2 , g1 , g2 ; A) satisfies the (SFRC ). Since epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R) is w ∗ -closed, it follows that
cl (K ∩ ({0} × R)) ⊆ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R).
(3.12)
To verify the converse inclusion, let (0, r ) ∈ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ . By Lemma 3.1(i), v(P ) ≥ −r and so
v(D) = v(P ) ≥ −r by the C -duality. Let ϵ > 0. Then, for each (u∗ , v ∗ ) ∈ H ∗ , there exists y∗ ∈ Y ∗ such that
−f1∗ (u∗ − A∗ y∗ ) + f2∗ (u∗ ) − g1∗ (y∗ + v ∗ ) + g2∗ (v ∗ ) ≥ −r − ϵ,
D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
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which implies that (0, r + ϵ) ∈ K ∩ ({0} × R) thanks to Lemma 3.1(ii). Hence, (0, r ) ∈ cl (K ∩ ({0} × R)), which shows the
converse inclusion of (3.12). Therefore, the (AFRC ) holds.
Conversely, suppose that the family (f1 , f2 , g1 , g2 ; A) satisfies the (AFRC ). Then by Remark 3.1, the (SFRC ) holds. Hence,
by Proposition 3.1(i), we have that v(D) ≤ v(P ). To show the converse inequality, suppose on the contrary that v(P ) > v(D).
Then there exists r ∈ R such that v(P ) > −r > v(D). This implies that (0, r ) ∈ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ , thanks
to Lemma 3.1(i). Since the family (f1 , f2 , g1 , g2 ; A) satisfies the (AFRC ), it follows that (0, r ) ∈ cl [K ∩ ({0} × R)]. Therefore,
there exists a net {(0, rτ )} ⊆ K such that (0, rτ ) → (0, r ). Hence, by Lemma 3.1(ii), for each (u∗ , v ∗ ) ∈ H ∗ there is y∗ ∈ Y ∗
such that
− f1∗ (u∗ − A∗ y∗ ) + f2∗ (u∗ ) − g1∗ (y∗ + v ∗ ) + g2∗ (v ∗ ) ≥ −rτ → −r .
(3.13)
This together with the definition of v(D) implies that v(D) ≥ −r, which contradicts with v(D) < −r. Hence, v(P ) ≤ v(D)
and so v(P ) = v(D). The proof is complete. For the following proposition on the characterization of the (ACC ), we recall the upper semicontinuous hull of a proper
function on X . Let h : X → R be a proper function. The upper semicontinuous hull of h is defined by
lim sup h(y) :=
y→x
inf sup h(y)
V ∈N (x) y∈V
for each x ∈ X ,
where N (x) denotes the set of the neighborhoods of x. Then h is said to be upper semicontinuous (usc) at x0 ∈ X if
lim supx→x0 h(x) = h(x0 ), and usc on X if h is usc at each point on X . Clearly,
h is usc at x0 ∈ dom h ⇐⇒ lim sup h(x) ≤ h(x0 ).
(3.14)
x→x0
Proposition 3.2. The function p → v(Pp ) is usc on X ∗ .
Proof. Let p0 ∈ X ∗ . By (3.14), we only need to show that lim supp→p0 v(Pp ) ≤ v(Pp0 ). To do this, let r < lim supp→p0 v(Pp )
be arbitrary and let V ∈ N (p0 ) be fixed. Then, there exists pV ∈ V such that r < v(PpV ), that is,
f1 (x) − f2 (x) + g1 (Ax) − g2 (Ax) − ⟨pV , x⟩ ≥ r
for each x ∈ X .
(3.15)
In particular, fixing x ∈ X and ϵ > 0, we choose Vx,ϵ := {p ∈ X : |⟨p − p0 , x⟩| < ϵ}. Then Vx,ϵ ∈ N (p0 ) and
⟨p0 , x⟩ < ⟨pVx,ϵ , x⟩ + ϵ for each pVx,ϵ ∈ Vx,ϵ . It follows from (3.15) (applied to pVx,ϵ in place of pV ) that
∗
f1 (x) − f2 (x) + g1 (Ax) − g2 (Ax) − ⟨p0 , x⟩ ≥ r − ϵ.
Hence, v(Pp0 ) ≥ r as x ∈ X and ϵ > 0 are arbitrary and we complete the proof.
Proposition 3.3. The family (f1 , f2 , g1 , g2 ; A) satisfies the (ACC ) if and only if
lim sup v(Dq ) = v(Pp )
q→ p
for each p ∈ X ∗ .
(3.16)
Proof. Suppose that the family (f1 , f2 , g1 , g2 ; A) satisfies the (ACC ). Let p ∈ X ∗ . Then, by Remark 3.1 and Proposition 3.1(ii),
we have that v(Dq ) ≤ v(Pq ) for each q ∈ X ∗ ; hence
lim sup v(Dq ) ≤ lim sup v(Pq ).
q→ p
(3.17)
q→ p
Since the function q → v(Pq ) is usc on X ∗ by Proposition 3.2, it follows that
lim sup v(Dq ) ≤ v(Pp ).
(3.18)
q→ p
To verify the converse inequality, let r < v(Pp ). Hence, by Lemma 3.1(i), (p, −r ) ∈ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ and,
(p, −r ) ∈ cl K by the (ACC ). Therefore, (V ×(−r −δ, −r +δ))∩K ̸= ∅ for each V ∈ N (p) and each δ > 0. Fix V ∈ N (p), δ > 0
and choose (pV , −rδ ) ∈ (V × (−r − δ, −r + δ)) ∩ K . Then, applying Lemma 3.1(ii), we have that v(DpV ) ≥ rδ ≥ r − δ , and
consequently,
inf sup v(Dp ) ≥
V ∈N (p) p∈V
inf v(DpV ) ≥ sup(r − δ) ≥ r .
V ∈N (p)
δ>0
This shows that lim supq→p v(Dq ) ≥ r, and so lim supq→p v(Dq ) ≥ v(Pp ) as r < v(Pp ) is arbitrary. Thus lim supq→p v(Dq ) =
v(Pp ) is proved by (3.18).
Conversely, suppose that (3.16) holds. Then, by definition, v(Dp ) ≤ v(Pp ) for each p ∈ X ∗ . Thus, by Proposition 3.1(ii),
the family (f1 , f2 , g1 , g2 ; A) satisfies the (SCC ). Since epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ is weak∗ -closed, it follows that
cl K ⊆ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ .
(3.19)
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D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
Thus to complete the proof, it suffices to verify the converse inclusion. For this end, let (p, r ) ∈ epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ .
Then v(Pp ) ≥ −r by Lemma 3.1(i), and so lim supq→p v(Dq ) = v(Pp ) ≥ −r by (3.16). Let ϵ > 0 and let V ∈ N (p). By
definition, there exists pV ∈ V such that v(DpV ) ≥ −r − ϵ holds and, consequently, for each (u∗ , v ∗ ) ∈ H ∗ , there exists
y∗ ∈ Y ∗ such that
−f1∗ (pV + u∗ − A∗ y∗ ) + f2∗ (u∗ ) − g1∗ (y∗ + v ∗ ) + g2∗ (v ∗ ) ≥ −r − ϵ,
which implies that (pV , r + ϵ) ∈ K thanks to Lemma 3.1(ii). Hence, (p, r ) ∈ cl K and the converse inclusion of (3.19) is
showed. Thus, the proof is complete. Our second main theorem below provides some complete characterizations for the stable C -duality between (P ) and (D).
Theorem 3.2. The following assertions are equivalent.
(i)
(ii)
(iii)
(iv)
The family (f1 , f2 , g1 , g2 ; A) satisfies the (ACC ) and the function p → v(Dp ) is usc on X ∗ .
For each p ∈ X ∗ , the family (f1 − p, f2 , g1 , g2 ; A) satisfies the (AFRC ).
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ = ∪p∈X ∗ cl[K ∩ ({p} × R)].
The stable C -duality holds between (P ) and (D).
Proof. (i) ⇒ (ii) Suppose that (i) holds. Let p0 ∈ X ∗ . Then by Proposition 3.3, we have that v(Pp0 ) = lim supp→p0 v(Dp ).
This together with the assumed upper semicontinuity implies that v(Pp0 ) = v(Dp0 ), that is the C -duality holds between
(Pp0 ) and (Dp0 ). Thus, applying Theorem 3.1 to f1 (·) − ⟨p0 , ·⟩ in place of f1 (·), we have that the family (f1 − p0 , f2 , g1 , g2 ; A)
satisfies the (AFRC ) and (ii) is seen to hold because p0 ∈ X ∗ is arbitrary.
(ii) ⇒ (iii) Suppose that (ii) holds. Let p ∈ X ∗ and let K (p) be the set defined by
K (p) :=

epi (f1 − p)∗ + (A∗ × IdR )(epi g1∗ ) − (u∗ , f2∗ (u∗ )) − (A∗ v ∗ , g2∗ (v ∗ )) .


(u∗ ,v ∗ )∈H ∗
Then by (ii), the following equality holds
epi (f1 − p − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R) = cl[K (p) ∩ ({0} × R)].
(3.20)
Noting by (2.8) that
K (p) = K + (−p, 0),
we have that
K (p) ∩ ({0} × R) = K ∩ ({p} × R) + (−p, 0).
(3.21)
Moreover, using (2.8), we conclude that
epi (f1 − p − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({0} × R) = epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({p} × R) + (−p, 0).
(3.22)
This together with (3.20) and (3.21) implies that
epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({p} × R) = cl[K ∩ ({p} × R)].
(3.23)
Hence,

p∈X ∗
cl[K ∩ ({p} × R)] =

epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ ∩ ({p} × R)

p∈X ∗
= epi (f1 − f2 + g1 ◦ A − g2 ◦ A)∗ .
(3.24)
Therefore, (iii) holds and implication (ii) ⇒ (iii) is proved.
(iii) ⇒ (iv) Suppose that (iii) holds and let p ∈ X ∗ . Then we have that
cl[K ∩ ({p} × R)] ∩ cl[K ∩ ({q} × R)] = ∅ for each q ∈ X ∗ \ {p}.
Therefore,

cl[K ∩ ({q} × R)] ∩ ({p} × R) =
q∈X ∗

cl[K ∩ ({q} × R)] ∩ cl[{p} × R]
q∈X ∗
=

cl[K ∩ ({q} × R)] ∩ cl[{p} × R]

q∈X ∗
= cl[K ∩ ({p} × R)].
Combining this and (iii), one sees that (3.23) holds. This together with (3.21) and (3.22) implies that (3.20) holds, that is, the
family (f1 − p0 , f2 , g1 , g2 ; A) satisfies the (AFRC ). Since p0 ∈ X ∗ is arbitrary, it follows from Theorem 3.1 (f1 (·) − ⟨p0 , ·⟩ in
place of f1 (·)) that (iv) holds.
D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
3679
(iv) ⇒ (i) Suppose that (iv) holds. Then
v(Pp ) = v(Dp ) for each p ∈ X ∗ .
(3.25)
Note by Proposition 3.2 that the function p → v(Pp ) is usc, it follows that the function p → v(Dp ) is also usc. Moreover, by
(3.25), we have that (3.16) holds. Therefore, the family (f1 , f2 , g1 , g2 ; A) satisfies the (ACC ), thanks to Proposition 3.3. The
proof is complete. Remark 3.2. As showed in Example 3.4, the (ACC ) and the stable C -duality are not equivalent in general even in the case
when f1 , f2 , g1 and g2 are lsc.
Let h : Y → R be a proper convex function and consider the following set
Kh∗ := {y∗ ∈ Y ∗ : ⟨y∗ , y⟩ ≤ h(y) for any y ∈ Y }.
Then Kh∗ is w ∗ -closed. Furthermore, it was proved in [31, Theorem 7A] that Kh∗ is w ∗ -compact if Y is a Banach space, h : Y → R
is lsc and continuous at 0. The following lemma does not necessarily require that Y is a Banach space and, even in the Banach
space setting, extends the above result in [31, Theorem 7A].
Lemma 3.2. Let h : Y → R be a proper convex function. Then Kh∗ is w ∗ -compact if one of the following conditions is satisfied:
(a) Y is a Banach space and 0 ∈ int(dom h).
(b) h is continuous at 0.
Proof. We first assume the condition (a) holds, that is Y is a Banach space and 0 ∈ int(dom h). By the well known
Banach–Alaoglu Theorem (cf. [32, Theorem 3.5, page 125]), it suffices to verify that Kh∗ is bounded. To do this, note that
cone(domh) = Y by the assumption 0 ∈ int(dom h). It follows that for each y∗ ∈ Kh∗ one has that
|⟨y∗ , y⟩| < +∞ for each y ∈ Y .
(3.26)
∗
This implies that Kh is bounded. Thus the proof for the case when Y is a Banach space is complete.
We now assume that h is continuous at 0. Then there exists a neighborhood U ∈ N (0) such that
|h(y)| ≤ |h(0)| + 1 for each y ∈ U .
Let U := {y∗ ∈ Y ∗ : ⟨y∗ , y⟩ ≤ 1 for all x ∈ U }. Then Kh∗ ⊆ (|h(0)| + 1)U 0 . Moreover, by the well known Alaoglu–Bourbaki
Theorem (cf. [33, page 66]), U 0 is w ∗ -compact. Hence, Kh∗ is w ∗ -compact as it is w ∗ -closed, and the proof is complete. 0
The following proposition provides some sufficient conditions ensuring the upper semicontinuity of the function p →
v(Dp ) on X ∗ .
Proposition 3.4. Suppose that one of the following conditions is satisfied.
(a) Y is a Banach space and A(dom f1 ) ∩ int(dom g1 ) ̸= ∅.
(b) A(dom f1 ) ∩ cont g1 ̸= ∅, where cont g1 denotes the set of all continuity points of g1 .
Then the function p → v(Dp ) is usc on X ∗ .
Proof. Let y0 ∈ A(dom f1 )∩int(dom g1 ) if assumption (a) holds and y0 ∈ A(dom f1 )∩cont g1 if assumption (b) holds. Without
loss of generality, we may assume that y0 = 0 (using the translation of independent variable of functions if necessary). To
show the upper semicontinuity of the function p → v(Dp ) on X ∗ , by (3.14), it is sufficient to show that
lim sup v(Dp ) ≤ v(Dp0 )
p→p0
for each p0 ∈ X ∗ .
(3.27)
To do this, let p0 ∈ X ∗ and let r < lim supp→p0 v(Dp ). Let ϵ > 0 and let (u∗0 , v0∗ ) ∈ H ∗ be such that
sup {−f1∗ (p0 + u∗0 − A∗ y∗ ) + f2∗ (u∗0 ) − g1∗ (y∗ + v0∗ ) + g2∗ (v0∗ )} ≤ v(Dp0 ) + ϵ.
(3.28)
y∗ ∈Y ∗
Consider the function F : X ∗ × Y ∗ → R defined by
F (p, y∗ ) := −f1∗ (p + u∗0 − A∗ y∗ ) + f2∗ (u∗0 ) − g1∗ (y∗ + v0∗ ) + g2∗ (v0∗ ) for each (p, y∗ ) ∈ X ∗ × Y ∗ .
(3.29)
Then we have by (3.28) that
sup F (p0 , y∗ ) ≤ v(Dp0 ) + ϵ
(3.30)
y∗ ∈Y ∗
and by definition that
v(Dp ) ≤ sup F (p, y∗ ) for any p ∈ X ∗ .
y∗ ∈Y ∗
(3.31)
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Let V ∈ N (p0 ) be fixed. Then, by definition, there exist pV ∈ V and y∗V ∈ Y ∗ such that
r ≤ v(DpV ) +
ϵ
2
≤ sup F (pV , y∗ ) +
y∗ ∈Y ∗
ϵ
2
≤ F (pV , y∗V ) + ϵ.
(3.32)
Note by the definition of conjugate functions and (3.29), we have that
F (pV , y∗V ) ≤ f1 (0) + f2∗ (u∗0 ) + g2∗ (v0∗ ) +
inf
y∈dom g1
{g1 (y) − ⟨y∗V + v0∗ , y⟩}.
This together with (3.32) implies that
r ≤ f1 (0) + f2∗ (u∗0 ) + g2∗ (v0∗ ) + ϵ +
inf
y∈dom g1
{g1 (y) − ⟨y∗V + v0∗ , y⟩}.
(3.33)
Now we define the function h : Y → R by
h(y) := g1 (y) − ⟨v0∗ , y⟩ − r + f1 (0) + f2∗ (u∗0 ) + g2∗ (v0∗ ) + ϵ
for each y ∈ Y .
Then h is a proper convex function with 0 ∈ int(dom h) because dom h = dom g1 , and, by assumption (a) or (b), Lemma 3.2
is applicable to concluding that Kh∗ is w ∗ -compact. Further, we have that (3.33) implies that {y∗V ∈ Y ∗ : V ∈ N (p0 )} ⊆ Kh∗ ;
consequently {y∗V ∈ Y ∗ : V ∈ N (p0 )} is w ∗ -compact. Since pV ∈ V for each V ∈ N (p0 ), it follows that there exist y∗0 ∈ Y ∗
and a net {Vτ } ⊆ N (p0 ) such that
w∗ − lim pVτ = p0 and w ∗ − lim y∗Vτ = y∗0 .
τ
(3.34)
τ
Noting that A∗ is continuous, and f1∗ and g1∗ are lsc, it is routine to check that F is usc at (p0 , y∗0 ). This together with (3.34)
implies, without loss of generality, that
F (pVτ , y∗Vτ ) ≤ F (p0 , y∗0 ) + ϵ ≤ v(Dp0 ) + 2ϵ
for each τ ,
(3.35)
where the last inequality holds by (3.30). Combining this and (3.32), we see that
r ≤ F (pVτ , y∗Vτ ) + ϵ ≤ v(Dp0 ) + 3ϵ.
Hence, by the arbitrariness of ϵ > 0, one has that r ≤ v(Dp0 ) and so (3.27) is showed because p0 ∈ X ∗ and r <
lim supp→p0 v(Dp ) are arbitrary. The proof is complete. The following example shows that neither condition (a) nor condition (b) is necessary for the upper semicontinuity of
the function p → v(Dp ).
Example 3.3. Let X = Y := R, and let A be the identity. Defined f1 , f2 , g1 , g2 : R → R by f1 := δ{0} g1 := δ(0,+∞) , and
f2 = g2 := 0. Then f1 , f2 , g1 and g2 are proper convex functions. Clearly,
f1∗ = 0,
g1∗ = δ(−∞,0] ,
and
f2∗ = g2∗ = δ{0} .
Hence, dom f2∗ = dom g2∗ = {0}. Therefore, for each p ∈ X ∗ ,
v(Dp ) = sup {−f1∗ (p − y∗ ) − g1∗ (y∗ )} = 0.
y∗ ∈R
This implies that the function p → v(Dp ) is usc. However, none of conditions (a) and (b) in Proposition 3.4 holds because
dom f1 ∩ cont g1 = dom f1 ∩ int(dom g1 ) = {0} ∩ (0, +∞) = ∅.
By Proposition 3.4 and the equivalence of (i) and (iv) of Theorem 3.2, we get the following theorem straightforwardly.
Theorem 3.3. Suppose that one of the conditions (a) and (b) in Proposition 3.4 is satisfied. Then the stable C -duality holds if and
only if the family (f1 , f2 , g1 , g2 ; A) satisfies the (ACC ).
The following example shows that, in general, the equivalence between the stable C -duality and the (ACC ) does not
necessarily hold, if both the conditions (a) and (b) in Proposition 3.4 are dropped.
Example 3.4. Let X = Y := l2 , the Hilbert space consisting of all sequences x = (xn )n∈N such that n=1 x2n < +∞, and let
A be the identity. Consider the functions f1 , g1 : l2 → R as in [34, Example 3.3], that is, f1 := δA , g1 (x) := x1 + δB (x) for each
x ∈ l2 , where,
∞
A := {x ∈ l2 : x2n−1 + x2n = 0 for each n ∈ N}
and
B := {x ∈ l2 : x2n + x2n+1 = 0 for each n ∈ N}.
D.H. Fang et al. / Nonlinear Analysis 75 (2012) 3672–3681
3681
Then, f1 and g1 are proper, convex and lsc functions. By [9, Theorem 2.1] (see also [23, Lemma 2.1]), one has that
epi (f1 + g1 )∗ = cl(epi f1∗ + epi g1∗ ).
(3.36)
Furthermore, we choose f2 = g2 := 0. Then K = epi f1∗ + epi g1∗ by definition, and so the (ACC ) holds thanks to (3.36).
However the stable C -duality does not hold because v(P ) = 0 and v(D) = −∞ by [34, Example 3.3]. Therefore, the stable
C -duality and the (ACC ) are not equivalent.
Acknowledgments
We thank the two anonymous referees for their comments to help in improving the presentation of this paper. The first
author was supported in part by the National Natural Science Foundation of China (grant 11101186). The second author was
supported in part by the National Natural Science Foundation of China (grant 11171300) and by Zhejiang Provincial Natural
Science Foundation of China (grant Y6110006). The third author was supported in part by the Research Grants council of
Hong Kong (PolyU 5334/08E).
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
R.I. Boţ, E.R. Csetnek, G. Wanka, Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim. 19 (2008) 217–233.
R.I. Boţ, S.M. Grad, G. Wanka, Duality in Vector Optimizaition, Springer-Verlag, 2009.
R.B. Holmes, Geometric Functional Analysis, Springer-Verlag, Berlin, 1975.
C. Zǎlinescu, Convex Analysis in General Vector Spaces, World Scientific, New Jersey, 2002.
R.S. Burachik, V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal. 12 (2005) 279–290.
R.S. Burachik, V. Jeyakumar, A simpler closure condition for the normal cone intersection formula, Proc. Amer. Math. Soc. 133 (2005) 1741–1748.
R.I. Boţ, S.M. Grad, G. Wanka, On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl. 337 (2008) 1315–1325.
R.I. Boţ, S.M. Grad, G. Wanka, New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces, Nonlinear Anal.
69 (2008) 323–336.
R.I. Boţ, G. Wanka, A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal. 64
(2006) 2787–2804.
R.S. Burachik, V. Jeyakumar, Z.Y. Wu, Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal. 64 (2006) 1998–2006.
N. Dinh, B. Mordukhovich, T.T.A. Nghia, Qualification and optimality conditions for convex and DC programs with infinite constraints, Acta Math.
Vietnam. 34 (2009) 125–155.
N. Dinh, B. Mordukhovich, T.T.A. Nghia, Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite
programs, Math. Program. 123 (2010) 101–138.
N. Dinh, T.T.A. Nghia, G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization 59 (2010) 541–560.
N. Dinh, G. Vallet, T.T.A. Nghia, Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal. 15 (2008) 235–262.
D.H. Fang, C. Li, X.Q. Yang, Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM J. Optim. 21 (2011) 730–760.
M.A. Goberna, V. Jeyakumar, M.A. López, Necessary and sufficient conditions for solvability of systems of infinite convex inequalities, Nonlinear Anal.
68 (2008) 1184–1194.
V. Jeyakumar, The strong conical hull intersection property for convex programming, Math. Program. Ser. A 106 (2006) 81–92.
V. Jeyakumar, N. Dinh, G.M. Lee, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex
programs, SIAM J. Optim. 14 (2003) 534–547.
V. Jeyakumar, G.M. Lee, Complete characterization of stable Farkas’ lemma and cone-convex programming duality, Math. Program. Ser. A 114 (2008)
335–347.
C. Li, D.H. Fang, G. López, M.A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM J. Optim. 20
(2009) 1032–1051.
C. Li, K.F. Ng, T.K. Pong, The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear spaces, SIAM J. Optim.
18 (2007) 643–665.
C. Li, K.F. Ng, T.K. Pong, Constraint qualifications for convex inequality systems with applications in constrained optimization, SIAM J. Optim. 19 (2008)
163–187.
D.H. Fang, C. Li, K.F. Ng, Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming, SIAM J.
Optim. 20 (2009) 1311–1332.
L.T.H. An, P.D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization
problems, Ann. Oper. Res. 133 (2005) 23–46.
R.I. Boţ, G. Wanka, Duality for multiobjective optimization problems with convex objective functions and DC constraints, J. Math. Anal. Appl. 315
(2006) 526–543.
R.I. Boţ, I.B. Hodrea, G. Wanka, Some new Farkas-type results for inequality systems with DC functions, J. Global Optim. 39 (2007) 595–608.
R. Horst, N.V. Thoai, DC programming: overview, J. Optim. Theory Appl. 103 (1999) 1–43.
J.E. Martinez-Legaz, M. Volle, Duality in DC programming: the case of several constraints, J. Math. Anal. Appl. 237 (1999) 657–671.
J.F. Toland, Duality in nonconvex optimization, J. Math. Anal. Appl. 66 (1978) 399–415.
T. Strömberg, The operation of infimal convolution, Disscuss. Math. 352 (1996) 1–61.
R.T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966) 46–63.
T.J. Morrison, Function Analysis: An Introduction to Banach Space Theory, Wiley-Interscience, 2000.
W. Rudin, Functional Analysis, second ed., McGraw-Hill, MA, Boston, 1991.
M.S. Gowda, M. Teboulle, A comparison of constraint qualifications in infinite dimensional convex programming, SIAM J. Control Optim. 28 (1990)
925–935.