J Optim Theory Appl https://doi.org/10.1007/s10957-018-1219-3

Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions D. H. Fang1 · Y. Zhang1

Received: 28 July 2017 / Accepted: 9 January 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract The paper is devoted to the study of a new class of conic constrained optimization problems with objectives given as differences of a composite function and a convex function. We first introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we provide necessary and sufficient conditions for several versions of Farkas lemmas to hold. Similarly, we provide characterizations for conic constrained optimization problems to have the strong or stable strong dualities such as Lagrange, Fenchel–Lagrange or Toland–Fenchel–Lagrange duality. Keywords Farkas lemma · Strong duality · Composite functions · Constraint qualifications · Conic programming Mathematics Subject Classification 90C26 · 49N15 · 46N10

1 Introduction Farkas-type results for convex systems and the dual method are fundamental in nonlinear programming and in other fields such as game theory, set containment problems, etc., and have played very important roles in nonlinear programming. Usually, for the Farkas lemma for convex systems, one seeks conditions characterizing families

B

D. H. Fang [email protected] Y. Zhang [email protected]

1

College of Mathematics and Statistics, Jishou University, Jishou 416000, People’s Republic of China

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of inequalities which are consequences of a consistent convex system, while a main target of the dual method is the establishment of the so-called strong duality, which means that the values of the primal problem and the dual problem coincide, and the dual problem has at least an optimal solution. The literature on these areas is very rich, see, for example, [1–11] regarding cone-convex systems and [12–16] regarding systems defined by an arbitrary family of convex inequalities. Indeed, in mathematical programming, many of the problems naturally involve nonconvex and noncontinuous functions, for example, the difference of convex functions (DC) programming problem and the composite problem. It is well known that many real-world problems can be recast into a DC programming problem or a composite optimization problem, and hence, the study of these optimization problems has received much attention in the literatures, see, for example [12,13,17–28] and references therein. Here we just mention some work of [19–22]. In [19,20], the authors established various Farkas-type results for a system involving a general cone-convex constraint, a geometrical constraint and a constraint defined by a DC function. In [21], by using some closedness conditions, the authors proposed some extended stable Farkas lemmas for the conic constrained composite optimization problem and gave a sufficient and necessary condition to guarantee the strong duality between this composite optimization problem and its dual problem holds. Recently, in [22], the authors considered the optimization problem with objectives given as the difference of two composite functions and constraints described by an arbitrary number of convex inequalities, and proposed some Farkas lemmas and established the dualities between this optimization problem and its dual problem. Motivated by the works mentioned above, we are devoted to establish various Farkas-type results in this paper for a system involving a general cone-convex constraint, a geometrical constraint and a constraint defined by the difference of a composite function and a convex function. Then, these Farkas-type results are used to study the duality of a problem of minimizing a DC function involving a composite function under a convex-cone constraint and a set constraint (problem (Pg ) in Sect. 4). Our main aim in the present paper is to give some new constraint qualifications involving epigraphs which completely characterize the various Farkas-type results and the strong (stable strong) duality for problem (P). In general, we do not impose any topological assumption on the involving functions and the involving set, that is, the set is not necessarily closed, the functions are not necessarily lower semicontinuous (lsc in brief). This paper is organized as follows. The next section contains some necessary notations and preliminary results. In Sect. 3, some new constraint qualifications are provided and several relationships among them are given. By using the new constraint qualifications, some new versions of Farkas lemma for systems involving convex, composite and DC functions are given. In Sect. 4, we introduce the Lagrange dual problem, the Fenchel–Lagrange dual problem and the Toland–Fenchel–Lagrange dual problem, and provide some necessary and sufficient conditions for the strong dualities and the stable strong dualities to hold. Applications to convex conic programming and to DC programming are also given in Sect. 5, which extend and improve some recent known results in [2,19,20].

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2 Notations and Preliminary Results The notation used in the present paper is standard (cf. [29]). In particular, we assume throughout the whole paper that X , Y and Z are real locally convex Hausdorff topological vector spaces, and X ∗ , Y ∗ and Z ∗ denote the dual space of X , Y and Z , endowed with the weak∗ -topology w ∗ (X ∗ , X ), w ∗ (Y ∗ , Y ) and w ∗ (Z ∗ , Z ), respectively. Let S ⊆ Y and K ⊆ Z be closed convex cones and Y , Z be partially ordered by S and K , respectively. Denote Y • := Y ∪ {∞Y } and Z • := Z ∪ {∞ Z }, where {∞Y } and {∞ Z } are the greatest elements with respect to the partial orders ≤ S and ≤ K , respectively. Define an order on Y (resp. Z ) by saying that y ≤ S x (resp. y ≤ K x) if y − x ∈ −S (resp. y − x ∈ −K ). The following operations are defined on Y • (resp. Z • ): for any y ∈ Y (resp. Z ), y + ∞ = ∞ + y = ∞ and t∞ = ∞ for any t ≥ 0. By x ∗ , x we denote the value of the functional x ∗ ∈ X ∗ at x ∈ X , i.e., x ∗ , x = x ∗ (x). Let D be a nonempty subset in X . The interior (resp. closure ) of D is denoted by int D (resp. cl D). If D ⊆ X ∗ , then cl D denotes the weak∗ -closure of D. For the whole paper, we endow X ∗ × R with the product topology of w ∗ (X ∗ , X ) and the usual Euclidean topology. The positive polar cone D ⊕ and the indicator function δ D of the nonempty set D are defined, respectively, by D ⊕ := {x ∗ ∈ X ∗ : x ∗ , x ≥ 0 for all x ∈ D} and  δ D (x) :=

0, x ∈ D, +∞, otherwise.

Let f : X → R := R ∪ {+∞} be a proper convex function. The effective domain and the epigraph of f are, respectively, defined by dom f := {x ∈ X : f (x) < +∞} and epi f := {(x, r ) ∈ X × R : f (x) ≤ r }. As usual, the conjugate function f ∗ : X ∗ → R of f is defined by f ∗ (x ∗ ) := sup{ x ∗ , x − f (x) : x ∈ X } for each x ∗ ∈ X ∗ . Clearly, f ∗ is a proper convex lsc function and epi f ∗ is weak∗ -closed. By definition, the Young–Fenchel inequality below holds: f (x) + f ∗ (x ∗ ) ≥ x, x ∗ for each pair (x, x ∗ ) ∈ X × X ∗ .

(1)

If g, h are proper convex functions with domg ∩ domh = ∅, then

and

epi g ∗ + epi h ∗ ⊆ epi (g + h)∗ ,

(2)

g ≤ h ⇒ g ∗ ≥ h ∗ ⇔ epi g ∗ ⊆ epi h ∗ .

(3)

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Moreover, for each p ∈ X ∗ and r ∈ R, the following facts are clear: ( f + p + r )∗ (x ∗ ) = f ∗ (x ∗ − p) − r for each x ∗ ∈ X ∗ ; ∗

∗

epi( f + p + r ) = epi f + ( p, −r ).

(4) (5)

Recall that a function ψ : Z → R is said to K -increasing, if for any x, y ∈ Z such that y ≤ K x one has ψ(y) ≤ ψ(x). For a function φ : X → Y • , the effective domain and the S-epigraph of φ is defined, respectively, by dom φ = {x ∈ X : φ(x) ∈ Y } and epi S φ = {(x, y) ∈ X × Y : y ∈ φ(x) + S}. Then, φ is called proper if dom φ = ∅, and is S-epi-closed if epi S φ is closed. Moreover, φ is said to be S-convex if for any x1 , x2 ∈ X and any t ∈ [0, 1], φ(t x1 + (1 − t)x2 ) ≤ S tφ(x1 ) + (1 − t)φ(x2 ) (see [26,29]) and φ is star S-lsc if (λφ) : X → R is lsc for each λ ∈ S ⊕ . Note that if φ is star S-lsc, then it is also S-epi-closed. The following lemma, which is taken from [29, Theorem 2.8.10], will be used in the sequel. Lemma 2.1 Let g : X → R be proper convex functions, ϕ : domϕ ⊆ X → Z • be proper K -convex and f : Z → R be proper convex and K -increasing on ϕ(domϕ) + K . If there exists x0 ∈ domg + ϕ −1 (dom f ) such that f is continuous at ϕ(x0 ), then, for each x ∗ ∈ X ∗ , we have (g + f ◦ ϕ)∗ (x ∗ ) =

min ( f ∗ (β) + (g + βϕ)∗ (x ∗ )),

β∈dom f ∗

or equivalently, epi(g + f ◦ ϕ)∗ =

 β∈dom

epi(g + βϕ − f ∗ (β)). f∗

The following lemma is known in [14,29]. Lemma 2.2 Let g, h : X → R be proper convex functions satisfying dom g∩dom h = ∅. (i) If g, h are lsc, then epi (g + h)∗ = cl (epi g ∗ + epi h ∗ ).

(6)

(ii) If either g or h is continuous at some point of dom g ∩ dom h, then epi(g ∗ h ∗ ) = epi (g + h)∗ = epi g ∗ + epi h ∗ .

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3 Constraint Qualifications and Farkas Lemmas Throughout this paper, let C ⊆ X be a nonempty convex set, ϕ : X → Z • a proper, K -convex mapping, f : Z • → R a proper, convex, K -increasing function, and h : X → Y • a proper, S-convex mapping, where f (∞ Z ) = +∞. Then, f ◦ ϕ is proper convex. Set  ( f ◦ ϕ)(x) :=

f (ϕ(x)), if x ∈ dom ϕ, +∞, otherwise,

and for the sake of convenience, we write for each λ ∈ S ⊕ ,  (λh)(x) :=

λ, h(x) , if x ∈ dom h, +∞, otherwise.

Let A denote the solution set of the system {x ∈ C; h(x) ∈ −S}, that is, A := {x ∈ C : h(x) ∈ −S}. To avoid triviality, we always assume that A ∩ dom( f ◦ ϕ) = ∅. For simpleness, we denote Λ1 := epiδC∗ + Λ2 :=



epi(βϕ − f ∗ (β))∗ ;

β∈dom f ∗

λ∈S ⊕





epi(λh)∗ +



∗

epi(δC + λh) +

epi(βϕ − f ∗ (β))∗

β∈dom f ∗

λ∈S ⊕

and 

Λ3 :=

epi(δC + λh + βϕ − f ∗ (β))∗ .

λ∈S ⊕ ,β∈dom f ∗

Then, the following inclusions hold. Lemma 3.1 One always has Λ1 ⊆ Λ2 ⊆ Λ3 ⊆ epi( f ◦ ϕ + δ A )∗ .

(8)

Proof Note that epiδC∗ +



epi(λh)∗ =

λ∈S ⊕



(epiδC∗ + epi(λh)∗ )

λ∈S ⊕

and that Λ2 =



(epi(δC + λh)∗ + epi(βϕ − f ∗ (β))∗ ),

λ∈S ⊕ ,β∈dom f ∗

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it follows from the above equalities and (2) that the inclusions Λ1 ⊆ Λ2 ⊆ Λ3 hold. Below we show that (9) Λ3 ⊆ epi( f ◦ ϕ + δ A )∗ . To do this, since for each λ ∈ S ⊕ , β ∈ dom f ∗ and x ∈ X , δC (x) + (λh)(x) ≤ δ A (x) and β, ϕ(x) − f ∗ (β) ≤ f (ϕ(x)), one has that δC + λh + βϕ − f ∗ (β) ≤ f ◦ ϕ + δ A and hence, by (3), epi(δC + λh + βϕ − f ∗ (β))∗ ⊆ epi( f ◦ ϕ + δ A )∗ . Note that the above inclusion holds for each λ ∈ S ⊕ and β ∈ dom f ∗ , it follows that (9) holds, which completes the proof.   Regarding some possible reversed inclusions in (8), we introduce the following constraint qualifications, which are crucial in the sequel. (CQ1)

epi( f ◦ ϕ + δ A )∗ = Λ1 ,

(CQ2) (CQ3)

epi( f ◦ ϕ + δ A )∗ = Λ2 , epi( f ◦ ϕ + δ A )∗ = Λ3 .

Remark 3.1 (a) By (8), we see that (CQ1), (CQ2) and (CQ3) can be equivalently replaced by

and

epi( f ◦ ϕ + δ A )∗ ⊆ Λ1 , epi( f ◦ ϕ + δ A )∗ ⊆ Λ2 ,

(10) (11)

epi( f ◦ ϕ + δ A )∗ ⊆ Λ3 ,

(12)

respectively. (b) By (8) and (a), we see that the following implications hold: (CQ1) ⇒ (CQ2) ⇒ (CQ3). (c) Let contψ denote the set of all points at which ψ is continuous. If conth ∩ C = ∅ or domh ∩ intC = ∅, then, by Lemma 2.2, one sees that Λ1 = Λ2 and (CQ1) ⇐⇒ (CQ2); if contϕ ∩ A = ∅, then Λ2 = Λ3 and (CQ2) ⇐⇒ (CQ3); if conth ∩ A ∩ intC = ∅ or contϕ ∩ A ∩ intC = ∅, then Λ1 = Λ2 = Λ3 and hence (CQ1) ⇐⇒ (CQ2) ⇐⇒ (CQ3).

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(d) It is easy to see that (CQ3) is equivalent to the condition KS = JS , which was introduced in [22], where KS := {(x ∗ , 0, r ) : (x ∗ , r ) ∈ epi( f ◦ ϕ + δ A )∗ } and JS :=





{(x ∗ , −β, r ) : (x ∗ , r ) ∈ epi(βϕ + δC + λh)∗ }

λ∈S ⊕ β∈dom f ∗

 +{(0, y ∗ , r ) : (y ∗ , r ) ∈ epi f ∗ } .

Moreover, by [22, Remark 5.1], we see that (CQ3) is weaker than the following closure condition (CQ) in [21]: {0} × epi f ∗ +



  {(x ∗ , −β, r ) : (x ∗ , r ) ∈ epi(βϕ + δC + λh)∗ } is w∗ -closed.

λ∈S ⊕ β∈dom f ∗

(e) Let ψ : X → R be a proper convex function. Recall from [2,3,14] that the family {δC ; λh} satisfies the condition C1 (ψ, A) iff epi(ψ + δ A )∗ = epiψ ∗ + epiδC∗ +



epi(λh)∗ ,

λ∈S ⊕

the condition C2 (ψ, A) iff epi(ψ + δ A )∗ = epi(ψ + δC )∗ +



epi(λh)∗ ,

λ∈S ⊕

and the condition C3 (ψ, A) iff epi(ψ + δ A )∗ =



epi(ψ + δC + λh)∗ .

λ∈S ⊕

In the case when X = Z and ϕ is the identity operator on X , then the conditions (CQ1) and (CQ2) are reduced into the conditions C1 ( f, A) and C2 ( f, A), respectively (see Sect. 5.1), while, as mentioned in [14], conditions C1 ( f, A) and C2 ( f, A) generalize the corresponding notions in [2,3]. Proposition 3.1 Suppose that f is lsc, ϕ is star K -lsc, C is closed and h is S-epiclosed. Then, the following statements hold. (i) The condition (CQ1) holds if and only if Λ1 is w∗ -closed. (ii) The condition (CQ2) holds if and only if Λ2 is w∗ -closed. (iii) The condition (CQ3) holds if and only if Λ3 is w∗ -closed.

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Proof Since f is lsc, ϕ is star K -lsc, C is closed, and h is S-epi-closed, it follows from [18, Corollary 1] and Lemma 2.2(i) that ⎛ epi( f ◦ ϕ + δ A )∗ = cl ⎝

β∈dom

⎛ = cl ⎝ = cl ⎝

β∈dom



epi(βϕ − f ∗ (β))∗ + f∗



⎛ ⎜ ⊆ cl ⎜ ⎝

epi(βϕ + δ A − f ∗ (β))∗ ⎠ f∗



β∈dom

⎛

⎞



⎞ epi(δC + λh)∗ ⎠

λ∈S ⊕

epi(βϕ − f ∗ (β))∗ + epiδC∗ + f∗



β∈dom f ∗ λ∈S ⊕



⎞ epi(λh)∗ ⎠

λ∈S ⊕

⎞ ⎟ epi(λh + δC + βϕ − f ∗ (β))∗ ⎟ ⎠

⊆ epi( f ◦ ϕ + δ A )∗ , where the second equality holds by [14, Proposition 3.5] and the first inclusion holds by (2), and the last inclusion holds because of (8) and the fact that epi( f ◦ ϕ + δ A )∗ is w ∗ -closed. Hence, the result is seen to hold and the proof is complete.   The following proposition gives some conditions for these conditions (CQ1), (CQ2) and (CQ3) to hold. Proposition 3.2 Suppose that there exists x0 ∈ ϕ −1 (dom f ) ∩ A such that f is continuous at ϕ(x0 ). (i) If the family {δC ; λh} satisfies the condition C1 ( f ◦ ϕ, A), then the condition (CQ1) holds. (ii) If the family {δC ; λh} satisfies the condition C2 ( f ◦ ϕ, A), then the condition (CQ2) holds. (iii) If the family {δC ; λh} satisfies the condition C3 ( f ◦ ϕ, A), then the condition (CQ3) holds. Proof (i) Assume that the family {δC ; λh} satisfies the condition C1 ( f ◦ϕ, A). Then, epi( f ◦ ϕ + δ A )∗ = epi( f ◦ ϕ)∗ + epiδC∗ +



epi(λh)∗ ;

λ∈S ⊕

While, applying Lemma 2.1 with g = 0, we see that 

epi( f ◦ ϕ)∗ =

β∈epi f ∗

Thus, the condition (CQ1) holds.

123

epi(βϕ − f ∗ (β))∗ .

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(ii) Assume that the family {δC ; λg} satisfies the condition C2 ( f ◦ ϕ, A). Then, epi( f ◦ ϕ + δ A )∗ = epi( f ◦ ϕ)∗ +



epi(δC + λh)∗ .

λ∈S ⊕

This together with (13) implies that (CQ2) holds. (iii) Assume that the family {δC ; λh} satisfies the condition C3 ( f ◦ ϕ, A). Then, epi( f ◦ ϕ + δ A )∗ =



epi( f ◦ ϕ + δC + λh)∗ ;

λ∈S ⊕

While, applying Lemma 2.1 with g = δC + λh, we see that 

epi( f ◦ ϕ + δC + λh)∗ = Λ3 .

λ∈S ⊕

Thus, the condition (CQ3) holds and the proof is complete.

 

Proposition 3.3 Assume that either there exists x0 ∈ ϕ −1 (dom f ) ∩ int A such that f is continuous at ϕ(x0 ) or there exists x0 ∈ ϕ −1 (dom f )∩ A such that f is continuous at ϕ(x0 ) and ϕ is continuous at x0 . If the family {δC ; λh} satisfies the condition C1 (0, A), then the condition (CQ1) holds. Consequently, the conditions (CQ2) and (CQ3) hold. Proof Suppose that the family {δC ; λh} satisfies the condition C1 (0, A). Then, epiδ ∗A = epiδC∗ +



epi(λh)∗ .

(14)

λ∈S ⊕

While, by Lemma 2.2(ii), epi( f ◦ ϕ + δ A )∗ = epi( f ◦ ϕ)∗ + epiδ ∗A . Combine this with (14) and (13), we get epi( f ◦ ϕ + δ A )∗ =



epi(βϕ − f ∗ (β))∗ + epiδC∗ +

β∈dom f ∗



epi(λh)∗ = Λ1 ,

λ∈S ⊕

that is, the condition (CQ1) holds. The proof is complete.

 

Lemma 3.2 (i) The condition (CQ1) holds if and only if for each x ∗ ∈ X ∗ , ( f ◦ ϕ + δ A )∗ (x ∗ ) =

∗ ∗ δC (y ) + (λh)∗ (z ∗ )

+ (βϕ)∗ (x ∗ − y ∗ − z ∗ ) + f ∗ (β) . min

λ∈S ⊕ ,β∈dom f ∗

min

y ∗ ,z ∗ ∈X ∗

(15)

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(ii) The condition (CQ2) holds if and only if for each x ∗ ∈ X ∗ , ( f ◦ ϕ + δ A )∗ (x ∗ ) =

min

λ∈S ⊕ ,β∈dom f ∗



(δC + λh)∗ (y ∗ ) + (βϕ)∗ (x ∗ − y ∗ ) + f ∗ (β) .(16) min ∗ ∗

y ∈X

(iii) The condition (CQ3) holds if and only if for each x ∗ ∈ X ∗ , ( f ◦ ϕ + δ A )∗ (x ∗ ) =

min



inf (λh)(x) + (βϕ)(x) − f ∗ (β) − x ∗ , x .

λ∈S ⊕ ,β∈dom f ∗ x∈C

(17) Proof Note that the proofs of (ii) and (iii) are similar to that of (i); we only need to prove that (i) holds. In fact, for each x ∗ ∈ X ∗ , (15) means exactly that (x ∗ , r ) ∈ epi( f ◦ ϕ + δ A )∗ if and only if there exists (λ, β, y ∗ , z ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ × X ∗ such that δC∗ (y ∗ ) + (λh)∗ (z ∗ ) + (βϕ)∗ (x ∗ − y ∗ − z ∗ ) + f ∗ (β) ≤ r, or, equivalently, if and only if there exist (λ, β, y ∗ , z ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ × X ∗ and (r1 , r2 ) ∈ R2 such that δC∗ (y ∗ ) ≤ r1 , (λh)∗ (z ∗ ) ≤ r2 , (βϕ − f ∗ (β))∗ (x ∗ − y ∗ − z ∗ ) ≤ r − r1 − r2 , which is equivalent to (x ∗ , r ) = (y ∗ , r1 ) + (z ∗ , r2 ) + (x ∗ − y ∗ − z ∗ , r − r1 − r2 ) ∈ epiδC∗ + epi(λh)∗ + epi(βϕ − f ∗ (β))∗ ⊆ Λ1 . Thus, epi( f ◦ ϕ + δ A )∗ ⊆ Λ1 . Therefore, by Remark 3.1(a), the result is seen to hold. The proof is complete.   As in [29], we use Λ(X ) to denote the class of all proper convex functions on X and Γ (X ) to denote the class of proper lsc convex functions on X . Given g ∈ Λ(X ), consider the following statements: (A) f (ϕ(x)) ≥ g(x), ∀x ∈ A. (B) ∀x ∗ ∈ domg ∗ , ∃(λ, β, y ∗ , z ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ × X ∗ such that δC∗ (y ∗ ) + (λh)∗ (z ∗ ) + (βϕ)∗ (x ∗ − y ∗ − z ∗ ) + f ∗ (β) ≤ g ∗ (x ∗ ).

(18)

(C) ∀x ∗ ∈ domg ∗ , ∃(λ, β, y ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ such that (δC + λh)∗ (y ∗ ) + (βϕ)∗ (x ∗ − y ∗ ) + f ∗ (β) ≤ g ∗ (x ∗ ).

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(D) ∀x ∗ ∈ domg ∗ , ∃(λ, β) ∈ S ⊕ × dom f ∗ such that (δC + λh + βϕ)∗ (x ∗ ) + f ∗ (β) ≤ g ∗ (x ∗ ).

(20)

Remark 3.2 It is easy to see that (D) is equivalent to the following statement: (D ) ∀x ∗ ∈ domg ∗ , ∃(λ, β) ∈ S ⊕ × dom f ∗ such that −(λh)(x) − (βϕ)(x) + f ∗ (β) + x ∗ , x ≤ g ∗ (x ∗ ) for each x ∈ C. The following proposition gives some relationships among the above statements. Proposition 3.4 Let g ∈ Λ(X ). Then, the following implications hold: (B) ⇒ (C) ⇒ (D). Furthermore, if g ∈ Γ (X ), then (B) ⇒ (C) ⇒ (D) ⇒ (A). Proof (B) ⇒ (C). Suppose that (B) holds. Let x ∗ ∈ domg ∗ . Then, there exists (λ, β, y0∗ , z 0∗ ) ∈ S ⊕ × dom f ∗ × X ∗ × X ∗ such that δC∗ (y0∗ ) + (λh)∗ (z 0∗ ) + (βϕ)∗ (x ∗ − y0∗ − z 0∗ ) + f ∗ (β) ≤ g ∗ (x ∗ ). Let y ∗ = y0∗ + z 0∗ . Then, (δC + λh)∗ (y ∗ ) + (βϕ)∗ (x ∗ − y ∗ ) + f ∗ (β) = (δC + λh)∗ (y0∗ + z 0 ) + (βϕ)∗ (x ∗ − y0∗ − z 0∗ ) + f ∗ (β) ≤ δC∗ (y0∗ ) + (λh)∗ (z 0∗ ) + (βϕ)∗ (x ∗ − y0∗ − z 0∗ ) + f ∗ (β) ≤ g ∗ (x ∗ ).

Thus, (C) holds. (C) ⇒ (D). Suppose that (C) holds. Note that for each x ∗ ∈ domg ∗ and (λ, β, y ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ , (δC + λh + βϕ)∗ (x ∗ ) ≤ (δC + λh)∗ (y ∗ ) + (βϕ)∗ (x ∗ − y ∗ ). Then, the result is clear. (D) ⇒ (A). Suppose that g ∈ Γ (X ) and (D) holds. Then, by Remark 3.2, for each x ∗ ∈ domg ∗ , x ∗ , x − g ∗ (x ∗ ) ≤ (δC + λh)(x) + (βϕ)(x) − f ∗ (β) ≤ δ A (x) + f (ϕ(x)) for each x ∈ X.

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Taking the supremum over x ∗ ∈ domg ∗ and noting g is lsc, we see that g(x) = g ∗∗ (x) ≤ f (ϕ(x)) for each x ∈ A. Therefore, (A) holds and the proof is complete.

 

Next we give some new versions of Farkas lemma. Theorem 3.1 The following assertions are equivalent. (i) The condition (CQ1) holds. (ii) For each g ∈ Γ (X ), (A) ⇐⇒ (B). (iii) For each p ∈ X ∗ and r ∈ R, [( f ◦ ϕ)(x) − p, x ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, β, y ∗ , z ∗ ) ∈ S ⊕ ×dom f ∗ × X ∗ × X ∗ s.t. δC∗ (y ∗ ) + (λh)∗ (z ∗ ) + (βϕ)∗ ( p − y ∗ − z ∗ ) + f ∗ (β) ≤ −r ]. (21) Proof (i)⇒(ii) Suppose that (i) holds. To prove (ii), by Proposition 3.4, it is sufficient to prove (A) ⇒ (B). To do this, assume that (A) holds. Let g ∈ Γ (X ) and x ∗ ∈ domg ∗ . Then, by (3), (x ∗ , g ∗ (x ∗ )) ∈ epig ∗ ⊆ epi( f ◦ ϕ + δ A )∗ . This implies that ( f ◦ ϕ + δ A )∗ (x ∗ ) ≤ g ∗ (x ∗ ). Hence, by Lemma 3.2(i), we see that there exists (λ, β, y ∗ , z ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ × X ∗ such that (18) holds, that is, (B) holds. (ii)⇒(iii) For any p ∈ X ∗ and r ∈ R, let g = p(·) + r in (ii), then (iii) holds trivially. (iii)⇒(i) Suppose that (iii) holds. By Remark 3.1(a), to show (i), we only need to show (10) holds. For this purpose, take ( p, r ) ∈ epi( f ◦ ϕ + δ A )∗ . Then, ( f ◦ ϕ + δ A )∗ ( p) ≤ r and hence ( f ◦ ϕ)(x) − p, x ≥ −r for each x ∈ A. Since (iii) holds, it follows that there exists (λ, β, y ∗ , z ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ × X ∗ such that δC∗ (y ∗ ) + (λh)∗ (z ∗ ) + (βϕ)∗ ( p − y ∗ − z ∗ ) + f ∗ (β) ≤ r, which is equivalent to δC∗ (y ∗ ) + (λh)∗ (z ∗ ) + (βϕ − f ∗ (β))∗ ( p − y ∗ − z ∗ ) ≤ r.

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This implies that ( p, r ) = (y ∗ , δC∗ (y ∗ )) + (z ∗ , (λh)∗ (z ∗ )) + ( p − y ∗ − z ∗ , r − δC∗ (y ∗ ) − (λh)∗ (z ∗ )) ∈ epiδC∗ + epi(λh)∗ + epi(βϕ − f ∗ (β))∗ ⊆ Λ1 .  

Therefore, (10) holds and the proof is complete. Analogously, we can get the following theorems. Theorem 3.2 The following assertions are equivalent. (i) The condition (CQ2) holds. (ii) For each g ∈ Γ (X ), (A) ⇐⇒ (C). (iii) For each p ∈ X ∗ and r ∈ R,

[( f ◦ ϕ)(x) − p, x ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, β, y ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ s.t. (δC + λh)∗ (y ∗ ) + (βϕ)∗ ( p − y ∗ ) + f ∗ (β) ≤ −r ]. Theorem 3.3 The following assertions are equivalent. (i) The condition (CQ3) holds. (ii) For each g ∈ Γ (X ), (A) ⇐⇒ (D). (iii) For each p ∈ X ∗ and r ∈ R, [( f ◦ ϕ)(x) − p, x ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, β) ∈ S ⊕ × dom f ∗ s.t. (δC + λh + βϕ)∗ ( p) + f ∗ (β) ≤ −r ]. Remark 3.3 Recall that the authors in [22] proposed the extended Farkas lemma (see statement (iii) in Theorem 3.3) by using the constraint qualification KS = JS (see Remark 3.1(d)). As mentioned in Remark 3.1(d), the condition KS = JS is just equivalent to our condition (CQ3). Thus, our Theorem 3.3 improves the corresponding one in [22]. In the case when g is a constant function, we can get the following results. Note that proofs of Theorems 3.4–3.6 are similar to that of Theorems 3.1–3.3, so we omit them here. Theorem 3.4 The following assertions are equivalent. (i) epi( f ◦ ϕ + δ A )∗ ∩ ({0} × R) = Λ1 ∩ ({0} × R). (ii) For each r ∈ R, [( f ◦ ϕ)(x) ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, β, y ∗ , z ∗ ) ∈ S ⊕ ×dom f ∗ × X ∗ × X ∗ s.t. δC∗ (y ∗ ) + (λh)∗ (z ∗ ) + (βϕ)∗ (−y ∗ − z ∗ ) + f ∗ (β) ≤ −r ]. Theorem 3.5 The following assertions are equivalent.

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(i) epi( f ◦ ϕ + δ A )∗ ∩ ({0} × R) = Λ2 ∩ ({0} × R). (ii) For each r ∈ R, [( f ◦ ϕ)(x) ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, β, y ∗ ) ∈ S ⊕ × dom f ∗ × X ∗ s.t. (δC + λh)∗ (y ∗ ) + (βϕ)∗ (−y ∗ ) + f ∗ (β) ≤ −r ]. Theorem 3.6 The following assertions are equivalent. (i) epi( f ◦ ϕ + δ A )∗ ∩ ({0} × R) = Λ3 ∩ ({0} × R). (ii) For each r ∈ R, [( f ◦ ϕ)(x) ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, β) ∈ S ⊕ ×dom f ∗ s.t. (δC + λh + βϕ)∗ (0) + f ∗ (β) ≤ −r ]. The following example illustrates Theorems 3.1–3.6. Example 3.1 Let X = Y = C := R, K := (−∞, 0], S := [0, +∞). Define ϕ, h, f : R → R, respectively, by ϕ := IdR , h := δ(−∞,0] and ⎧ ⎪ x < 0, ⎨0, f (x) := 1, x = 0, ⎪ ⎩ +∞, x > 0

for each x ∈ R.

Then, f is a proper convex and K -increasing function, ϕ is a proper K -convex function, h is a proper S-convex function. It is easy to see that A := {x ∈ R : h(x) ∈ −S} = (−∞, 0]. Thus, f ◦ ϕ + δ A = f and ( f ◦ ϕ + δ A )∗ = δ[0,+∞) . Hence, epi( f ◦ ϕ + δ A )∗ = [0, +∞) × [0, +∞). Note that f ∗ = δ[0,+∞) and S ⊕ = [0, +∞), it follows that (λh)∗ = δ[0,+∞) for each λ ∈ S ⊕ and that for each β ≥ 0,  (βϕ − f ∗ (β))∗ (x ∗ ) =

0, x ∗ = β, +∞, x ∗ = β

for each x ∗ ∈ R.

Thus, 

Λ1 =

λ∈[0,+∞)

epi(λh)∗ +



epi(βϕ − f ∗ (β))∗ = [0, +∞) × [0, +∞).

β∈[0,+∞)

Hence, the condition (CQ1) holds and all statements in Theorem 3.1 and Theorem 3.4 hold. Moreover, by Remark 3.1(b), we see that the conditions (CQ2) and (CQ3) also hold. Therefore, several versions of Farkas lemmas in Theorems 3.2, 3.3, 3.5 and 3.6 are established.

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4 Strong Dualites and Stable Strong Dualities Let g ∈ Γ (X ). Consider the primal problem inf f (ϕ(x)) − g(x), s.t. x ∈ C, h(x) ∈ −S.

(Pg )

(22)

Define the Lagrange dual problem by (Dg )

inf

sup

x ∗ ∈domg ∗ (λ,β)∈S ⊕ ×dom f ∗

{g ∗ (x ∗ ) − (δC + λh + βϕ)∗ (x ∗ ) − f ∗ (β)}.

Moreover, we define the Fenchel–Lagrange dual problems by (D g )

inf

sup

x ∗ ∈domg ∗ (λ,β,y ∗ ,z ∗ )∈S ⊕ ×dom f ∗ ×X ∗ ×X ∗

{g ∗ (x ∗ ) − δC∗ (y ∗ ) − (λh)∗ (z ∗ )

−(βϕ)∗ (x ∗ − y ∗ − z ∗ ) − f ∗ (β)} and ( D˜ g )

inf

sup

x ∗ ∈domg ∗ (λ,β,y ∗ )∈S ⊕ ×dom f ∗ ×X ∗

{g ∗ (x ∗ )

−(δC + λh)∗ (y ∗ ) − (βϕ)∗ (x ∗ − y ∗ ) − f ∗ (β)}. In the case when g = 0, problem (Pg ) is reduced to the problem (P) inf f (ϕ(x)), s.t. x ∈ C, h(x) ∈ −S,

(P)

(23)

˜ that is, and problems (Dg ) (resp. (D g ), ( D˜ g )) are denoted by (D) (resp. (D), ( D)), (D) (D)

sup

(λ,β)∈S ⊕ ×dom f ∗

{−(δC + λh + βϕ)∗ (0) − f ∗ (β)},

sup

(λ,β,y ∗ ,z ∗ )∈S ⊕ ×dom f ∗ ×X ∗ ×X ∗ ∗ ∗ ∗

{−δC∗ (y ∗ ) − (λh)∗ (z ∗ ) − (βϕ)∗

×(−y − z ) − f (β)} and ˜ ( D)

sup

(λ,β,y ∗ )∈S ⊕ ×dom f ∗ ×X ∗

{−(δC + λh)∗ (y ∗ ) − (βϕ)∗ (−y ∗ ) − f ∗ (β)}.

As usual, we use v(Pg ) (resp. v(Dg ), v(D g ), v( D˜ g )) to denote the optimal value of ˜ Obviously, the weak duality between problem problem (Pg ) (resp. (Dg ), (D), ( D)). (Pg ) and its dual problem holds, that is, the optimal value of (Pg ) is always greater

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J Optim Theory Appl

than or equal to the optimal value of its dual problem. Moreover, by Proposition 3.4, the following inequalities hold: v(D g ) ≤ v( D˜ g ) ≤ v(Dg ) ≤ v(Pg ) for each g ∈ Γ (X ).

(24)

The following theorem gives some equivalent conditions for the stable strong duality holds between (P) and (D). Theorem 4.1 Each of (i)–(iii) of Theorem 3.3 is a necessary and sufficient condition for the following statements: (iv) For each g ∈ Γ (X ), the strong duality holds between (Pg ) and (Dg ), that is, v(Pg ) =

inf

max

x ∗ ∈domg ∗ (λ,β)∈S ⊕ ×dom f ∗

{g ∗ (x ∗ ) − (δC + λh + βϕ)∗ (x ∗ ) − f ∗ (β)}. (25)

(v) The stable strong duality holds between (P) and (D), that is, for each p ∈ X ∗ , v(Pp ) =

max

(λ,β)∈S ⊕ ×dom f ∗

{−(δC + λh + βϕ)∗ ( p) − f ∗ (β)}.

(26)

Proof (ii)⇒(iv). Suppose that (ii) of Theorem 3.3 holds. Let g ∈ Γ (X ) and r := v(Pg ). We assume that r = −∞ (if r = −∞, the result holds trivially). Then, r ∈ R as dom( f ◦ ϕ − g) ∩ A = ∅ and f (ϕ(x)) − g(x) ≥ r for each x ∈ A. Thus, applying (ii) of Theorem 3.3, for each x ∗ ∈ domg ∗ , there exists (λ, β) ∈ S ⊕ × dom f ∗ such that g ∗ (x ∗ ) − (δC + λh + βϕ)∗ (x ∗ ) − f ∗ (β) ≥ r. This together with the weak duality shows that (25) holds. (iv)⇒(v) Suppose that (iv) holds. Let g := p ∈ X ∗ . Then, g ∗ (x ∗ ) :=



0 x ∗ = p, +∞ otherwise,

and hence, (v) holds. (v)⇒(iii). Suppose that (v) holds. Let p ∈ X ∗ and r ∈ R be such that f (ϕ(x)) − p, x ≥ r for each x ∈ A. Then, v(Pp ) ≥ r and by (26), max

(λ,β)∈S ⊕ ×dom f ∗

{−(δC + λh + βϕ)∗ ( p) − f ∗ (β)} = v(Pp ) ≥ r.

This implies that there exists (λ, β) ∈ S ⊕ × dom f ∗ such that (δC + λh + βϕ)∗ ( p) + f ∗ (β) ≤ −r. Thus, the implication “⇒” in (3.3) holds and so does (3.3) since the converse implication of (3.3) holds automatically by Proposition 3.4. The proof is complete.  

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Remark 4.1 Recall that the authors in [22] established the stable strong duality between (P) and (D) (see statement (v) in Theorem 4.1) by using the constraint qualification KS = JS . As stated in Remark 3.1(d), the condition KS = JS is just equivalent to our condition (CQ3). Thus, our Theorem 4.1 improves the corresponding one in [22]. By the same way, we can get the following theorems, which characterize the stable ˜ strong duality between (P) and (D), and between (P) and ( D). Theorem 4.2 Each of (i)–(iii) of Theorem 3.1 is a necessary and sufficient condition for the following statements: (iv) For each g ∈ Γ (X ), the strong duality holds between (Pg ) and (D g ), that is, v(Pg ) =

inf

max

x ∗ ∈domg ∗ (λ,β,y ∗ ,z ∗ )∈S ⊕ ×dom f ∗ ×X ∗ ×X ∗ ∗ ∗ ∗ ∗ ∗

{g ∗ (x ∗ ) − δC∗ (y ∗ ) − (λh)∗ (z ∗ )

−(βϕ) (x − y − z ) − f (β)}.

(v) The stable strong duality holds between (P) and (D), that is, for each p ∈ X ∗ , v(Pp ) =

{−δC∗ (y ∗ ) − (λh)∗ (z ∗ )

max

(λ,β,y ∗ ,z ∗ )∈S ⊕ ×dom f ∗ ×X ∗ ×X ∗ ∗ ∗ ∗ ∗

−(βϕ) ( p − y − z ) − f (β)}.

Theorem 4.3 Each of (i)–(iii) of Theorem 3.2 is a necessary and sufficient condition for the following statements: (iv) For each g ∈ Γ (X ), the strong duality holds between (Pg ) and ( D˜ g ), that is, v(Pg ) =

inf

max

{g ∗ (x ∗ )

x ∗ ∈domg ∗ (λ,β,y ∗ )∈S ⊕ ×dom f ∗ ×X ∗ −(δC + λh)∗ (y ∗ ) − (βϕ)∗ (x ∗ −

y ∗ ) − f ∗ (β)}.

˜ that is, for each p ∈ X ∗ , (v) The strong duality holds between (P) and ( D), v(Pp ) =

max

(λ,β,y ∗ )∈S ⊕ ×dom f ∗ ×X ∗

{−(δC + λh)∗ (y ∗ ) − (βϕ)∗ ( p − y ∗ ) − f ∗ (β)}.

˜ As usual, we say that the strong duality holds between (P) and (D) (resp. (D), ( D)) ˜ ˜ if v(P) = v(D) (resp. v(D), v( D)) and problem (D) (resp. (D), ( D)) has an optimal solution. The following theorems establish the strong dualities of problem (P). Note that those proofs are similar to that of Theorem 4.1, so we omit those here. Theorem 4.4 Each of (i)–(ii) of Theorem 3.6 is a necessary and sufficient condition for the strong duality between (P) and (D), that is, inf f (ϕ(x)) =

x∈A

max

(λ,β)∈S ⊕ ×dom f ∗

{−(δC + λh + βϕ)∗ (0) − f ∗ (β)}.

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Theorem 4.5 Each of (i)–(ii) of Theorem 3.4 is a necessary and sufficient condition for the strong duality between (P) and (D), that is, inf f (ϕ(x)) =

x∈A

max

(λ,β,y ∗ ,z ∗ )∈S ⊕ ×dom f ∗ ×X ∗ ×X ∗ ∗ ∗ ∗ ∗

{−δC∗ (y ∗ ) − (λh)∗ (z ∗ )

−(βϕ) (−y − z ) − f (β)}.

Theorem 4.6 Each of (i)–(ii) of Theorem 3.5 is a necessary and sufficient condition ˜ that is, for the strong duality between (P) and ( D), inf f (ϕ(x)) =

x∈A

max

(λ,β,y ∗ )∈S ⊕ ×dom f ∗ ×X ∗

{−(δC + λh)∗ (y ∗ ) − (βϕ)∗ (−y ∗ ) − f ∗ (β)}.

By Theorems 4.1–4.6 and Remark 3.1, we get the following corollaries directly. Corollary 4.1 Consider the following assertions: (i) The stable strong duality holds between (P) and (D). ˜ (ii) The stable strong duality holds between (P) and ( D). (iii) The stable strong duality holds between (P) and (D). Then (i)⇒(ii)⇒(iii). Consequently, if the condition (CQ1) holds, then (i), (ii) and (iii) hold. Moreover, if contϕ ∩ A ∩ intC = ∅, then (i)⇐⇒(ii)⇐⇒(iii). Corollary 4.2 Consider the following assertions: (i) The strong duality holds between (P) and (D). ˜ (ii) The strong duality holds between (P) and ( D). (iii) The strong duality holds between (P) and (D). Then (i)⇒(ii)⇒(iii). Consequently, if epi( f ◦ ϕ + δ A )∗ ∩ ({0} × R) = Λ1 ∩ ({0} × R), then (i), (ii) and (iii) hold. Furthermore, if contϕ ∩ A ∩ intC = ∅, then (i)⇐⇒(ii)⇐⇒(iii). Example 4.1 Let X, Y, C, K , S and f, ϕ, h : R → R be defined as Example 3.1. Then, f, ϕ and h are proper convex functions and A = (−∞, 0]. As showed in Example 3.1, the conditions (CQ1), (CQ2) and (CQ3) hold. Thus, by Theorems 4.1–4.3, we see that the strong duality holds between (Pg ) and (Dg ) (resp. (D g ), ( D˜ g )) for each g ∈ Γ (X ). For example, let g(x) = x. Then, v(Pg ) =

inf

{ f (ϕ(x)) − g(x)} = 0.

x∈(−∞,0]

Clearly, f ∗ = δ[0,+∞) , g ∗ = δ{1} and for each λ, β ≥ 0, (λh)∗ = δ[0,+∞) and (βϕ)∗ = δ{β} . Thus, v(D g ) =

123

sup

λ,β≥0,z ∗ ∈R

{−(λh)∗ (z ∗ ) − (βϕ)∗ (1 − z ∗ ) − f ∗ (β)} = 0,

J Optim Theory Appl

and hence, the strong duality holds between (Pg ) and (D g ). Moreover, by (24), we see that the strong dualities between (Pg ) and (Dg ), and between (Pg ) and ( D˜ g ) also hold. The following example shows that the condition g ∈ Γ (X ) can not be dropped. Example 4.2 Let X, Y, C, K , S and f, ϕ, h : R → R be defined as Example 3.1. Then, as showed in Example 3.1, the conditions (CQ1), (CQ2) and (CQ3) hold. Define g : R → R by ⎧ ⎪0, x < 0, ⎨ g(x) := 2, x = 0, for each x ∈ R. ⎪ ⎩ +∞, x > 0 Then, g is a proper convex function (not lsc). Thus, v(Pg ) =

inf

x∈(−∞,0]

{ f (ϕ(x)) − g(x)} = −1.

Note that f ∗ = g ∗ = δ[0,+∞) and for each λ, β ≥ 0,  ∗

∗

(λh + βϕ) (x ) =

0, x ∗ ≥ β, +∞, x ∗ < β

for each x ∗ ∈ R.

It follows that v(Dg ) = inf sup {g ∗ (x ∗ ) − (λh + βϕ)∗ (x ∗ ) − f ∗ (β)} = 0. ∗ x ≥0 λ,β≥0

This implies that the strong duality between (Pg ) and (Dg ) (resp. (D g ), ( D˜ g )) does not hold.

5 Applications 5.1 Applications to Conic Programming Let X = Z , ϕ := Id X , the identity operator on X . Then, the problem (P) turns into the classical convex conical programming problem (P)

inf f (x), s.t. x ∈ C, h(x) ∈ −S.

Note that for each x ∗ , β ∈ X ∗ , (βϕ)∗ (x ∗ ) :=



0 +∞

x ∗ = β, otherwise.

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˜ are reduced to the problem It follows that problems (D) and ( D) (D)

sup

(λ,x ∗ )∈S ⊕ ×X ∗

{− f ∗ (−x ∗ ) − (δC + λh)∗ (x ∗ )},

and problem (D) is reduced to (D)

sup

(λ,x ∗ ,y ∗ )∈S ⊕ ×X ∗ ×X ∗

{− f ∗ (−x ∗ − y ∗ ) − δC∗ (x ∗ ) − (λh)∗ (y ∗ )}.

Moreover, in this case, 

epi(βϕ − f ∗ (β))∗ = epi f ∗ ,

β∈dom f ∗

we see that the condition (CQ1) and (CQ2) coincide with C1 ( f, A) and C2 ( f, A), respectively. Thus, by Theorems 4.2, 4.3, 4.5 and 4.6, we get the following theorems straightforwardly. Theorem 5.1 The following statements are equivalent. (i) For each r ∈ R, [ f (x) ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, x ∗ , y ∗ ) ∈ S ⊕ × X ∗ × X ∗ s.t. δC∗ (x ∗ ) + (λh)∗ (y ∗ ) + f ∗ (−x ∗ − y ∗ ) ≤ −r ]. (ii) epi( f + δ A )∗ ∩ ({0} × R) = (epi f ∗ + epiδC∗ + (iii) The strong duality holds between (P) and (D).



λ∈S ⊕

epi(λh)∗ ) ∩ ({0} × R).

Theorem 5.2 The following statements are equivalent. (i) For each g ∈ Γ (X ), [ f (x) ≥ g(x), ∀x ∈ A] ⇐⇒ [∀z ∗ ∈ ×domg ∗ , ∃(λ, x ∗ , y ∗ ) ∈ S ⊕ × X ∗ × X ∗ s.t. δC∗ (x ∗ ) + (λh)∗ (y ∗ ) + f ∗ (z ∗ − x ∗ − y ∗ ) ≤ g ∗ (z ∗ )]. (ii) For each p ∈ X ∗ and r ∈ R, [ f (x) − p, x ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, x ∗ , y ∗ ) ∈ S ⊕ × X ∗ × X ∗ s.t. δC∗ (x ∗ ) + (λh)∗ (y ∗ ) + f ∗ ( p − x ∗ − y ∗ ) ≤ −r ]. (iii) The family {δC ; λh} satisfies the condition C1 ( f, A). (iv) The stable strong duality holds between (P) and (D), that is, for each p ∈ X ∗ , inf { f (x) − p, x } =

x∈A

123

max

{− f ∗ ( p − x ∗ − y ∗ )

(λ,x ∗ ,y ∗ )∈S ⊕ ×X ∗ ×X ∗ −δC∗ (x ∗ ) − (λh)∗ (y ∗ )}.

J Optim Theory Appl

Theorem 5.3 The following statements are equivalent. (i) For each r ∈ R, [ f (x) ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, x ∗ ) ∈ S ⊕ × X ∗ s.t. (δC + λh)(x ∗ ) + f ∗ (−x ∗ ) ≤ −r ].  (ii) epi( f + δ A )∗ ∩ ({0} × R) = (epi f ∗ + λ∈S ⊕ epi(δC + λh)∗ ) ∩ ({0} × R). (iii) The strong duality holds between (P) and (D). Theorem 5.4 The following statements are equivalent. (i) For each g ∈ Γ (X ), [ f (x) ≥ g(x), ∀x ∈ A] ⇐⇒ [∀z ∗ ∈ ×domg ∗ , ∃(λ, x ∗ ) ∈ S ⊕ × X ∗ s.t. (δC + λh)∗ (x ∗ ) + f ∗ (z ∗ − x ∗ ) ≤ g ∗ (z ∗ )]. (ii) For each p ∈ X ∗ and r ∈ R, [ f (x) − p, x ≥ r, ∀x ∈ A] ⇐⇒ [∃(λ, x ∗ ) ∈ S ⊕ × X ∗ s.t. (δC + λh)(x ∗ ) + f ∗ ( p − x ∗ ) ≤ −r ]. (iii) The family {δC ; λh} satisfies the condition C2 ( f, A). (iv) The stable strong duality holds between (P) and (D), that is, for each p ∈ X ∗ , inf { f (x) − p, x } =

x∈A

max

(λ,x ∗ )∈S ⊕ ×X ∗

{− f ∗ ( p − x ∗ ) − (δC + λh)∗ (x ∗ )}.

By Proposition 3.4 and Theorems 5.2 and 5.4, we get the following corollary directly. Corollary 5.1 Suppose that the family {δC ; λh} satisfies the condition C1 ( f, A) or C2 ( f, A). Then, the following statements hold: (i) For each g ∈ Γ (X ), [ f (x) ≥ g(x), ∀x ∈ A] ⇐⇒ [∀z ∗ ∈ ×domg ∗ , ∃λ ∈ S ⊕ s.t. ( f + δC + λh)∗ (z ∗ ) ≤ g ∗ (z ∗ )]. (ii) For each p ∈ X ∗ and r ∈ R, [ f (x) − p, x ≥ r, ∀x ∈ A] ⇐⇒ [∃λ ∈ S ⊕ s.t. ( f + δC + λh)∗ ( p) ≤ −r ]. (iii) For each p ∈ X ∗ , inf { f (x) − p, x } = max inf { f (x) + (λh)(x) − p, x }.

x∈A

λ∈S ⊕ x∈C

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Remark 5.1 (a) Under the assumptions that f is lsc, C is closed and h is S-epi-closed,

(27)

Bo¸t et. al. established in [2, Theorem 2] the strong duality between (P) and (D) via the following closedness condition epi f ∗ +



epi(δC + λh)∗ is w ∗ -closed,

(28)

λ∈S ⊕

and in [2, Theorem 1] the strong duality between (P) and (D) via the closedness condition (CC) 

epi f ∗ + epiδC∗ +

epi(λh)∗ is w ∗ -closed.

(29)

λ∈S ⊕

Another strong duality result between (P) and (D) was established in [8, Theorem 3.1] under the assumption that f and h are continuous. Note that, in the case when (27) holds, the condition (28) is equivalent to the condition C2 ( f, A) and (29) is equivalent to C1 ( f, A). Thus, our results extend and improve the corresponding ones in [2,8]. (b) Under the assumptions that f is lsc, C is closed and h is star S − lsc,

(30)

Dinh. et al. established the extended Farkas lemmas for problem (P) in [19, Theorem 5.1] and [20, Corollary 6.2] via the condition (CC). Note that, in this case, the condition (29) is equivalent to C1 ( f, A). Thus, our results extend and improve the corresponding ones in [19,20]. (c) Recall that the authors in [19,20] established the statement (iii) of Corollary 5.1 via the condition (CC) under the assumption that (30) holds, and the authors in [14] showed that the statement (iii) of Corollary 5.1 is equivalent to the following condition C( f, A) epi( f + δ A )∗ =



epi( f + δC + λh)∗ .

λ∈S ⊕

In fact, it is easy to prove that statements (i)–(iii) of Corollary 5.1 are equivalent to the condition C( f, A). On the other hand, assume that (27) holds, the authors in [10,11] showed that the statement (ii) of Corollary 5.1 is equivalent to 

epi( f + δC + λh)∗ is w ∗ -closed.

(31)

λ∈S ⊕

In fact, in this case, the condition C( f, A) and (31) are equivalent. Thus, our results extend and improve the corresponding ones in [10,11,19,20].

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(d) The statement (iii) of Corollary 5.1 was also obtained in [4, Theorem 2.1] via the following constraint qualification (CQ)

0 ∈ ri(h(C) + S).

As mentioned in [30, Example 4.10], the condition C1 ( f, A) is strictly weaker than the condition (CQ). Thus, our Corollary 5.1 extend and improve the corresponding ones in [4]. 5.2 Applications to DC Programming Consider the following optimization problem with the objective function is given as a DC function inf f (x) − g(x), (P) s.t. x ∈ C, h(x) ∈ −S, where f, h, C, S are defined as Sect. 3 and g ∈ Γ (X ) with dom( f − g) ∩ A = ∅. Following [20], we define the Fenchel–Lagrange dual problems of (P) by (D)

inf

sup

x ∗ ∈domg ∗ (λ,y ∗ )∈S ⊕ ×X ∗

{g ∗ (x ∗ ) − f ∗ (x ∗ − y ∗ ) − (δC + λh)∗ (y ∗ )},

and (D)

inf

sup

x ∗ ∈domg ∗ (λ,y ∗ ,z ∗ )∈S ⊕ ×X ∗ ×X ∗

{g ∗ (x ∗ )− f ∗ (x ∗ − y ∗ −z ∗ )−δC∗ (y ∗ )−(λh)∗ (z ∗ )}.

As usual, we use v(P), v(D), v(D) to denote the optimal value of problem (P), (D) and (D), respectively. Obviously, the weak duality between (P) and (D) and, between(P) and (D) holds. Especially, we have v(D) ≤ v(D) ≤ v(P). Theorem 5.5 Suppose that the family {δC , λh} satisfies the condition C1 ( f, A). Then, (i) For each r ∈ R, [ f (x) − g(x) ≥ r, ∀x ∈ A] ⇐⇒ [∀x ∗ ∈ domg ∗ , ∃(λ, y ∗ , z ∗ ) ∈ S ⊕ × X ∗ × X ∗ s.t. g ∗ (x ∗ ) − f ∗ (x ∗ − y ∗ − z ∗ ) − δC∗ (y ∗ ) − (λh)∗ (z ∗ ) ≥ r. (ii) The strong duality holds between (P) and (D), that is, inf { f (x) − g(x)} =

x∈A

inf

max

x ∗ ∈domg ∗ (λ,y ∗ ,z ∗ )∈S ⊕ ×X ∗ ×X ∗ ∗ ∗

{g ∗ (x ∗ ) − f ∗ (x ∗ − y ∗ − z ∗ )

−δC∗ (y ∗ ) − (λh) (z )}.

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Proof (i) Let r ∈ R and ψ(x) = g(x)+r . Then, ψ ∈ Γ (X ) and ψ ∗ (x ∗ ) = g ∗ (x ∗ )−r for each x ∗ ∈ X ∗ . Thus, applying Theorem 5.2 to ψ in place of g, we see that (i) holds. (ii) Let r = inf x∈A { f (x) − g(x)}. Without loss of generality, we assume that r = −∞. Then r ∈ R as dom( f − g) ∩ A = ∅ and f (x) − g(x) ≥ r for each x ∈ A. Thus, by (i), for each x ∗ ∈ domg ∗ , there exists (λ, y ∗ , z ∗ ) ∈ S ⊕ × X ∗ × X ∗ such that g ∗ (x ∗ ) − f ∗ (x ∗ − y ∗ − z ∗ ) − δC∗ (y ∗ ) − (λh)∗ (z ∗ ) ≥ r. This together with the weak duality implies that (ii) holds. The proof is complete.   Analogously, we can get the following theorem. Theorem 5.6 Suppose that the family {δC , λh} satisfies the condition C2 ( f, A). Then, (i) For each r ∈ R, [ f (x) − g(x) ≥ r, ∀x ∈ A] ⇐⇒ [∀x ∗ ∈ domg ∗ , ∃(λ, y ∗ ) ∈ S ⊕ × X ∗ s.t. g ∗ (x ∗ ) − f ∗ (x ∗ − y ∗ ) − (δC + λh)∗ (y ∗ ) ≥ r. (ii) The strong duality holds between (P) and (D), that is, inf { f (x) − g(x)} =

x∈A

inf

max

x ∗ ∈domg ∗ (λ,y ∗ )∈S ⊕ ×X ∗ −(δC + λh)∗ (y ∗ )}.

{g ∗ (x ∗ ) − f ∗ (x ∗ − y ∗ )

Remark 5.2 Under the assumptions that f and g are lsc, C is closed, and h is star S-lsc, Dinh. et al. established the extended Farkas lemmas for problem (P) in [20, Theorem 4.3] and the strong duality between (P) and (D), between (P) and (D) in [19, Theorem 4.2] via the condition (CC). As stated in Remark 5.1(b), the condition (CC) and C1 ( f, A) are equivalent under those assumptions. Thus, our theorems extend and improve the corresponding ones in [19,20].

6 Conclusions In this paper, a generalized class of conic constrained optimization problems with objectives given by the differences of a composite function and a convex function is considered, and simultaneously, some constraint qualifications are proposed to provide a complete characterization for several versions of Farkas lemmas. As applications, some characterizations of strong duality and stable strong duality properties for conic constrained optimization problems are provided. Acknowledgements The authors thank the editor and the referees for their valuable comments and constructive suggestions which improved the presentation of this manuscript. Research work of the first author is supported in part by the National Natural Science Foundation of China (Grant 11461027), Hunan Provincial Natural Science Foundation of China (Grant 2016JJ2099) and the Scientific Research Fund of Hunan

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J Optim Theory Appl Provincial Education Department (Grant 17A172). The second authors is supported in part by the Scientific Research Fund of Hunan Provincial Education Department (Grant 15C1156).

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