Optimality conditions for strict minimizers of higher ... - Springer Link

Yu Journal of Inequalities and Applications (2016) 2016:263 DOI 10.1186/s13660-016-1209-7

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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-called semi-infinite multiobjective optimization problems (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 perspectives. 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 discussed in [–]. The density of the set of all stable convex semi-infinite vector optimization problems has been established in []. However, the work on optimality conditions 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 index t belonging to a compact set T. 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 constraint 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 © 2016 Yu. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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proposing the concepts of (strictly) generalized convex functions defined by using the limiting 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 mentioned above, a meaningful solution concept called a strict efficient solution of higherorder (also called a strict minimizer of 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 optimality conditions for strict efficiency of higher-order in multi-objective optimization under the basic regularity condition and generalized 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 (possibly infinite) inequality constraints. For the purpose of investigating this new solution concept, we found that the notion of convexity that appears to be most appropriate in the development 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 semistrict 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-KuhnTuchker 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 let Rn be the n-dimensional Euclidean space endowed with the Euclidean norm  · , X be a convex subset of Rn , and m ≥  be a positive integer. Let W be a subset of Rn . We use cl W , co W , and cone W to denote the closure of W , the convex hull of W , and the conic hull of W (i.e., the smallest convex cone containing W ), respectively. Definition . (see [–]) Let W be a nonempty subset of Rn . The tangent cone to W at x¯ ∈ cl W is the set defined by    T(W ; x¯ ) := h ∈ Rn : h = lim tn xn – x¯ such that xn ∈ W , n→∞  lim xn = x¯ and tn >  for all n = , , . . . . n→∞

Recall that a function ϕ : X → R is Lipschitz at x¯ ∈ X if there exists a positive constant K such that   ϕ(x) – ϕ(¯x) ≤ Kx – x¯ 

for all x ∈ X,

where K is called the rank of ϕ at x¯ . ϕ is said to be Lipschitz on X if ϕ is Lipschitz at each x ∈ X. Suppose that ϕ is Lipschitz at x¯ ∈ X, then Clarke’s generalized directional derivative

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of ϕ at x¯ ∈ X in the direction v ∈ Rn , denoted by ϕ  (¯x, v), is defined as ϕ  (¯x, v) = lim sup (x,t)→(¯x,+ )

ϕ(x + tv) – ϕ(x) . t

Clarke’s generalized gradient of ϕ at x¯ ∈ X, denoted by ∂ϕ(¯x), is defined as   ∂ϕ(¯x) = ξ ∈ Rn : ϕ  (¯x, v) ≥ ξ , v for all v ∈ Rn . It is well known that ∂ϕ(¯x) is a nonempty convex compact set in Rn . Definition . (see []) Let ϕ : Rn → R be Lipschitz at x¯ ∈ Rn . It is said that ϕ admits a strict derivative at x¯ , an element of Rn , denoted by Ds ϕ(¯x), provided that, for each x ∈ Rn , the following holds: lim

(x ,t)→(¯x,)

ϕ(x + tx) – ϕ(x ) = Ds ϕ(¯x), x . t

If ϕ admits a strict derivative at x¯ , then ϕ is called strictly differentiable at x¯ . Lemma . (see []) Let ϕ, ϕ , and ϕ be Lipschitz from X to R, and x¯ ∈ X. Then the following properties hold: (a) ϕ  (¯x, v) = max{ξ , v : ξ ∈ ∂ϕ(¯x)}, for all v ∈ Rn . (b) ∂(λϕ(¯x)) = λ∂ϕ(¯x), for all λ ∈ R. (c) ∂(ϕ + ϕ )(¯x) ⊂ ∂ϕ (¯x) + ∂ϕ (¯x). Now, we recall the definition of the strong convexity of order m for a Lipschitz function. Definition . (see [, ]) Let ϕ : X → R be Lipschitz at x¯ ∈ X. (a) ϕ is said to be strongly convex of order m at x¯ if there exists a constant c >  such that for each x ∈ X and ξ ∈ ∂ϕ(¯x) ϕ(x) – ϕ(¯x) ≥ ξ , x – x¯ + cx – x¯ m . (b) ϕ is said to be strongly quasiconvex of order m at x¯ if there exists a constant c >  such that, for each x ∈ X and ξ ∈ ∂ϕ(¯x), ϕ(x) ≤ ϕ(¯x)

⇒

ξ , x – x¯ + cx – x¯ m ≤ .

Based upon the above definition of a strongly convex function of order m, we define the following generalized strong convexities of order m for a Lipschitz function. Definition . Let ϕ : X → R be Lipschitz at x¯ ∈ X. (a) ϕ is strictly strong convex of order m at x¯ if there exists a constant c >  such that, for each x ∈ X with x = x¯ and ξ ∈ ∂ϕ(¯x), ϕ(x) – ϕ(¯x) > ξ , x – x¯ + cx – x¯ m .

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(b) ϕ is strictly strong quasiconvex of order m at x¯ if there exists a constant c >  such that, for each x ∈ X with x = x¯ and any ξ ∈ ∂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 . Let ϕi : X → R be Lipschitz at x¯ ∈ X, i = , , , . . . , s. Suppose that ϕ is a strictly strong convex function of order m and ϕ , ϕ , . . . , ϕs are strongly convex functions of  order m at x¯ . If λ >  and λi ≥  for i = , , . . . , s, then si= λi ϕi is strictly strong convex of order m at x¯ . Proof It is evident that the function ∂

s

s

i= λi ϕi

is Lipschitz at x¯ . Thus, we get

 λi ϕi (¯x) = ∅.

i=

Taking ξ ∈ ∂( ∂

s

s

x). i= λi ϕi )(¯

 λi ϕi (¯x) ⊂

i=

s

It follows from Lemma . that

∂(λi ϕi )(¯x) =

i=

s

λi ∂ϕi (¯x).

i=

This means that there exist ξi ∈ ∂ϕi (¯x), i = , , . . . , s, such that ξ=

s

λi ξi .

i=

Since ϕ is strictly strong convex of order m at x¯ and ϕi , i = , , . . . , s, is strongly convex of order m at x¯ , we derive that there exist ci > , i = , , . . . , s, such that, for all x ∈ Rn , ⎧ ⎨ϕ (x) – ϕ (¯x) > ξ , x – x¯ + c x – x¯ m ,     ⎩ϕi (x) – ϕi (¯x) ≥ ξi , x – x¯ + ci x – x¯ m for all i = , , . . . , s, which implies that  s s s

  λi ϕi (x) – ϕi (¯x) > λi ξi , x – x¯ + λi ci x – x¯ m . i=

i=

i=

Therefore, we get

s

 λi ϕi (x) –

i=

where c =



s

 λi ϕi (¯x) > ξ , x – x¯ + cx – x¯ m ,

i=

s

i= λi ci .

This completes the proof of the proposition.



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Consider the following semi-infinite multi-objective optimization problem: (P)

  Minimize f (x) = f (x), f (x), . . . , fp (x) subject to gt (x) ≤  for t ∈ T, x ∈ X,

where fi , i ∈ P = {, , . . . , p}, and gt , t ∈ T are Lipschitz from X to R, and the index set T is arbitrary, not necessarily finite (but nonempty). The feasible set of (P) is denoted by ,    := x ∈ X : gt (x) ≤ , ∀t ∈ T . For a given x¯ ∈ , set   ˆ x) := t ∈ T : gt (¯x) =  . T(¯ In the sequel, we use the following notations. For x, y ∈ Rn . (i) f (x) < f (y) ⇔ fi (x) < fi (y) for every i ∈ P; (ii) f (x) ≮ f (y) is the negation of f (x) < f (y); (iii) f (x) ≤ f (y) ⇔ fi (x) ≤ fi (y) for every i ∈ P, but there is at least one i ∈ P such that fi (x) < fi (y); (iv) f (x)  f (y) is the negation of f (x) ≤ f (y). Definition . (see []) A point x¯ ∈  is said to be a strict minimizer of order m for (P) if there exists c = (c , c , . . . , cp ) ∈ Rp with ci > , i ∈ P, such that f (x) ≮ f (¯x) + cx – x¯ m

for all x ∈ .

Definition . A point x¯ ∈  is said to be a semi-strict minimizer of order m for (P) if there exists c = (c , c , . . . , cp ) ∈ Rp with ci > , i ∈ P, such that f (x)  f (¯x) + cx – x¯ m

for all x ∈ .

Remark . It is obvious that if x¯ ∈  is a semi-strict minimizer of order m for (P), then x¯ ∈  is a strict minimizer of order m for (P). Example . The functions fi : R → R, i = , , defined by

f (x) =

⎧ ⎨x + ex ⎩x +  

if x ≥ , if x < ,

f (x) =

⎧ ⎨x + x

if x ≥ ,

⎩x – x

if x < ,



are Lipschitz at x¯ = . It is easy to verify that x¯ is a semi-strict minimizer of order  with c = (, ) for the following optimization problem:   (P ) Minimize f (x) = f (x), f (x) subject to x ∈ R.

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Motivated by the notion of a linearizing cone at a point to the feasible set of a differentiable 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. Let x¯ ∈  be a semi-strict minimizer of order m for (P), and define   Qi (¯x) := x ∈ Rn : fk (x) ≤ fk (¯x) + ck x – x¯ m , k ∈ P and k = i ∩ . It is obvious that x¯ ∈ Qi (¯x), i ∈ P. Definition . Let x¯ ∈ . The linearizing cone at x¯ is the set defined by  C(¯x) = z ∈ Rn : ξ , z ≤  for all ξ ∈ ∂fi (¯x), i ∈ P and  ˆ x) . ζ , z ≤  for all ζ ∈ ∂gt (¯x), t ∈ T(¯

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 the generalized Guignard constraint qualification at a given point x¯ ∈  which is a semi-strict minimizer of order m for (P) if the following p holds: C(¯x) ⊆ i= cl[co(T(Qi ; x¯ ))], where Qi := Qi (¯x). Definition . The problem (P) satisfies the generalized Abadie constraint qualification at a given point x¯ ∈  which is a semi-strict minimizer of order m for (P) if the following p holds: C(¯x) ⊆ i= T(Qi ; x¯ ). Next, we recall the generalized Motzkin theorem discussed in []. Lemma . (see []) Let A be a compact set in Rn , B an arbitrary set in Rn . Suppose that the set cone B 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 integers μ and ν, with  ≤ ν ≤ n + , such that there exist μ points ai ∈ A (i = , , . . . , μ), ν points bm ∈ B (m = , , . . . , ν), μ nonnegative numbers ui , with ui >  for at least one i ∈ {, , . . . μ}, and ν positive numbers vm for m ∈ {, , . . . ν}, such that μ

ui ai +

ν

i=

but never both.

m=

vm bm = ,

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The following lemma is a generalization of the classical Tucker theorem of the alternative. 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 in Rn . Suppose  that, for each i ∈ P, the set cone(B ∪ [ j∈P,j =i Aj ]) is closed. Then either the system ⎧ ⎪ one i ∈ P and for all a ∈ Ai , ⎪ ⎨a, z <  for at least  p a, z ≤  for all a ∈ i= Ai , ⎪ ⎪ ⎩ b, z ≤  for all b ∈ B, p has a solution z ∈ Rn , or there exist u ∈ U ≡ {u ∈ Rp : u > , i= ui = }, ai ∈ Ai for i ∈ P, and integer ν with  ≤ ν ≤ n + , such that there exist ν points bm ∈ B, and ν positive numbers vm , m ∈ {, , . . . , ν}, such that p

ui ai +

ν

vm bm = ,

m=

i=

but never both. Lemma . Let x¯ be a semi-strict minimizer of order m for (P), let fi (x), i ∈ P, be Lipschitz at x¯ of rank Ki , for all t ∈ T, let the functions gt (x) be Lipschitz at x¯ . If the generalized Guignard constraint qualification holds at x¯ and fi (x), i ∈ P, are strictly differentiable at x¯ or the generalized Abadie constraint qualification holds at x¯ , then the system ⎧ ⎪ one i ∈ P and for all ξ ∈ ∂fi (¯x), ⎪ ⎨ξ , z <  for at least  p ξ , z ≤  for all ξ ∈ i= ∂fi (¯x), ⎪ ⎪ ⎩ ˆ x), ζ , z ≤  for all ζ ∈ ∂gt (¯x), t ∈ T(¯

(.)

has no solution z ∈ Rn . Proof Suppose to the contrary that (.) has a solution z. Then z =  and z ∈ C(¯x). Without loss of generality, we can assume that ξ , z <  for all ξ ∈ ∂f (¯x), ξ , z ≤ 

for all ξ ∈

p 

∂fi (¯x).

i=

By our generalized Guignard constraint qualification assumption, z ∈ cl[co(T(Q ; x¯ ))], and  ¯ )) such that hence there exists a sequence {zm }∞ m= ⊂ co(T(Q ; x lim zm = z.

(.)

m→∞

For each zm , m = , , . . . , there exist numbers Lm and λml ≥ , and zml ∈ T( ; x¯ ), zml = , l = , , . . . , Lm , such that Lm

l=

λml = ,

Lm

l=

λml zml = zm .

(.)

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Since, for each m = , , . . . and l = , , . . . , Lm , zml ∈ T(Q ; x¯ ), there exist sequences ∞  {xmln }∞ n= ⊂ Q and {tmln }n= ⊂ R, with tmln >  for all n, such that lim xmln = x¯ ,

n→∞

  lim tmln xmln – x¯ = zml .

(.)

n→∞

Noticing that xmln ∈ Q for all n, we have xmln ∈  and fi (xmln ) ≤ fi (¯x) + ci xmln – x¯ m for i = , , . . . , p. Since x¯ is a semi-strict minimizer of order m for (P), for all n we may assume  m   f xmln ≥ f (¯x) + c xmln – x¯  .

(.)

Since zml = , for each m = , , . . . and l = , , . . . , Lm , we must have tmln → +∞ as n → +∞, and hence xmln – t  zml → x¯ as n → +∞. Therefore, mln

  f (y + tzml ) – f (y) f x¯ ; zml = lim sup t y→¯x t↓

≥ lim sup

f (xmln –

 ml z tmln

+

n→∞

≥ lim sup

f (xmln ) – f (¯x) + f (¯x) – f (xmln –

f (xmln ) – f (¯x)  tmln

n→∞

≥ lim sup

–

 ml z ) tmln

 ml z ) tmln

 tmln

n→∞

≥ lim sup

 ml z ) – f (xmln tmln  tmln

 tmln

|f (¯x) – f (xmln –

– lim sup

 ml z )| tmln

 tmln

n→∞

c xmln – x¯ m

n→∞

– lim sup

|f (¯x) – f (xmln –

n→∞

 ml z )| tmln

 tmln

(by (.)) ≥ – lim sup

|f (¯x) – f (xmln –

n→∞

 ml z )| tmln

 tmln

    ml  mln  ≥ – lim sup tmln K x¯ – x + z tmln  n→∞ (by the Lipschitz continuity of f )     = – lim sup K tmln x¯ – xmln + zml  n→∞

=  (by (.)).

(.)

Because f (x) is strictly differentiable at x¯ , we know f (x; v) = Ds f (¯x), v . By Proposition .. in [], we also know that ∂f (¯x) = {Ds f (¯x)}. In view of (.) we obtain Ds f (¯x), zm = f (¯x; zm ) ≥ , which further gives us Ds f (¯x), z = f (¯x; z) ≥  because of (.), contradicting the assumption that z is a solution of the system (.). Therefore, under the assumption that the generalized Guignard constraint qualification holds and fi (x), i ∈ P, are strictly differentiable at x¯ , (.) has no solution z ∈ Rn . Now let us show that (.) has no solution z ∈ Rn under the generalized Abadie constraint qualification. Suppose to the contrary that (.) has a solution z. Then z =  and

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z ∈ C(¯x). By the generalized Abadie constraint qualification, without loss of general ity we may assume that z ∈ T(Q ; x¯ ), and hence there exist sequences {xn }∞ n= ⊂ Q and {tn }∞ n= ⊂ R, with tn >  for all n, such that lim xn = x¯ ,

n→∞

  lim tn xn – x¯ = z.

n→∞

Replacing tmln by tn , xmln by xn , and zml by z in (.), we arrive at f (¯x; z) ≥ . By Proposition .. in [], we know that there is a ξ ∈ ∂f (¯x) such that ξ , z = f (¯x; z) ≥ , contradicting the assumption that z is a solution of the system (.). Therefore, (.) has no solution z ∈ Rn .  Now we are ready to prove the following necessary optimality condition for (P). Theorem . Let x¯ ∈  and let the functions fi (x) for i ∈ P and gt (x) for t ∈ T be Lipschitz at x¯ . If x¯ 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, ˆ x)} ∪ {ξ ∈ ∂fi (¯x) : i ∈ P, i = i }) is closed, then there exist the set cone({ζ ∈ ∂gt (¯x) : t ∈ T(¯ p ∗ p u ∈ U ≡ {u ∈ R : u > , i= ui = }, integers ν ∗ , with  ≤ ν ∗ ≤ n + , such that there exist ˆ x) and v∗m > , m ∈ {, , . . . , ν ∗ }, with the property that t m ∈ T(¯ ∈

p

∗

u∗i ∂fi (¯x) +

i=

ν

v∗m ∂gtm (¯x).

(.)

m=

Proof In Lemma ., set Ai = ∂fi (¯x), i ∈ P,  B= ∂gt (¯x). ˆ x) t∈T(¯

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) for i ∈ P, integers ν ∗ with  ≤ ν ∗ ≤ n + , such that there exist ν ∗ points ˆ x) and ν ∗ positive numbers v∗m > , m ∈ {, , . . . , ν ∗ }, such that (.) holds. t m ∈ T(¯  Remark . If we modify the definition of Qi (¯x) as follows:   Qi (¯x) := x ∈ Rn : fk (x) < fk (¯x) + ck x – x¯ , k ∈ P and k = i ∩ , and define the generalized Guignard constraint qualification and generalized Abadie constraint qualification accordingly, we can prove a similar necessary result for x¯ to be a strict minimizer of order m for (P).

4 Sufficient conditions In this section we discuss sufficient optimality results under various generalized higherorder strong convexity (introduced in Section ) hypotheses imposed on the involved functions.

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ˆ x) = ∅. Suppose that Theorem . (Sufficient optimality conditions I) Let x¯ ∈  and T(¯ p ˆ x) with βt =  for there exist scalars αi ≥ , i = , , . . . , p with i= αi = , and βt ≥ , t ∈ T(¯ finitely many indices t, such that ∈

p

αi ∂fi (¯x) +

βt ∂gt (¯x).

(.)

ˆ x) t∈T(¯

i=

ˆ x) and If the functions fi , i = , , . . . , p, are strongly convex of order m at x¯ , and gt for t ∈ T(¯ βt = , are strongly quasiconvex of order m at x¯ , then x¯ is a strict minimizer of order m for (P). ˆ x) : βt = }. Because of (.), we derive that there exist ξi ∈ ∂fi (¯x) Proof Let J(¯x) := {t ∈ T(¯ for i ∈ {, , . . . , p} and ζt ∈ ∂gt (¯x) for t ∈ J(¯x) such that p

αi ξi +

i=

βt ζt = .

(.)

t∈J(¯x)

Since fi for i ∈ {, , . . . , p} are strongly convex of order m at x¯ , we see that there exist c¯i >  for i ∈ {, , . . . , p} such that fi (x) – fi (¯x) ≥ ξi , x – x¯ + c¯i x – x¯ m

for all x ∈ , ξi ∈ ∂fi (¯x), i = , , . . . , p.

Noticing αi ≥  for i ∈ {, , . . . , p}, we have  p  p p

  αi fi (x) – fi (¯x) ≥ αi ξi , x – x¯ + αi c¯i x – x¯ m . i=

i=

(.)

i=

On the other hand, gt (x) ≤ gt (¯x) =  for x ∈ , t ∈ J(¯x). By the strong quasiconvexity of order m at x¯ for gt with t ∈ J(¯x), we see that there exist c¯t >  for t ∈ J(¯x) such that, for ζt ∈ ∂gt (¯x), ζt , x – x¯ + c¯t x – x¯ m ≤ , furthermore, it follows from βt ≥  for t ∈ J(¯x) that 

 βt ζt , x – x¯ +

t∈J(¯x)

βt c¯t x – x¯ m ≤ .

(.)

t∈J(¯x)

Adding (.) to (.), we get p

  αi fi (x) – fi (¯x) i=

≥



p

i=

αi ξi +

t∈J(¯x)

 βt ζt , x – x¯ +

p

i=

αi c¯i +

t∈J(¯x)

 βt c¯t · x – x¯ m .

(.)

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It follows from (.) that  p p

  αi fi (x) – fi (¯x) – αi c¯i + βt c¯t · x – x¯ m ≥ . i=

Let c¯ =

i=

p

i= αi c¯i

+



t∈J(¯x) βt c¯t

t∈J(¯x)

and ci = αi c¯ . Noticing that

p

i= αi

= , we obtain

p p

  αi fi (x) – fi (¯x) ≥ αi cx – x¯ m . i=

i=

This implies

α, f (x) – f (¯x) – cx – x¯ m ≥ , where c = (¯c, c¯ , . . . , c¯ ) with c¯ > . Since αi ≥  and x∈

p

i= αi

= , we further see that for all

f (x) ≮ f (¯x) + cx – x¯ m , which implies that x¯ is a strict minimizer of order m for (P).



ˆ x) = ∅. Suppose that Theorem . (Sufficient optimality conditions II) Let x¯ ∈  and T(¯ p ˆ x) with βt =  for there exist scalars αi ≥ , i = , , . . . , p with i= αi = , and βt ≥ , t ∈ T(¯ finitely many indices t, such that ∈

p

αi ∂fi (¯x) +

βt ∂gt (¯x).

ˆ x) t∈T(¯

i=

If the functions fi , i ∈ {, , . . . , p : αi = }, are strongly convex of order m at x¯ and at least one ˆ x) and βt = , are strongly of them is strictly strong convex of order m at x¯ , and gt for t ∈ T(¯ quasiconvex of order m at x¯ , then x¯ is a semi-strict minimizer of order m for (P). Proof In the proof of Theorem ., we derived that there exist ξi ∈ ∂fi (¯x) for i ∈ {, , . . . , p} and ζt ∈ ∂gt (¯x) for t ∈ J(¯x), such that p

αi ξi +

i=

βt ζt = ,

(.)

t∈J(¯x)

and that there exist c¯t >  for t ∈ J(¯x), such that, for ζt ∈ ∂gt (¯x), we have  t∈J(¯x)

 βt ζt , x – x¯ + βt c¯t x – x¯ m ≤ .

(.)

t∈J(¯x)

Since the functions fi for i ∈ {, , . . . , p} are strongly convex of order m at x¯ and there is at least one i ∈ {, , . . . , p : αi = } such that fi is strictly strong convex of order m at x¯ ,

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using the same argument as in the proof of Theorem ., we arrive at the conclusion that there exist c¯i >  for i ∈ {, , . . . , p} such that  p  p p

  αi fi (x) – fi (¯x) > αi ξi , x – x¯ + αi c¯i x – x¯ m . i=

i=

i=

Adding the above inequality to (.), we obtain p

  αi fi (x) – fi (¯x) i=

>



p

αi ξi +

i=

 βt ζt , x – x¯ +

p

αi c¯i +

i=

t∈J(¯x)

 βt c¯t x – x¯ m .

(.)

t∈J(¯x)

It follows from (.) that  p p

  αi fi (x) – fi (¯x) – αi c¯i + βt c¯t x – x¯ m > . i=

Let c¯ =

i=

p

i= αi c¯i

+



t∈J(¯x) βt c¯t

t∈J(¯x)

and ci = αi c¯ . Noticing that

p

i= αi

= , we get

p p

  αi fi (x) – fi (¯x) > αi cx – x¯ m . i=

i=

Hence, with c = (¯c, c¯ , . . . , c¯ ) we have

α, f (x) – f (¯x) – cx – x¯ m > . Because αi ≥  for i ∈ {, , . . . , p} and

p

i= αi

= , we know that, for all x ∈ ,

f (x)  f (¯x) + cx – x¯ m , which implies that x¯ is a semi-strict minimizer of order m for (P).



ˆ x) = ∅. Suppose that Theorem . (Sufficient optimality conditions III) Let x¯ ∈  and T(¯ p ˆ x) with βt =  for there exist scalars αi ≥ , i = , , . . . , p with i= αi = , and βt ≥ , t ∈ T(¯ finitely many indices t, such that ∈

p

i=

αi ∂fi (¯x) +

βt ∂gt (¯x).

ˆ x) t∈T(¯

 If the functions fi , i = , , . . . , p, are strongly convex of order m at x¯ , and t∈T(¯ ˆ x) βt gt is strictly strong quasiconvex of order m at x¯ , then x¯ is a semi-strict minimizer of order m for (P). Proof The proof is similar to that of Theorems . and ..



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5 Conclusions We have defined a strict minimizer of higher-order and a semi-strict minimizer of higherorder for a semi-infinite multi-objective optimization problem in this paper. We have presented a non-smooth semi-infinite version of the generealized Guignard and Abadie constraint qualifications. Under those constraint qualifications, utilizing the method in [, ] we have proved necessary optimality conditions for a semi-strict minimizer of higherorder and a strict minimizer of 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 References 1. Chuong, TD: Lower semicontinuity of the Pareto solution in quasiconvex semi-infinite vector optimization. J. Math. Anal. Appl. 388, 443-450 (2012) 2. Chuong, TD, Huy, NQ, Yao, JC: Stability of semi-infinite vector optimization problems under functional perturbations. J. Glob. Optim. 45, 583-595 (2009) 3. Chuong, TD, Huy, NQ, Yao, JC: Pseudo-Lipschitz property of linear semi-infinite vector optimization problems. Eur. J. Oper. Res. 200, 639-644 (2010) 4. Chuong, TD, Yao, JC: Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems. Nonlinear Anal. 71, 6312-6322 (2009) 5. Huy, NQ, Kim, DS: Lipschitz behavior of solutions to nonconvex semi-infinite vector optimization problems. J. Glob. Optim. 56, 431-448 (2013) 6. Fan, X, Cheng, C, Wang, H: Density of stable convex semi-infinite vector optimization problems. Oper. Res. Lett. 40, 140-143 (2012) 7. Caristi, G, Ferrara, M, Stefanescu, A: Semi-infinite multiobjective programming with generalized invexity. Math. Rep. 12, 217-233 (2010) 8. Kanzi, N, Nobakhtian, S: Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8, 1517-1528 (2014) 9. Chuong, TD, Kim, DS: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748-762 (2014) 10. Jiménez, B: Strict efficiency in vector optimization. J. Math. Anal. Appl. 265, 264-284 (2002) 11. Auslender, A: Stability in mathematical programming with non-differentiable data. SIAM J. Control Optim. 22, 239-254 (1984) 12. Ward, DE: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. Optim. Theory Appl. 80, 551-571 (1994) 13. Bhatia, G: Optimality and mixed saddle point criteria in multiobjective optimization. J. Math. Anal. Appl. 342, 135-145 (2008) 14. Lin, GH, Fukushima, M: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 118, 67-80 (2003) 15. Maeda, T: Constraint qualifications in multiobjective optimization problems: differentiable case. J. Optim. Theory Appl. 80, 483-500 (1994) 16. Preda, V, Chitescu, I: On constraint qualification in multiobjective optimization problems: semidifferentiable case. J. Optim. Theory Appl. 100, 417-433 (1999) 17. Zalmai, GJ, Zhang, QH: Global semiparametric sufficient efficiency conditions for semiinfinite multiobjective fractional programming problems containing generalized (α , η, ρ )-V-invex functions. Southeast Asian Bull. Math. 32, 573-599 (2008) 18. Clarke, FH: Optimization and Nonsmooth Analysis. Wiley, New York (1983) 19. Clarke, FH, Ledyaev, YS, Stern, RJ, Wolenski, PR: Nonsmooth Analysis and Control Theory. Springer, New York (1998) 20. Goberna, MA, López, MA: Linear Semi-Infinite Optimization. Wiley, New York (1998)