R E S E A R C H
Open Access
Optimality conditions for strict minimizers
of higher-order in semi-infinite
multi-objective optimization
Guolin Yu
**Correspondence: [email protected]
Institute of Applied Mathematics, Beifang University of Nationalities, Yinchuan, Ningxia 750021, P.R. China
Abstract
This paper is devoted to the study of optimality conditions for strict minimizers of higher-order for a non-smooth semi-infinite multi-objective optimization problem. We propose a generalized Guignard constraint qualification and a generalized Abadie constraint qualification for this problem under which necessary optimality conditions are proved. Under the assumptions of generalized higher-order strong convexity for the functions appearing in the formulation of the non-smooth semi-infinite multi-objective optimization problem, three sufficient optimality conditions are derived.
MSC: Primary 90C34; 90C40; secondary 49J52
Keywords: optimality conditions; multi-objective optimization; semi-infinite optimization; strict minimizer of higher-order
1 Introduction
In recent years, there has been considerable interest in the so-calledsemi-infinite multi-objective optimizationproblems (SIMOPs), which is the simultaneous minimization of finitely many scalar objective functions subject to an infinitely many constraints. SIMOPs have been investigated intensively by many researchers from several different perspec-tives. For example, the pseudo-Lipschitz property and the semicontinuity of the efficient solution map under some types of perturbation with respect to a parameter have been dis-cussed in [–]. The density of the set of all stable convex semi-infinite vector optimiza-tion problems has been established in []. However, the work on optimality condioptimiza-tions for SIMOPs is limited. Here we should mention that the authors in [] have examined the optimality conditions and duality relations in SIMOPs involving differentiable functions, whose constraints are required to depend continuously on an indextbelonging to a com-pact setT. For non-smooth semi-infinite multi-objective optimization problems, work has been done to obtain necessary optimality conditions for weakly efficient solutions and sufficient optimality conditions for efficient solutions by presenting several kinds of con-straint qualifications and imposing assumptions of generalized convexity (see []), and to establish necessary and sufficient conditions for (weakly) efficient solutions of SIMOPs by applying some advanced tools of variational analysis and generalized differentiation and
proposing the concepts of (strictly) generalized convex functions defined by using the lim-iting subdifferential of locally Lipschitz functions (see []). It is worth noticing that all of the above mentioned literature studies only weakly efficient solutions or efficient solutions of SIMOPs.
On the other hand, a continuing interest in the theory of multi-objective optimization is to define and characterize its solutions. Besides the weak efficiency and efficiency men-tioned above, a meaningful solution concept called a strict efficient solution of higher-order (also called a strict minimizer of higher-higher-order) was recently extended by Jiménez in [] from the strict minimizer of higher-order in scalar optimization given by Auslender in [] and Ward in []. Recently, Bhatia [] established necessary and sufficient optimal-ity conditions for strict efficiency of higher-order in multi-objective optimization under thebasic regularity conditionandgeneralized higher-order strong convexity assumption, respectively.
In this paper, we introduce the notion of a semi-strict minimizer of higher-order for a semi-infinite multi-objective optimization problem, which includes arbitrary many (possi-bly infinite) inequality constraints. For the purpose of investigating this new solution con-cept, we found that the notion of convexity that appears to be most appropriate in the de-velopment of sufficient optimality conditions is the strong convexity of higher-order []. The rest of this paper is organized as follows. In Section , some basic notations and results of non-smooth and convex analysis are reviewed, and the concept of a semi-strict minimizer for a semi-infinite multi-objective optimization problem is presented. In Section , we introduce the generalized Guignard constraint qualification and Abadie constraint qualification for SIMOPs. Necessary optimality conditions of Karush-Kuhn-Tuchker type are derived under these two constraint qualifications. Finally, in Section , three sufficient optimality conditions for SIMOPs are obtained under the assumption of some generalized strong convexity of higher-order.
2 Notations and preliminaries
Throughout the paper, we letRnbe then-dimensional Euclidean space endowed with the
Euclidean norm · ,Xbe a convex subset ofRn, andm≥ be a positive integer. LetWbe
a subset ofRn. We useclW,coW, andconeWto denote the closure ofW, the convex hull
ofW, and the conic hull ofW (i.e., the smallest convex cone containingW), respectively.
Definition .(see [–]) LetW be a nonempty subset ofRn. Thetangent conetoW
atx¯∈clW is the set defined by
T(W;x¯) :=h∈Rn:h= lim n→∞tn
xn–x¯such thatxn∈W,
lim n→∞x
n=x¯andt
n> for alln= , , . . .
.
Recall that a functionϕ:X→RisLipschitzatx¯∈Xif there exists a positive constant
Ksuch that
ϕ(x) –ϕ(x¯)≤Kx–x¯ for allx∈X,
whereKis called the rank ofϕatx¯.ϕis said to be Lipschitz onXifϕis Lipschitz at each
ofϕatx¯∈Xin the directionv∈Rn, denoted byϕ(x¯,v), is defined as
ϕ(x¯,v) = lim sup
(x,t)→(x¯,+)
ϕ(x+tv) –ϕ(x)
t .
Clarke’s generalized gradient ofϕatx¯∈X, denoted by∂ϕ(x¯), is defined as
∂ϕ(x¯) =ξ∈Rn:ϕ(x¯,v)≥ ξ,v for allv∈Rn.
It is well known that∂ϕ(x¯) is a nonempty convex compact set inRn.
Definition .(see []) Letϕ:Rn→Rbe Lipschitz atx¯∈Rn. It is said thatϕadmits a strict derivativeatx¯, an element ofRn, denoted byDsϕ(x¯), provided that, for eachx∈Rn, the following holds:
lim
(x,t)→(¯x,)
ϕ(x+tx) –ϕ(x)
t = Dsϕ(x¯),x
.
Ifϕadmits a strict derivative atx¯, thenϕis calledstrictly differentiableatx¯.
Lemma .(see []) Letϕ,ϕ,andϕ be Lipschitz from X toR,andx¯∈X.Then the
following properties hold:
(a) ϕ(x¯,v) =max{ξ,v :ξ∈∂ϕ(x¯)},for allv∈Rn.
(b) ∂(λϕ(x¯)) =λ∂ϕ(x¯),for allλ∈R. (c) ∂(ϕ+ϕ)(x¯)⊂∂ϕ(x¯) +∂ϕ(x¯).
Now, we recall the definition of thestrong convexity of order mfor a Lipschitz function.
Definition .(see [, ]) Letϕ:X→Rbe Lipschitz atx¯∈X.
(a) ϕis said to bestrongly convex of ordermatx¯if there exists a constantc> such that for eachx∈Xandξ∈∂ϕ(x¯)
ϕ(x) –ϕ(x¯)≥ ξ,x–x¯ +cx–x¯m.
(b) ϕis said to bestrongly quasiconvexof ordermatx¯if there exists a constantc> such that, for eachx∈Xandξ∈∂ϕ(x¯),
ϕ(x)≤ϕ(x¯) ⇒ ξ,x–x¯ +cx–x¯m≤.
Based upon the above definition of a strongly convex function of orderm, we define the following generalized strong convexities of ordermfor a Lipschitz function.
Definition . Letϕ:X→Rbe Lipschitz atx¯∈X.
(a) ϕisstrictly strong convexof ordermatx¯if there exists a constantc> such that, for eachx∈Xwithx=x¯andξ∈∂ϕ(x¯),
(b) ϕisstrictly strong quasiconvexof ordermatx¯if there exists a constantc> such that, for eachx∈Xwithx=x¯and anyξ∈∂ϕ(x¯),
ϕ(x)≤ϕ(x¯) ⇒ ξ,x–x¯ +cx–x¯m< .
The next Lemma gives a basic property of generalized higher-order strong convexities, which will be used in Section .
Proposition . Letϕi:X→Rbe Lipschitz atx¯∈X,i= , , , . . . ,s.Suppose thatϕis a
strictly strong convex function of order m andϕ,ϕ, . . . ,ϕsare strongly convex functions of order m atx¯.Ifλ> andλi≥for i= , , . . . ,s,then
s
i=λiϕiis strictly strong convex of order m atx¯.
Proof It is evident that the functionsi=λiϕiis Lipschitz atx¯. Thus, we get
∂
s
i=
λiϕi
(x¯)=∅.
Takingξ∈∂(si=λiϕi)(x¯). It follows from Lemma . that
∂
s
i=
λiϕi
(x¯)⊂
s
i=
∂(λiϕi)(x¯) = s
i=
λi∂ϕi(x¯).
This means that there existξi∈∂ϕi(x¯),i= , , . . . ,s, such that
ξ=
s
i=
λiξi.
Sinceϕis strictly strong convex of ordermatx¯andϕi,i= , , . . . ,s, is strongly convex of
ordermatx¯, we derive that there existci> ,i= , , . . . ,s, such that, for allx∈Rn,
⎧ ⎨ ⎩
ϕ(x) –ϕ(x¯) >ξ,x–x¯ +cx–x¯m,
ϕi(x) –ϕi(x¯)≥ ξi,x–x¯ +cix–x¯m for alli= , , . . . ,s,
which implies that
s
i=
λi
ϕi(x) –ϕi(x¯)
>
s
i=
λiξi,x–x¯ +
s
i=
λici
x–x¯m.
Therefore, we get
s
i=
λiϕi
(x) –
s
i=
λiϕi
(x¯) >ξ,x–x¯ +cx–x¯m,
Consider the followingsemi-infinite multi-objective optimizationproblem:
(P) Minimizef(x) =f(x),f(x), . . . ,fp(x)
subject togt(x)≤ fort∈T,
x∈X,
wherefi,i∈P={, , . . . ,p}, andgt,t∈Tare Lipschitz fromXtoR, and the index setTis arbitrary, not necessarily finite (but nonempty). The feasible set of (P) is denoted by,
:=x∈X:gt(x)≤,∀t∈T. For a givenx¯∈, set
ˆ
T(x¯) :=t∈T:gt(x¯) = .
In the sequel, we use the following notations. Forx,y∈Rn.
(i) f(x) <f(y)⇔fi(x) <fi(y)for everyi∈P; (ii) f(x)≮f(y)is the negation off(x) <f(y);
(iii) f(x)≤f(y)⇔fi(x)≤fi(y)for everyi∈P, but there is at least onei∈Psuch that
fi(x) <fi(y);
(iv) f(x)f(y)is the negation off(x)≤f(y).
Definition .(see []) A pointx¯∈is said to be a strict minimizer of ordermfor (P) if there existsc= (c,c, . . . ,cp)∈Rpwithci> ,i∈P, such that
f(x)≮f(x¯) +cx–x¯m for allx∈.
Definition . A pointx¯∈is said to be a semi-strict minimizer of ordermfor (P) if there existsc= (c,c, . . . ,cp)∈Rpwithci> ,i∈P, such that
f(x)f(x¯) +cx–x¯m for allx∈.
Remark . It is obvious that ifx¯∈is a semi-strict minimizer of ordermfor (P), then
¯
x∈is a strict minimizer of ordermfor (P).
Example . The functionsfi:R→R,i= , , defined by
f(x) =
⎧ ⎨ ⎩
x+ex ifx≥,
x+ ifx< , f(x) =
⎧ ⎨ ⎩
x+x ifx≥,
x–x ifx< ,
are Lipschitz atx¯= . It is easy to verify thatx¯is a semi-strict minimizer of order with
c= (, ) for the following optimization problem:
(P) Minimizef(x) =
f(x),f(x)
Motivated by the notion ofa linearizing coneat a point to the feasible set of a differen-tiable multi-objective optimization problem, which was introduced by Maeda in [], we give the definition of a linearizing cone for the semi-infinite multi-objective optimization problem (P). We first need to define a set.
Letx¯∈be a semi-strict minimizer of ordermfor (P), and define
Qi(x¯) :=x∈Rn:fk(x)≤fk(x¯) +ckx–x¯m,k∈Pandk=i∩.
It is obvious thatx¯∈Qi(x¯),i∈P.
Definition . Letx¯∈. Thelinearizing coneatx¯is the set defined by
C(x¯) =z∈Rn:ξ,z ≤ for allξ∈∂fi(x¯),i∈Pand
ζ,z ≤ for allζ ∈∂gt(x¯),t∈ ˆT(x¯).
3 Necessary conditions
In this section, we shall examine necessary optimality conditions for a semi-strict (strict) minimizer of order m for (P). we begin with presenting two constraint qualifications, which are the non-smooth, semi-infinite version of the generalized Guignard constraint qualification and generalized Abadie constraint qualification presented in [] and [].
Definition . The problem (P) satisfies thegeneralized Guignard constraint qualification
at a given pointx¯∈which is a semi-strict minimizer of ordermfor (P) if the following holds:C(x¯)⊆pi=cl[co(T(Qi;x¯))], whereQi:=Qi(x¯).
Definition . The problem (P) satisfies thegeneralized Abadie constraint qualification
at a given pointx¯∈which is a semi-strict minimizer of ordermfor (P) if the following holds:C(x¯)⊆ip=T(Qi;x¯).
Next, we recall thegeneralized Motzkin theoremdiscussed in [].
Lemma .(see []) Let A be a compact set inRn,B an arbitrary set inRn.Suppose that the setconeB is closed.Then either the system
⎧ ⎨ ⎩
a,z < for all a∈A,
b,z ≤ for all b∈B,
has a solution z∈Rn,or there exist integersμandν,with≤ν≤n+ ,such that there exist μpoints ai∈A(i= , , . . . ,μ),νpoints bm∈B(m= , , . . . ,ν),μnonnegative numbers ui, with ui> for at least one i∈ {, , . . .μ},andνpositive numbers vmfor m∈ {, , . . .ν}, such that
μ
i=
uiai+ ν
m=
vmbm= ,
The following lemma is a generalization of the classical Tucker theorem of the alterna-tive. We use this lemma in the proof of our necessary efficiency result. The proof of the lemma is similar to that of Lemma . in [], hence it is omitted.
Lemma . Let Ai⊂Rn,i∈P,be compact convex sets,B an arbitrary set inRn.Suppose that,for each i∈P,the setcone(B∪[j∈P,j=iAj])is closed.Then either the system
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
a,z < for at least one i∈P and for all a∈Ai,
a,z ≤ for all a∈pi=Ai,
b,z ≤ for all b∈B,
has a solution z∈Rn,or there exist u∈U≡ {u∈Rp:u> ,p
i=ui= },ai∈Aifor i∈P, and integerν with≤ν≤n+ ,such that there existν points bm∈B,andν positive numbers vm,m∈ {, , . . . ,ν},such that
p
i=
uiai+
ν
m=
vmbm= ,
but never both.
Lemma . Letx be a semi-strict minimizer of order m for¯ (P),let fi(x),i∈P,be Lipschitz atx of rank Ki¯ ,for all t∈T,let the functions gt(x)be Lipschitz atx¯.If the generalized Guignard constraint qualification holds atx and f¯ i(x),i∈P,are strictly differentiable atx¯ or the generalized Abadie constraint qualification holds atx¯,then the system
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
ξ,z < for at least one i∈P and for allξ∈∂fi(x¯),
ξ,z ≤ for allξ∈pi=∂fi(x¯),
ζ,z ≤ for allζ∈∂gt(x¯),t∈ ˆT(x¯),
(.)
has no solution z∈Rn.
Proof Suppose to the contrary that (.) has a solutionz. Thenz= andz∈C(x¯). Without loss of generality, we can assume that
ξ,z < for allξ∈∂f(x¯),
ξ,z ≤ for allξ∈ p
i=
∂fi(x¯).
By our generalized Guignard constraint qualification assumption,z∈cl[co(T(Q;x¯))], and hence there exists a sequence{zm}∞
m=⊂co(T(Q;x¯)) such that
lim m→∞z
m=z. (.)
For eachzm,m= , , . . . , there exist numbersL
mandλml≥, andzml∈T(;x¯),zml= , l= , , . . . ,Lm, such that
Lm
l=
λml= , Lm
l=
Since, for each m= , , . . . and l = , , . . . ,Lm, zml ∈T(Q;x¯), there exist sequences
{xmln}∞
n=⊂Qand{tmln}∞n=⊂R, withtmln> for alln, such that
lim n→∞x
mln=x¯, lim n→∞tmln
xmln–x¯=zml. (.)
Noticing thatxmln∈Qfor alln, we havexmln∈andf
i(xmln)≤fi(x¯) +cixmln–x¯mfor i= , , . . . ,p.
Sincex¯is a semi-strict minimizer of ordermfor (P), for allnwe may assume
f
xmln≥f(x¯) +cxmln–x¯m. (.)
Sincezml= , for eachm= , , . . . andl= , , . . . ,Lm, we must havetmln→+∞asn→
+∞, and hencexmln–t
mlnz
ml→ ¯xasn→+∞. Therefore,
fx¯;zml=lim sup y→¯x t↓
f(y+tzml) –f(y)
t
≥lim sup n→∞
f(xmln–tmln zml+tmln zml) –f(xmln–tmln zml)
tmln
≥lim sup n→∞
f(xmln) –f(x¯) +f(x¯) –f(xmln–t
mlnz
ml)
tmln
≥lim sup n→∞
f(xmln) –f(x¯)
tmln
–lim sup n→∞
|f(x¯) –f(xmln–tmln zml)|
tmln
≥lim sup n→∞
cxmln–x¯m
tmln
–lim sup n→∞
|f(x¯) –f(xmln–t
mlnz
ml)|
tmln
(by (.))
≥–lim sup n→∞
|f(x¯) –f(xmln–t
mlnz
ml)|
tmln
≥–lim sup n→∞tmlnK
x¯–xmln+
tmlnz ml
(by the Lipschitz continuity off)
= –lim sup n→∞
Ktmln
¯
x–xmln+zml
= (by (.)). (.)
Because f(x) is strictly differentiable at x¯, we know f(x;v) =Dsf(x¯),v. By
Propo-sition .. in [], we also know that ∂f(x¯) ={Dsf(x¯)}. In view of (.) we obtain
Dsf(x¯),zm =f(x¯;zm)≥, which further gives usDsf(x¯),z =f(x¯;z)≥ because of
(.), contradicting the assumption thatzis a solution of the system (.). Therefore, un-der the assumption that the generalized Guignard constraint qualification holds andfi(x), i∈P, are strictly differentiable atx¯, (.) has no solutionz∈Rn.
z ∈C(x¯). By the generalized Abadie constraint qualification, without loss of general-ity we may assume thatz∈T(Q;x¯), and hence there exist sequences{xn}∞
n=⊂Q and
{tn}∞n=⊂R, withtn> for alln, such that
lim n→∞x
n=x¯, lim n→∞tn
xn–x¯=z.
Replacingtmlnbytn,xmlnbyxn, andzmlbyzin (.), we arrive atf
(x¯;z)≥. By
Propo-sition .. in [], we know that there is aξ∈∂f(x¯) such thatξ,z =f(x¯;z)≥,
con-tradicting the assumption thatzis a solution of the system (.). Therefore, (.) has no
solutionz∈Rn.
Now we are ready to prove the following necessary optimality condition for (P).
Theorem . Letx¯∈ and let the functions fi(x)for i∈P and gt(x)for t∈T be Lips-chitz at x¯.Ifx is a semi-strict minimizer of order m for¯ (P),if the generalized Guignard constraint qualification holds at x and fi¯ (x)for each i∈P is strictly differentiable at x¯
(or the generalized Abadie constraint qualification holds at x¯), and if for each i∈P,
the setcone({ζ ∈∂gt(x¯) :t∈ ˆT(x¯)} ∪ {ξ ∈∂fi(x¯) :i∈P,i=i})is closed,then there exist
u∗∈U≡ {u∈Rp:u> ,pi=ui= },integersν∗,with≤ν∗≤n+ ,such that there exist tm∈ ˆT(x¯)and vm∗ > ,m∈ {, , . . . ,ν∗},with the property that
∈
p
i=
u∗i∂fi(x¯) + ν∗
m=
v∗m∂gtm(x¯). (.)
Proof In Lemma ., set
Ai=∂fi(x¯), i∈P, B=
t∈ ˆT(x¯)
∂gt(x¯).
By Proposition .. in [], we know that Ai is a compact convex set. According to
Lemma ., the system (.) has no solution and, therefore, by Lemma ., there exist
u∗∈U,ai∈∂fi(x) fori∈P, integersν∗with ≤ν∗≤n+ , such that there existν∗points tm∈ ˆT(x¯) andν∗positive numbersv∗
m> ,m∈ {, , . . . ,ν∗}, such that (.) holds. Remark . If we modify the definition ofQi(x¯) as follows:
Qi(x¯) :=x∈Rn:fk(x) <fk(x¯) +ckx–x¯,k∈Pandk=i∩,
and define the generalized Guignard constraint qualification and generalized Abadie con-straint qualification accordingly, we can prove a similar necessary result forx¯to be a strict minimizer of ordermfor (P).
4 Sufficient conditions
Theorem .(Sufficient optimality conditions I) Letx¯∈andTˆ(x¯)=∅.Suppose that there exist scalarsαi≥,i= , , . . . ,p with
p
i=αi= ,andβt≥,t∈ ˆT(x¯)withβt= for finitely many indices t,such that
∈
p
i=
αi∂fi(x¯) +
t∈ ˆT(x¯)
βt∂gt(x¯). (.)
If the functions fi,i= , , . . . ,p,are strongly convex of order m atx¯,and gtfor t∈ ˆT(x¯)and βt= ,are strongly quasiconvex of order m atx¯,thenx is a strict minimizer of order m¯ for(P).
Proof LetJ(x¯) :={t∈ ˆT(x¯) :βt= }. Because of (.), we derive that there existξi∈∂fi(x¯)
fori∈ {, , . . . ,p}andζt∈∂gt(x¯) fort∈J(x¯) such that
p
i=
αiξi+
t∈J(x¯)
βtζt= . (.)
Sincefifori∈ {, , . . . ,p}are strongly convex of ordermatx¯, we see that there existci¯ > fori∈ {, , . . . ,p}such that
fi(x) –fi(x¯)≥ ξi,x–x¯ +ci¯x–x¯m for allx∈,ξi∈∂fi(x¯),i= , , . . . ,p.
Noticingαi≥ fori∈ {, , . . . ,p}, we have p
i=
αi
fi(x) –fi(x¯)≥
p
i=
αiξi,x–x¯
+
p
i=
αici¯x–x¯m. (.)
On the other hand,gt(x)≤gt(x¯) = forx∈,t∈J(x¯). By the strong quasiconvexity of
ordermatx¯ forgt witht∈J(x¯), we see that there existct¯ > fort∈J(x¯) such that, for
ζt∈∂gt(x¯),
ζt,x–x¯ +c¯tx–x¯m≤,
furthermore, it follows fromβt≥ fort∈J(x¯) that
t∈J(x¯)
βtζt,x–x¯
+
t∈J(x¯)
βtct¯x–x¯m≤. (.)
Adding (.) to (.), we get
p
i=
αi
fi(x) –fi(x¯)
≥
p
i=
αiξi+
t∈J(x¯)
βtζt,x–x¯
+
p
i=
αic¯i+
t∈J(x¯)
βtc¯t
It follows from (.) that
p
i=
αi
fi(x) –fi(x¯)
–
p
i=
αic¯i+
t∈J(x¯)
βtc¯t
· x–x¯m≥.
Let¯c=pi=αici¯+
t∈J(x¯)βtct¯ andci=αi¯c. Noticing that
p
i=αi= , we obtain
p
i=
αi
fi(x) –fi(x¯)
≥
p
i=
αicx–x¯m.
This implies
α,f(x) –f(x¯) –cx–x¯m≥,
wherec= (c¯,¯c, . . . ,¯c) withc¯> . Sinceαi≥ and
p
i=αi= , we further see that for all x∈
f(x)≮f(x¯) +cx–x¯m,
which implies thatx¯is a strict minimizer of ordermfor (P).
Theorem .(Sufficient optimality conditions II) Letx¯∈andTˆ(x¯)=∅.Suppose that there exist scalarsαi≥,i= , , . . . ,p with
p
i=αi= ,andβt≥,t∈ ˆT(x¯)withβt= for finitely many indices t,such that
∈
p
i=
αi∂fi(x¯) +
t∈ ˆT(x¯)
βt∂gt(x¯).
If the functions fi,i∈ {, , . . . ,p:αi= },are strongly convex of order m atx and at least one¯ of them is strictly strong convex of order m atx¯,and gtfor t∈ ˆT(x¯)andβt= ,are strongly quasiconvex of order m atx¯,thenx is a semi-strict minimizer of order m for¯ (P).
Proof In the proof of Theorem ., we derived that there existξi∈∂fi(x¯) fori∈ {, , . . . ,p}
andζt∈∂gt(x¯) fort∈J(x¯), such that p
i=
αiξi+
t∈J(x¯)
βtζt= , (.)
and that there existc¯t> fort∈J(x¯), such that, forζt∈∂gt(x¯), we have
t∈J(x¯)
βtζt,x–x¯
+
t∈J(x¯)
βtc¯tx–x¯m≤. (.)
Since the functionsfifori∈ {, , . . . ,p}are strongly convex of ordermatx¯and there is
using the same argument as in the proof of Theorem ., we arrive at the conclusion that there existci¯> fori∈ {, , . . . ,p}such that
p
i=
αi
fi(x) –fi(x¯)
>
p
i=
αiξi,x–x¯
+
p
i=
αic¯ix–x¯m.
Adding the above inequality to (.), we obtain
p
i=
αi
fi(x) –fi(x¯)
>
p
i=
αiξi+
t∈J(x¯)
βtζt,x–x¯
+
p
i=
αici¯+
t∈J(¯x)
βtct¯
x–x¯m. (.)
It follows from (.) that
p
i=
αi
fi(x) –fi(x¯)
–
p
i=
αic¯i+
t∈J(x¯)
βtc¯t
x–x¯m> .
Let¯c=pi=αic¯i+
t∈J(x¯)βtc¯tandci=αi¯c. Noticing that
p
i=αi= , we get
p
i=
αifi(x) –fi(x¯)>
p
i=
αicx–x¯m.
Hence, withc= (¯c,c¯, . . . ,¯c) we have
α,f(x) –f(x¯) –cx–x¯m> .
Becauseαi≥ fori∈ {, , . . . ,p}and
p
i=αi= , we know that, for allx∈, f(x)f(x¯) +cx–x¯m,
which implies thatx¯is a semi-strict minimizer of ordermfor (P).
Theorem .(Sufficient optimality conditions III) Letx¯∈andTˆ(x¯)=∅.Suppose that there exist scalarsαi≥,i= , , . . . ,p with
p
i=αi= ,andβt≥,t∈ ˆT(x¯)withβt= for finitely many indices t,such that
∈
p
i=
αi∂fi(x¯) +
t∈ ˆT(x¯)
βt∂gt(x¯).
If the functions fi, i= , , . . . ,p,are strongly convex of order m at x¯,andt∈ ˆT(¯x)βtgt is strictly strong quasiconvex of order m atx¯,thenx is a semi-strict minimizer of order m¯ for(P).
5 Conclusions
We have defined a strict minimizer of order and a semi-strict minimizer of higher-order for a semi-infinite multi-objective optimization problem in this paper. We have pre-sented a non-smooth semi-infinite version of the generealized Guignard and Abadie con-straint qualifications. Under those concon-straint qualifications, utilizing the method in [, ] we have proved necessary optimality conditions for a semi-strict minimizer of higher-order and a strict minimizer of higher-higher-order. Three sufficient optimality conditions have been proved under the assumption of strong convexities.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This research was supported by Natural Science Foundation of China under Grant No. 11361001; Natural Science Foundation of Ningxia under Grant No. NZ14101.
Received: 27 August 2016 Accepted: 13 October 2016
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