Duality and Farkas-type results for DC fractional programming with DC constraints

Mathematical and Computer Modelling 53 (2011) 1026–1034

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Duality and Farkas-type results for DC fractional programming with DC constraints Hai-Jun Wang ∗ , Cao-Zong Cheng Department of Mathematics, Beijing University of Technology, Beijing 100124, PR China

article

info

Article history: Received 15 July 2010 Received in revised form 8 November 2010 Accepted 9 November 2010 Keywords: Fractional programming DC functions DC constraints Conjugate duality Farkas-type results

abstract The aim of this paper is to discuss the fractional programming problem (P ) that an objective function is a ratio of two DC (difference of convex) functions with finitely many DC constraints. A type of dual problem is constructed and the duality assertions are obtained. By using the obtained duality assertions, some Farkas-type results which characterize the optimal value of the problem (P ) are given. Some programming problems considered in recent literature are the special cases of the problem (P ). © 2010 Elsevier Ltd. All rights reserved.

1. Introduction For both theoretical and practical requirements, fractional programming problems have been paid an increasing amount of attention in recent decades. People have focused on constructing dual problems of various fractional programming problems and discussing the conditions that the weak or strong duality holds (see, for example, [1–5]). Some necessary and sufficient optimality conditions which characterize the optimal value of the fractional programming problems are also given such as in [6,7]. In this paper, we consider the following fractional programming problem that an objective function is a ratio of two DC functions with finitely many DC constraints:

(P )

inf

x∈F (P )

f (x) − g (x) u(x) − v(x)

where X ⊆ Rn is a nonempty convex set, f , g , −u, −v, φi and ψi (i = 1, . . . , m) : Rn → R = R ∪ {±∞} are proper convex functions, and F (P ) = {x ∈ X : φi (x) − ψi (x) ≤ 0, i = 1, . . . , m}, m ∈ N. We will assume that f − g, and −u + v are proper and u − v is positive on the F (P ). Let λ ∈ R. For the above fractional programming problem, we associate it with the following problem by using the idea due to Dinkelbach [3]

(P λ )

inf (f (x) − g (x) − λu(x) + λv(x)).

x∈F (P )

To the problem (P λ ), we will determine its Fenchel–Lagrange type dual problem which was first considered in [8] and further studied in [9–13]. By using the similar approach in [14,9], we construct a dual problem for (P λ ) and give a constraint qualification guaranteeing the strong duality between (P λ ) and its dual problem. According to the duality assertions,

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (H.-J. Wang), [email protected] (C.-Z. Cheng).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.059

H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

1027

we obtain some Farkas-type results for the primal fractional programming problem, that is, obtain some equivalent characterizations for the optimal value of the primal problem. We will give an example to illustrate our main results. Finally, we will show that some recently obtained Farkas-type results in the literature are the corollaries of the results of this paper. Note that the problem (P ) is based on the finite dimensional spaces. By the way, for the case of the infinite dimensional spaces, some convex programming problems (not the problem (P )) are also considered by some authors. We mention the Refs. [15–17]. For convenience’s sake, we first recall some notations and some known facts. Let X ⊆ Rn . We denote the relative interior, the convex hull and the closure of the set  X by ri(X ), co(X ) and cl(X ) respectively; denote the cone and the convex cone generated by the set X by cone(X ) = λ≥0 λX and coneco(X ) =

λcoX respectively. The support function σX : Rn → R of X is defined by σX (u) = supx∈X uT x. The indicator function δX : R → R of X is defined by δX (x) = 0 if x ∈ X , and δX (x) = +∞ if x ̸∈ X . For a function f : Rn → R, the effective domain and epigraph of f are given by dom(f ) = {x ∈ Rn : f (x) < +∞} and epi(f ) = {(x, r ) ∈ Rn × R : f (x) ≤ r }. We say that f is proper if dom(f ) ̸= ∅ and f (x) > −∞ for all x ∈ Rn . The subdifferential of f at x is defined by ∂ f (x) = {x∗ ∈ Rn : f (y) − f (x) ≥ (y − x)T x∗ , ∀y ∈ Rn }. We say that f is subdifferentiable at x ∈ Rn if ∂ f (x) ̸= ∅, and f is subdifferentiable on the set X ⊆ Rn if f is subdifferentiable at x for each x ∈ X . The conjugate function fX∗ : Rn → R of f relative to the set X is defined by fX∗ (p) = sup{pT x − f (x) : x ∈ X }. Note that if X = Rn , the conjugate function of f relative to the set X is just the (Fenchel–Moreau) conjugate function of f , denoted by f ∗ . It is known that x∗ ∈ ∂ f (x) if and only if f (x) + f ∗ (x∗ ) = x∗ T x, and that epi((α f )∗ ) = α epi(f ∗ ) for a function f and a positive real number α . 

λ≥0

n

In this paper, we adopt the following conventions as in [7]:



(±∞) − (±∞) = (±∞) + (∓∞) = +∞, 0 × (±∞) = 0, r × (±∞) = ∓∞ for r < 0.

r × (±∞) = ±∞ for r > 0,

Let  fi : Rn → R (i = 1, . . . , m) be proper convex functions. By Theorem 16.4 in [18] and Corollary 2.2 in [9], we know m that if i=1 ri(dom(fi )) is nonempty, then (i) ( i=1 fi )∗ (p) = inf{ i=1 fi∗ (pi ) : p = i=1 pi }; (ii) for all∑ p ∈ Rn , the infimum in (i) is attained. ∑ m m ∗ (iii) epi(( i=1 fi )∗ ) = i=1 epi(fi ).

∑m

∑m

∑m

We will denote val(P ) as the optimal value of an optimization problem (P ) in this paper. 2. Main results In this section, we always assume that

 X



dom(f − g )



m 

 (φi − ψi ) (−R+ ) ̸= ∅. −1

i=1

Then F (P ) ̸= ∅. Moreover, we suppose that ψi (i = 1, . . . , m) are subdifferentiable on F (P ). It is obvious that the following relation between val(P ) and val(P λ ) holds. Lemma 2.1. The inequality val(P ) ≥ λ holds if and only if the inequality val(P λ ) ≥ 0 holds. According to the subdifferentiability of ψi (i = 1, . . . , m), we can give another composition of the feasible set F (P ) of the problem (P ) as the following lemma due to Boţ–Hodrea–Wanka [9]. Lemma 2.2. Let φi , ψi (i = 1, . . . , m) assumed as above. We have

F (P ) =



{x ∈ X : φi (x) − y∗i T x + ψi∗ (y∗i ) ≤ 0, i = 1, . . . , m}.

y∗ ∈dom(ψ ∗ ) i i i=1,...,m

Throughout the paper we shall denote y∗ ∈ i=1 dom(ψi∗ ) for y∗i ∈ dom(ψi∗ ) (i = 1, . . . , m), where y∗ is the m-tuple (y1 , . . . , y∗m ). Motivated by the approach of DC optimization problems (see [9,11,13,19]), we will give a dual problem for (P λ ). It is easy to see that the objective function f − g − λu + λv of the problem (P λ ) can always be viewed as the difference of two convex functions: f − λu and g − λv are two convex functions when λ is nonnegative; f + λv and g + λu are two convex functions when λ is negative. Thus we discuss the duality as two cases: λ ≥ 0 and λ < 0.

∏m

∗

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H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

2.1. When λ ≥ 0 In this case, we assume further that

 ri(dom(f ))



ri(dom(−u))



ri(X )



m 

 ri(dom(φi ))

̸= ∅,

i=1

and that g − λv is lower semicontinuous on the feasible set F (P ). Let F , G : Rn → R by F (x) = f (x) − λu(x) and G(x) = g − λv(x). Then the problem (P λ ) can be written as

(P λ )

inf {F (x) − G(x)}.

x∈F (P )

Since ri(dom(f ))∩ ri(dom(−u)) ̸= ∅, by using Theorem 6.5 in [18] we have that ri(dom(f ))∩ ri(dom(−u)) = ri(dom(f )∩ dom(−u)) = ri(dom(F )) whenever λ > 0. If λ  = 0, we have that dom(F ) = dom(f ), and so ri(dom(f )) ∩ ri(dom(−u)) ⊆ m ri(dom(F )). Therefore, ri(dom(F )) ∩ ri(X ) ∩ ( i=1 ri(dom(φi ))) ̸= ∅. In this case, the dual problem of (P λ ) has been constructed in [9] as

 λ

(D )

inf

G (z ) + ∗

sup

z ∗ ∈domG∗ x∗ ∈Rn m m ∏ y∗ ∈ dom(ψ ∗ ) α∈R+ i i=1

Noticing that ri(dom(f ))

∗

m −

 αi ψi (yi ) − F (x ) − ∗

∗

∗

∗

i=1

m −

∗  αi φi

i=1

X

m −

 αi yi + z − x ∗

∗

∗

.

i =1

ri(dom(−u)) ̸= ∅, one gets that



F (x ) = (f − λu) (x ) = inf{ f ∗ (p∗ ) + (−λu)∗ (q∗ ) : p∗ + q∗ = x∗ } ∗

∗

∗

∗

= inf{ f ∗ (p∗ ) + λ(−u)∗ (q∗ ) : p∗ + λq∗ = x∗ }, and the infimum as above is attained for any x∗ ∈ Rn . Therefore the dual (Dλ ) has the form

 λ

(D )

 −

(g − λv)∗ (z ∗ ) − f ∗ (p∗ ) − λ(−u)∗ (q∗ ) +

sup

inf

z ∗ ∈dom(g −λv)∗ p∗ ,q∗ ∈Rn m ∏ α∈Rm dom(ψ ∗ ) y∗ ∈ + i i=1

m −

∗  αi φi

i =1

X

m −

m −

αi ψi∗ (y∗i )

i =1

 αi yi − p − λq + z ∗

∗

∗

∗

.

i=1

In order to obtain the duality assertions, we introduce the following generalized interior point constraint qualification:

(CQy∗ ) ∃ x ∈ ri(dom(f )) ′



ri(dom(−u))

 m  

 

ri (dom(φi ))

ri(X ), such that

i=1

 φi (x′ ) − y∗i T x′ + ψi∗ (y∗i ) ≤ 0, φi (x′ ) − y∗i T x′ + ψi∗ (y∗i ) < 0,

i ∈ L, i ∈ N,

where L := {i ∈ {1, . . . , m} : φi is an affine function} and N := {1, . . . , m} \ L. From [9, Theorems 3.5 and 3.6] and the above discussion, one can state the weak and strong duality assertions as follows. Theorem 2.1. Between (P λ ) and (Dλ ), (i) the weak duality holds, that is, val(P λ ) ≥ val(Dλ ); ∏m (ii) the strong duality holds, that is, val(P λ ) = val(Dλ ), if (CQy∗ ) is fulfilled for all y∗ ∈ i=1 dom(ψi∗ ). Now, we turn to give the following Farkas-type results for λ ≥ 0. Theorem 2.2. Suppose that (CQy∗ ) holds for all y∗ ∈ (F1) x ∈ F (P ) H⇒

∏m

i=1

dom(ψi∗ ) and λ ≥ 0. Then the following assertions are equivalent:

f (x)−g (x) u(x)−v(x)

≥ λ; ∏m (F2) for any z ∈ dom((g − λv)∗ ) and y∗ ∈ i=1 dom(ψi∗ ), there exist p∗ , q∗ ∈ Rn and α ∈ Rm + , such that  ∗   m m m − − − ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ f (p ) + λ(−u) (q ) − (g − λv) (z ) + αi φi αi yi − p − λq + z − αi ψi∗ (y∗i ) ≤ 0; ∗

i =1

(F3) for any z ∈ dom((g − λv) ) and y ∈ ∗

∗

∗

∏m

i=1

X

i=1

i=1

dom(ψi ), there exist p , q , r ∈ R and α ∈ R+ , such that ∗

∗

∗

∗

n

m

H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

 f ∗ (p∗ ) + λ(−u)∗ (q∗ ) − (g − λv)∗ (z ∗ ) +

m −

∗  αi φi

i=1

−

m −

m −

1029

 αi y∗i − p∗ − λq∗ + z ∗ − r ∗ + σX (r ∗ )

i =1

αi ψi∗ (y∗i ) ≤ 0.

i=1

Proof. (F1)H⇒(F2): By Lemma 2.1, we have val(P λ ) =

inf {F (x) − G(x)} ≥ 0.

x∈F (P )

By Theorem 4.1 in [9], for any z ∗ ∈ dom(G∗ ) and y∗ ∈

(G) (z ) + ∗

∗

m −

 αi ψi (yi ) − F (x ) − ∗

∗

∗

∗

m −

i=1

∏m

i=1

∗  αi φi

i=1

dom(ψi∗ ) there exist x∗ ∈ Rn and α ∈ Rm + such that

m −

 αi yi + z − x ∗

∗

≥ 0.

∗

i=1

X

Since ri(dom(f )) ri(dom(−u)) ̸= ∅, then there exist p∗ , q∗ ∈ Rn with p∗ +λq∗ = x∗ such that F ∗ (x∗ ) = (f −λu)∗ (x∗ ) = f ∗ (p∗ ) + λ(−u)∗ (q∗ ). Thus, (F2) is true. (F2)H⇒ (F1): If (F2) is fulfilled, it follows that val(Dλ ) ≥ 0. By weak duality, one gets that val(P λ ) ≥ val(Dλ ) ≥ 0. Thus, the assertion (F1) holds by Lemma 2.1. m  ri(X ) ̸= ∅. Thus, From the condition (CQy∗ ), we have that ( i=1 ri(dom(φi )))





m −

∗  αi φi

i =1

m −

 αi yi − p − λq + z ∗

∗

∗

∗

= ∗infn r ∈R

i=1

X

 m −

∗  αi φi

i=1

m −

 αi yi − p − λq + z − r ∗

∗

∗

∗

∗

 + σX (r ) , ∗

i =1

and the infimum is obtained. So, the equivalence of (F2) and (F3) is obvious.



Next, we will give an equivalent assertion to the statement (F2) in Theorem 2.2 by using the epigraphs of the involved functions. Theorem 2.3. In the case of λ ≥ 0, the statement (F2)in Theorem 2.2 is equivalent to



   m  ∗ ∗ ∗ ∗ epi(f ) + λepi((−u) ) + coneco (epi(φi ) − (yi , ψi (yi ))) + epi(σX ) .



epi((g − λv) ) ⊆ ∗

y∗ ∈

m ∏

i=1

∗

∗

i =1

dom(ψi∗ )

∗ Proof. By the proof of Theorem ∏m 2.2, we have that the statement (F2) in Theorem 2.2 holds if and only if for any z ∈ dom((g − λv)∗ ) and y∗ ∈ i=1 dom(ψi∗ ), there exist x∗ ∈ Rn and α ∈ Rm , such that +

(G) (z ) + ∗

∗

m −

 αi ψi (yi ) − F (x ) − ∗

∗

∗

∗

i=1

m −

∗  αi φi

i=1

X

m −

 αi yi + z − x ∗

∗

∗

≥ 0.

i=1

In view of Theorem 4.3 in [9], we just need to prove that epi(F ∗ ) = epi(f ∗ ) + λepi((−u)∗ ). If λ = 0, the conclusion is obviously true. If λ > 0, from the assumption given in this subsection we have that epi(F ∗ ) = epi((f − λu)∗ ) = epi(f ∗ ) + epi((−λu)∗ ) = epi(f ∗ ) + λepi((−u)∗ ). The proof is completed.



2.2. When λ < 0 In this case, we assume further that ri(dom(f ))



ri(dom(−v))



ri(X )

 m  

 ri(dom(φi ))

i =1

and that g + λu is lower semicontinuous on the feasible set F (P ).

̸= ∅,

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H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

By using the similar discussion to the case of λ ≥ 0, if only λ, u and v are replaced by −λ, v and u respectively, we can construct the following dual problem for (P λ ):

 (D ) ′λ

(g + λu)∗ (z ∗ ) − f ∗ (p∗ ) + λ(−v)∗ (q∗ ) +

sup

inf

z ∗ ∈dom(g +λu)∗ ∗ ∗ n p ,q ∈R m ∏ m y∗ ∈ dom(ψ ∗ ) α∈R+ i i=1

 −

m −

∗  αi φi

i =1

X

m −

m −

αi ψi∗ (y∗i )

i=1

 αi yi − p + λq + z ∗

∗

∗

.

∗

i=1

The generalized interior point constraint qualification is formed:

(CQ

′

y∗

) ∃ x ∈ ri(dom(f )) ′



ri(dom(−v))

 m  

 ri(dom(φi ))



ri(X ), such that

i =1

 φi (x′ ) − y∗i T x′ + ψi∗ (y∗i ) ≤ 0, φi (x′ ) − y∗i T x′ + ψi∗ (y∗i ) < 0,

i∈L i ∈ N,

where L := {i ∈ {1, . . . , m} : φi is an affine function} and N := {1, . . . , m} \ L. Theorem 2.4. Between (P λ ) and (D′λ ), (1) the weak duality holds, that is, val(P λ ) ≥ val(D′λ ); ∏m (2) the strong duality holds, that is, val(P λ ) = val(D′λ ), if (CQy′∗ ) is fulfilled for all y∗ ∈ i=1 dom(ψi∗ ). As in the previous subsection, we can also give the Farkas-type results for the case of λ < 0 as follows. Theorem 2.5. Suppose (CQ ′ y∗ ) is fulfilled for all y∗ ∈ f (x)−g (x) u(x)−v(x)

(F1) x ∈ F (P ) ⇒

i=1

dom(ψi∗ ), and λ < 0. Then the following assertions are equivalent:

≥ λ;

(F2) for any z ∈ dom(g + λu)∗ and y∗ ∈ ∗

∏m

∏m

i=1

dom(ψi∗ ), there exist p∗ , q∗ ∈ Rn and α ∈ Rm + , such that

 f ∗ (p∗ ) − λ(−v)∗ (q∗ ) − (g + λu)∗ (z ∗ ) +

m −

∗  αi φi

i =1

(F3) for any z ∗ ∈ dom(g + λu)∗ and y∗ ∈

X

αi y∗i − p∗ + λq∗ + z ∗ −

i=1

m −

αi ψi∗ (y∗i ) ≤ 0;

i=1

∗ ∗ ∗ ∗ n m i=1 dom(ψi ), there exist p , q , r ∈ R and α ∈ R+ , such that

 m −

∗  αi φi

i=1 m −



∏m

f ∗ (p∗ ) − λ(−v)∗ (q∗ ) − (g + λv)∗ (z ∗ ) +

−

m −

m −

 αi y∗i − p∗ + λq∗ + z ∗ − r ∗ + σX (r ∗ )

i =1

αi ψi∗ (y∗i ) ≤ 0.

i=1

Theorem 2.6. In the case of λ < 0, the statement (F2)in Theorem 2.5 is equivalent to

 

epi((g + λu) ) ⊆ ∗

y∗ ∈

m ∏

i=1

dom(ψi∗ )



  m  ∗ ∗ ∗ ∗ epi(f ) − λepi((−v) ) + coneco (epi(φi ) − (yi , ψi (yi ))) + epi(σX ) . ∗

∗

i=1

3. An example In this section, we will give an example to illustrate the constraint qualification, the strong duality assertions and the Farkas-type results obtained in the above section. We consider the fractional programming problem as follows

(P )

inf x∈X

φ(x)−ψ(x)≤0

f (x) − g (x) u(x) − v(x)

,

where X = R, f (x) = 4|x|, g (x) = 2x2 − 6, u(x) = −2|x|, v(x) = −(x2 + 2) and φ(x) = 12 x2 − 1, ψ(x) = |x|. It is obvious that the functions f − g, u − v and φ − ψ are all DC functions, but neither convex nor concave functions, and that u(x) − v(x) > 0 for all x ∈ R. First, we can calculate the optima value val(P ) = 21 .

H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

1031

For any λ ∈ R, the problem (P λ ) will be

(P λ )

inf

x∈ R 1 x2 −1−|x|≤0 2

{4|x| − (2x2 − 6) + 2λ|x| − λ(x2 + 2)},

and the optimal value of (P λ ) can be calculated as val(P λ ) =



2 − 4λ, 8 − λ,

if λ ≥ −2, if λ < −2.

The conjugate function of ψ can be formulated as ψ (y ) = 0 if y ∈ [−1, 1], and ψ ∗ (y∗ ) = +∞ if y∗ ∈ (−∞, −1) ∪ (1, +∞). Then the feasible set ∗



F (P ) =

 x∈R:

y∗ ∈[−1,1]

1 2

∗

∗



x2 − 1 − y∗ x ≤ 0 .

Now, we consider the case of λ ≥ 0. We can easily calculate the conjugate functions of f , −u, g − λv and αφ (α ≥ 0). f ∗ (p∗ ) =



0,

+∞,

(g − λv)∗ (z ∗ ) =

p∗ ∈ [−4, 4], otherwise,

(−u)∗ (q∗ ) =

4(2 + λ)(6 − 2λ) + z

∗2

4(2 + λ)

,



0,

+∞,

q∗ ∈ [−2, 2], otherwise,

 2 2   2α + r , ∗ 2α (αφ) (r ) =  0, +∞,

α > 0, α = 0, r = 0, α = 0, r ̸= 0.

Then for any z ∗ ∈ dom((g − λv)∗ ) = R and y∗ ∈ dom(ψ ∗ ) = [−1, 1], f ∗ (p∗ ) + λ(−u)∗ (q∗ ) − (g − λv)∗ (z ∗ ) + (αφ)∗ (α y∗ − p∗ − λq∗ + z ∗ ) − αψ ∗ (y∗ )

 2α 2 + (α y∗ − p∗ − λq∗ + z ∗ )2 4(2 + λ)(6 − 2λ) + z ∗ 2    + , if p∗ ∈ [−4, 4], q∗ ∈ [−2, 2], α ̸= 0, −   4(2 + λ) 2α = 4(2 + λ)(6 − 2λ) + z ∗ 2  − , if p∗ ∈ [−4, 4], q∗ ∈ [−2, 2], α = 0, and α y∗ − p∗ − λq∗ + z ∗ = 0,   4(2 + λ)   +∞, if α = 0, α y∗ − p∗ − λq∗ + z ∗ ̸= 0 or p∗ ̸∈ [−4, 4], or q∗ ̸∈ [−2, 2].    It is easy to see that ri(dom(f )) ri(dom(−u)) ri(dom(φ)) ri(X ) = R, and that φ(x′ ) − y∗ x′ + ψ(y∗ ) = −1 < 0 for all ∗ ′ y ∈ [−1, 1] when taking x = 0. This shows that the constraint qualification (COy∗ ) holds for all y∗ ∈ [−1, 1]. The dual problem (Dλ ) will be   (Dλ ) inf sup ( g − λv)∗ (z ∗ ) − (αφ)∗ (α y∗ − p∗ − λq∗ + z ∗ ) . ∗ z ∈R y∗ ∈[−1,1]

p∗ ∈[−4,4] q∗ ∈[−2,2],α∈R+

Since for any given z ∗ ∈ R and y∗ ∈ [−1, 1], sup p∗ ∈[−4,4] q∗ ∈[−2,2],α∈R+



 (g − λv)∗ (z ∗ ) − (αφ)∗ (α y∗ − p∗ − λq∗ + z ∗ )

  4(2 + λ)(6 − 2λ) + z ∗ 2  ∗ ∗  + ( 4 + 2 λ − z )( y + y∗ 2 + 2),    4(2 + λ)   4(2 + λ)(6 − 2λ) + z ∗ 2 = ,  4(2 + λ)    2   4(2 + λ)(6 − 2λ) + z ∗   − (4 + 2λ + z ∗ )(y∗ − y∗ 2 + 2), 4(2 + λ)

if z ∗ > 4 + 2λ, if − (4 + 2λ) ≤ z ∗ ≤ 4 + 2λ, if z ∗ < −4 − 2λ,

then val(Dλ ) = 2 − 4λ = val(P λ )(λ ≥ 0), i.e. the strong duality between (P λ ) and (Dλ ) holds. f (x)−g (x) If 0 ≤ λ ≤ 12 , it is obvious that x ∈ R, φ(x) − ψ(x) ≤ 0 H⇒ u(x)−v(x) ≥ λ. Meanwhile for any z ∗ ∈ R, and y∗ ∈ [−1, 1], the inequality f ∗ (p∗ ) + λ(−u)∗ (q∗ ) − (g − λv)∗ (z ∗ ) + (αφ)∗ (α y∗ − p∗ − λq∗ + z ∗ ) − αψ ∗ (y∗ ) ≤ 0 has a solution (p∗ , q∗ , α) ∈ R × R × R+ by being taken as follows

   α    α      α α

z ∗ − 4 − 2λ =  , q∗ = 2, p∗ = 4, y∗ 2 + 2 −z ∗ − 4 − 2λ =  , q∗ = −2, p∗ = −4, y∗ 2 + 2 = 0, q∗ = 2, p∗ = z ∗ − 2λ, ∗ = 0, q = −2, p∗ = z ∗ + 2λ,

if z ∗ > 4 + 2λ, if z ∗ < −4 − 2λ, if 0 ≤ z ∗ ≤ 4 + 2λ, if − 4 − 2λ ≤ z ∗ < 0.

This shows that both of the assertions (F1) and (F2) in Theorem 2.2 are satisfied when 0 ≤ λ ≤

1 . 2

1032

H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034 f (x)−g (x)

If λ > 12 , the assertion ‘‘x ∈ R, φ(x) − ψ(x) ≤ 0 H⇒ u(x)−v(x) ≥ λ’’ is not true. In this case, for z ∗ = 2(1 + and y∗ = 1 ∈ [−1, 1], it is easy to see that for any p∗ , q∗ ∈ R and any α ∈ R+ , we have

√

3)(2 + λ) ∈ R

f ∗ (p∗ ) + λ(−u)∗ (q∗ ) − (g − λv)∗ (z ∗√ ) + (αφ)∗ (α y∗ − p∗ − λq∗ + z ∗ ) − αψ ∗ (y∗ ) √ 4(2 + λ)(6 − 2λ) + 4(1 + 3)2 (2 + λ)2 2α 2 + (α − 4 − 2λ + 2(1 + 3)(2 + λ))2

≥−

+

4(2 + λ)

≥ 4λ − 2 > 0.

2α

This shows that neither (F1) nor (F2) in Theorem 2.2 is fulfilled when λ > 12 . Next we consider the case of λ < 0. The conjugate functions of −v and g + λu can be calculated as q∗ 2

(−v) (q ) = ∗

∗

4

 (z ∗ + 2λ)2   ,  6 + 8 (g + λu)∗ (z ∗ ) = 6,  ∗ 2   6 + (z − 2λ) ,

− 2,

8

z ∗ > −2λ, 2λ ≤ z ∗ ≤ −2λ, z ∗ < 2λ.

Then dom((g + λu)∗ ) = R and the dual problem (D′λ ) will be

(D′λ )

inf

z ∗ ∈R y∗ ∈[−1,1]

sup



p∗ ∈[−4,4] q∗ ∈R,α∈R+

 λ(−v)∗ (q∗ ) + (g + λu)∗ (z ∗ ) − (αφ)∗ (α y∗ − p∗ + λq∗ + z ∗ ) .

In this case, one can see that (CQy′∗ ) also holds for every y∗ ∈ dom((ψ)∗ ) = [−1, 1]. We can also calculate that val(D′λ ) =



2 − 4λ, 8 − λ,

if − 2 ≤ λ < 0, ifλ < −2.

That is to say the strong duality between (P λ ) and

(D′λ )(λ < 0) holds. It is obvious that x ∈ R, φ(x) − ψ(x) ≤ 0 H⇒

f (x) − g (x) u(x) − v(x)

≥ λ(λ < 0).

For any z ∗ ∈ R and y∗ ∈ [−1, 1], taking p∗ , q∗ ∈ R and α ∈ R+ as follows

α     α         α        α             α      

= 0,

p∗ = z ∗ ,

= 0,

p∗ = 4 ,

= 0,

p∗ = −4,

q∗ = 0, z∗ − 4 , q∗ =

if − 4 ≤ z ∗ ≤ 4, if 4 < z ∗ ≤ −2λ(y∗ +

λ

q∗ =

z∗ + 4

λ  −4 + z ∗ + 2λ(y∗ + y∗ 2 + 2)  , = y∗ 2 + 2

, p∗ = 4,

if − 2λ(y∗ −



q∗ = 2(y∗ +



=

−4 − z ∗ − 2λ(y∗ − y∗ 2 + 2)  , y∗ 2 + 2

p∗ = −4,

q∗ = 2(y∗ −

y∗ 2 + 2) + 4,

y∗ 2 + 2) − 4 ≤ z ∗ < −4,

y∗ 2 + 2),

if z ∗ > −2λ(y∗ +







y∗ 2 + 2) + 4,



y∗ 2 + 2),

if z ∗ < −2λ(y∗ −



y∗ 2 + 2) − 4,

we get f ∗ (p∗ ) − λ(−v)∗ (q∗ ) − (g + λu)∗ (z ∗ ) + (αφ)∗ (α y∗ − p∗ + λq∗ + z ∗ ) − αψ ∗ (y∗ ) ≤ 0. This shows that the assertions (F1) and (F2) in Theorem 2.5 hold when λ < 0. From the above discussion about the problem (P ), we can see that for a fixed real number λ, both of the assertions (F1) and (F2) in Theorem 2.2 (resp. in Theorem 2.5) are satisfied or not satisfied. Since X = R, the equivalence of the assertions (F2) and (F3) in Theorem 2.2 (resp. in Theorem 2.5) is obvious. 4. Some corollaries In this section, we will give some special cases of our general results, which have been treated in the previous papers. When u ≡ 1, v ≡ 0, the problem (P ) becomes the following optimization problem with a DC objective function and finitely many DC constraint functions:

(P1 )

inf

x∈ X φi (x)−ψi (x)≤0 i=1,...,m

(f (x) − g (x)).

H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

1033

Let λ = 0. The (CQy∗ ) becomes the following condition

 (

CQy0∗



) ∃ x ∈ ri(dom(f )) ′

m 

 

ri(dom(φi ))

ri(X ), such that

i =1



φi (x′ ) − y∗i T x′ + ψi∗ (y∗i ) ≤ 0, φi (x′ ) − y∗i T x′ + ψi∗ (y∗i ) < 0,

i ∈ L, i ∈ N,

where L := {i ∈ {1, . . . , m} : φi is an affine function} and N := {1, . . . , m} \ L. The following corollary follows from Theorems 2.2 and 2.3. Corollary 4.1 ([9]). Suppose the condition (CQ 0y∗ ) holds for all y∗ ∈

∏m

i =1

dom(ψi∗ ). Then the following assertions are equivalent:

(i) x ∈ F (P ) ⇒ f (x) − g (x) ≥ 0;∏ m (ii) for any z ∗ ∈ dom(g ∗ ) and y∗ ∈ i=1 dom(ψi∗ ), there exist p∗ ∈ Rn and α ∈ Rm + , such that

 f (p ) − g (z ) + ∗

∗

∗

∗

m −

∗  αi φi

i=1

(iii) epi(g ∗ ) ⊆

 αi yi − p + z ∗

∗

∗

−

m −

i =1

X

∗ ∏ ∗ {epi(f ) y∗ ∈ m i=1 dom(ψi )



m −

+ coneco[

αi ψi∗ (y∗i ) ≤ 0;

i=1

∗ ∗ ∗ ∗ i=1 (epi(φi ) − (yi , ψi (yi )))] + epi(σX )}.

m

When ψi ≡ 0, φi : Rn → R (i = 1, . . . , m), the problem (P ) now becomes the following fractional programming problem which is discussed in [7].

(P2 )

inf

x∈X φi (x)≤0 i=1,...,m

f (x) − g (x) u(x) − v(x)

.

It is obvious that for all i = 1, . . . , m, we have that ψi∗ (0) = 0, and ψi∗ (x∗ ) = +∞ for any x∗ ̸= 0, and dom(φi ) = Rn . When λ ≥ 0, the constraint qualification (CQy∗ ) is simplified into the following

(CQ 0 ) ∃ x′ ∈ ri(dom(f ))  φi (x′ ) ≤ 0, i ∈ L, φi (x′ ) < 0, i ∈ N,



ri(dom(−u))



ri(X ), such that

where L := {i ∈ {1, . . . , m} : φi is an affine function} and N := {1, . . . , m} \ L. According to Theorems 2.2 and 2.3, we can have the following results at once. Corollary 4.2 ([7]). Suppose that condition (CQ 0 ) holds and λ ≥ 0. Then the following assertions are equivalent: f (x)−g (x)

(i) x ∈ X , φi (x) ≤ 0, i = 1, . . . , m ⇒ u(x)−v(x) ≥ λ; (ii) for any z ∗ ∈ dom((g − λv)∗ ), there exist p∗ , q∗ ∈ Rn and α ∈ Rm + , such that

 f (p ) + λ(−u) (q ) − (g − λv) (z ) + ∗

∗

∗

∗

∗

∗

m −

∗ αi φi

i =1

(−p∗ − λq∗ + z ∗ ) ≤ 0; X

m (iii) epi((g − λv) ) ⊆ epi(f ) + λepi((−u) ) + coneco( i=1 epi(φi∗ )) + epi(σX ). ∗

∗

∗

When λ < 0, the corresponding constraint qualification is as follows

(CQ ′ 0 ) ∃ x′ ∈ ri(dom(f ))  φi (x′ ) ≤ 0, i ∈ L, φi (x′ ) < 0, i ∈ N ,



ri(dom(−v))



ri(X ), such that

where L := {i ∈ {1, . . . , m} : φi is an affine function} and N := {1, . . . , m} \ L. According to Theorems 2.5 and 2.6, we can have the following results. Corollary 4.3 ([7]). Suppose condition (CQ ′ 0 ) holds and λ < 0. Then the following assertions are equivalent: f (x)−g (x)

(i) x ∈ X , φi (x) ≤ 0, i = 1, . . . , m ⇒ u(x)−v(x) ≥ λ; (ii) for any z ∗ ∈ dom((g + λu)∗ ), there exist p∗ , q∗ ∈ Rn and α ∈ Rm + , such that

 f (p ) − λ(−v) (q ) − (g + λu) (z ) + ∗

∗

∗

∗

∗

∗

m −

∗ αi φi

i =1

(iii) epi((g + λu)∗ ) ⊆ epi(f ∗ ) − λepi((−v)∗ ) + coneco(

(−p∗ + λq∗ + z ∗ ) ≤ 0; X

∗ i=1 epi(φi )) + epi(σX ).

m

1034

H.-J. Wang, C.-Z. Cheng / Mathematical and Computer Modelling 53 (2011) 1026–1034

Remark. When ψi ≡ 0 (i = 1, . . . , m), g ≡ 0 and v ≡ 0, the problem (P ) is actually the fractional programming problem with convex constraints which is treated in [6]. Furthermore, when u ≡ 1, the problem (P ) becomes a more special case, namely the optimization problem with convex objective function and convex constraints which is studied in [12]. It is easy to see that some Farkas-type results obtained in [6,12] for these special cases are also the corollaries of the results in this paper. In addition, it is showed in Remark 4.7 of [12] that some results obtained in [20] are their special case. Thus these results in [20] are also the corollaries of our results. Acknowledgement The authors are grateful to the anonymous reviewers for their valuable comments and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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