R E S E A R C H
Open Access
Vector critical points and generalized
quasi-efficient solutions in nonsmooth
multi-objective programming
Zhen Wang, Ru Li and Guolin Yu
**Correspondence:
Institute of Applied Mathematics, North Minzu University, Yinchuan, Ningxia 750021, P.R. China
Abstract
In this work, several extended approximately invex vector-valued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higher-order (weak) quasi-efficiency with respect to a function are proposed for a multi-objective programming. Under the introduced generalization of higher-order approximate invexities assumptions, we prove that the solutions of generalized vector
variational-like inequalities in terms of the generalized Jacobian are the generalized quasi-efficient solutions of nonsmooth multi-objective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function.
MSC: Primary 90C29; 90C46; secondary 26B25
Keywords: vector variational-like inequality; multi-objective programming; approximate invexity; quasi-efficiency; vector critical point
1 Introduction
Convexity and its generalizations played a critical role in multi-objective programming problems. In many generalizations, approximate convexity and invexity are two significant generalized versions of convexity, which tried to weaken the convexity hypotheses thus to study the relations between vector variational-like inequalities and multi-objective pro-gramming problems. Invexity was firstly put forward by Hanson []. Then Osuna-Gómez et al.[] introduced the notions of generalized invexity for differentiable functions in a finite-dimensional contex. And this generalized invexity has been extended to locally Lipschitz functions using the generalized Jacobian (see [, ]). Ben-Israel and Mond [] presented pseudoinvex functions which generalized pseudoconvex functions in the same manner as invex functions generalized convex functions. Mishraet al.[] and Ngaiet al. [] introduced the concept of approximately convex functions. Inspired and motivated by this ongoing research work, we present the concept of approximately invex function of higher order.
The notion of an efficient solution in multi-objective programming is widely used. Con-sidering the complexity of optimization problems, several variants of the efficient solu-tions have been introduced (see [–]). Recently, researchers have shown great interests
in quasi-efficiency of multi-objective programming (see [, ]). In this work, we give the notion of a quasi-efficient solution of higher order for a class ofnonsmooth multi-objective programming problems(NMPs) with respect to a function.
The vector variational inequality was initially introduced by Giannessi []. Since then vector variational inequalities, which were used as an efficient tool to study multi-objective programming, have attracted much attention and have been extended togeneralized vec-tor variational-like inequalities(GVVI). Recently, a great quantity of work focused on the study of relations between (GVVI) and multi-objective programming under different con-vexity assumptions (see [–]). Motivated by the previous contributions, in this note, our purpose is to obtain the relations between (GVVI) and (NMP) under approximate invexity of higher order.
The rest of this work is organized as follows. In Section , we recall some basic defi-nitions and preliminary results. Besides, the notions of approximately invex function of higher order with respect to vector-valued functions and (weakly) quasi-efficient solu-tion of higher order for (NMP) with respect to a vector-valued funcsolu-tion are introduced, and examples are provided to illustrate their existence. In Section , the relations between (GVVI) and (NMP) are established under the approximate invexity of higher-order as-sumptions. In Section , we study the relations between vector critical points and weakly quasi-efficient solutions of higher order for (NMP) with respect to a vector-valued func-tion.
2 Notations and preliminaries
Throughout the current paper, unless otherwise stated,R, Rn, Rn
+ stand for the set of
all real numbers, then-dimensional Euclidean space and the nonnegative orthant ofRn, respectively. For anyx,y∈Rn, the inner product ofxandyis denotedxTy, where the
superscriptTrepresents the transpose of a vector. LetX⊆Rnbe a nonempty subset and
m≥ be a positive integer, the symbolsco(X) andint(X) represent the convex hull ofX and the interior ofX, respectively. We employ the following conventions for vectors inRn:
x=y ⇔ xi=yi, ∀i= , . . . ,n;
xy ⇔ xi≤yi, ∀i= , . . . ,n;
x≤y ⇔ xi≤yi, ∀i= , . . . ,n,i=jandxj<yjfor somej;
x<y ⇔ xi<yi, ∀i= , . . . ,n.
For the sake of convenience, we firstly recall some notations that will be used in the sequel.We always suppose that f :X→Rp,η:X×X→Rn andψ :X×X→Rnare vector-valued functions in the rest of this paper.
Definition .(see []) The functionf :X→Rpis said to be locally Lipschitz onX, if
for everyx∈Xthere exist a neighborhoodUx⊆Xofxand a constantL> such that, for
ally,z∈Ux,
Rademacher’s theorem (see Corollary . in []) indicates that a functionf satisfy-ing the Lipschitz condition (.) is Fréchet differentiable. Based on this fact, Clarke [] presented the following concept of the generalized Jacobian off at some point.
Definition .(see []) Letx∈XandJf(x) represent the usual Jacobian matrix off at
xwheneverf is Fréchet differentiable atx. The generalized Jacobian off atxis
∂Jf(x) =co
lim
n→∞Jf(xn) :xn→x,Jf(xn) exists
.
Let a scalar functionϕ:X→Rbe locally Lipschitz atx, then the upper Clarke
direc-tional derivative ofϕatxin the directionv∈Rnis given by
ϕ(x,v) = lim
x→x
sup
t↓
ϕ(x+tv) –ϕ(x)
t ,
and the Clarke subdifferential ofϕatx, denoted by∂ϕ(x), is defined as follows:
∂ϕ(x) =
ξ∈Rn:ϕ(x,v)≥ ξ,v,∀v∈Rn
.
For a vector-valued function f = (f, . . . ,fp)T :X→Rp, its Clarke subdifferential is the
cartesian product of Clarke subdifferentials of the componentsfi:X→R,i= , , . . . ,p
off, that is,∂f(x) =∂f(x)× · · · ×∂pf(x). It has been shown in [] that, for a scalar
func-tionϕ:X→R,∂Jϕ(x) =∂ϕ(x), but for the vector-valued functionf,∂Jf(x) is contained
and is different from∂f(x).
Definition .(see []) The subset∅ =X⊆Rnis said to be invex with respect toη: X×X→Rn, if for everyx,y∈X,λ∈[, ], we have
y+λη(x,y)∈X.
From now on, we always assume that the subset X⊆Rnis a nonempty invex set with
respect to someηunless otherwise specified.
The generalized invexity of differentiable functions in a finite-dimensional space (see []) has been extended to locally Lipschitz functions using the generalized Jacobian as follows (see [, ]).
Definition .(see []) Letx∈Xandη:X×X→Rn. The functionf :X→Rpis said
to be invex atxwith respect toη, if for allx∈X,
f(x)f(x) +Aη(x,x), ∀A∈∂Jf(x).
f is said to be invex with respect toηonX, if for everyx∈X,f is invex atxwith respect toη.
Definition .(see []) Letx∈Xandη:X×X→Rn. The functionf :X→Rpis said
to be pseudoinvex atxwith respect toη, if for allx∈X,
or equivalently,
f(x) <f(x) ⇒ Aη(x,x) < , ∀A∈∂Jf(x).
f is said to be pseudoinvex with respect toηonX, if for everyx∈X,f is pseudoinvex atx with respect toη.
In the generalized convexity of functions, the study of approximately convex functions (see [, , , ]) is a hot spot. Mishra and Upadhyay [] introduced the following concept of vector-valued approximately convex functions.
The functionf:X→Rpis said to be approximately convex atx
∈X, if for allα∈int(Rp+)
such that
f(x)f(x) +ξT(x–x) –αx–x, ∀x∈X,∀ξ∈∂f(x).
Motivated by above definitions, we give the notions of approximate invexity of orderm with respect toηandψ, strictly approximate invexity of ordermwith respect toηandψ
and approximate pseudoinvexity of type I of ordermwith respect toηandψ as follows.
Definition . Letx∈Xandm≥ be a positive integer. The functionf :X→Rp is
said to be approximately invex of order matxwith respect toηandψ, if there exist η:X×X→Rn,ψ:X×X→Rnandα∈int(Rp
+) such that, for allx∈X,
f(x)f(x) +Aη(x,x) –αψ(x,x)
m
, ∀A∈∂Jf(x).
f is said to be approximately invex of ordermwith respect toηandψ onX, if for every x∈X,f is approximately invex of ordermatxwith respect toηandψ.
Remark . ReplacingA∈∂Jf(x) byξ∈∂f(x), settingη(x,x) =x–x,ψ(x,x) =x–x
andm= in Definition ., then we arrive at the notion of approximately convex function, defined by Mishra and Upadhyay [].
Remark . A function which is invex atxwith respect toηis also approximately invex
of ordermatxwith respect toηandψ, but in the contrary case, it does not hold. The
following example is given to illustrate this fact.
Example . Consider the vector-valued functionf :R→R, defined byf(x) = (x,ϕ(x))T,
where
ϕ(x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–x ifx< ,
if ≤x≤,
–x ifx> .
It can easily be seen that
∂Jf(x) =
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
{(, –)T} ifx< orx> , {(,a)T: –≤a≤} ifx= orx= ,
We firstly prove thatf is not invex with respect toη:R×R→RonR. Indeed, choose ¯
x= andx= , then there exists nou=η(x,¯ x)∈Rsuch thatf(x) –¯ f(x)Aufor every
A= (,a)T∈∂
Jf(x). In other words, if (–, )T(u,au)T,∀a∈[–, ], this implies –u
and au. Fora= –, from above inequality auwe obtainu, which contradicts –u. Finally, we show thatf is approximately invex of ordermatx= with respect to η:R×R→Randψ:R×R→R. Takeη(x,x) = ,ψ(x,x) =x–x– =x– ,m=
andα= (α,α)T= (, )T. From the above we haveA= (,a)T∈∂Jf(), –≤a≤. then
we can see the following inequality holds true invariably:
f(x) –f(x) –Aη=
x– ,ϕ(x) –aT
–|x– |, –|x– |T = –αψ(x,x).
Definition . Letx∈Xandm≥ be a positive integer. The functionf:X→Rpis said
to be strictly approximately invex of ordermatxwith respect toηandψ, if there exist η:X×X→Rn,ψ:X×X→Rnandα∈int(Rp
+) such that, for allx∈X,
f(x) >f(x) +Aη(x,x) –αψ(x,x)
m
, x=x,∀A∈∂Jf(x).
f is said to be strictly approximately invex of ordermwith respect toηandψ onX, if for everyx∈X,f is strictly approximately invex of ordermatxwith respect toηandψ.
Definition . Letx∈Xandm≥ be a positive integer. The functionf:X→Rpis said
to be approximately pseudoinvex type I of ordermatxwith respect toηandψ, if there
existη:X×X→Rn,ψ:X×X→Rnandα∈int(Rp
+) such that, for allx∈X,
Aη(x,x), for someA∈∂Jf(x) ⇒ f(x) –f(x)–αψ(x,x)
m
,
or, equivalently,
f(x) +αψ(x,x)m<f(x) ⇒ Aη(x,x) < , ∀A∈∂Jf(x).
f is said to be approximately pseudoinvex type I of ordermwith respect toηandψonX, if for everyx∈X,f is approximately pseudoinvex type I of ordermatxwith respect toη
andψ.
The following example illustrates the existence of approximate invexity of ordermwith respect toηandψand of an approximately pseudoinvex type I function of ordermwith respect toηandψ.
Example . LetX=R,α= (α,α)T> andm≥ be a positive integer. Consider the
following functions:f :X→R,f:X→R,η:X×X→Randψ:X×X→Rdefined
by
f(x) =x,max{–x, ,x– }T, f(x) =–x,max{–x, ,x– }T,
η(x,x) =x–x and ψ(x,x) =
x
+x ,
T
For any positive integerm≥, it is easy to verify thatf is approximately invex of orderm atx= – with respect toηandψ. In fact, we can easily obtainA= (, –)T∈∂Jf(x). By
direct calculation, we derive
f(x) –f(x) =
x,max{–x, ,x– }T– (–, )T=x+ ,max{–x, ,x– }– T, Aη(x,x) –αψ(x,x)m= (, –)T(x+ ) – (α,α)T
x √ + m
=x+ , –(x+ )T– (α,α)T x √ m . Obviously,
x+ ,max{–x, ,x– }– Tx+ , –(x+ )T. Furthermore, because ofα= (α,α)T> , we can arrive at
x+ ,max{–x, ,x– }– Tx+ , –(x+ )T– (α,α)T x √ m . That is,
f(x)f(x) +Aη(x,x) –αψ(x,x)
m
, ∀x∈X,∀A∈∂Jf(x).
So, we have verified thatf is approximatelyly invex of ordermatx= – with respect toη
andψ.
Now, let us prove thatf(x) = (–x,max{–x, ,x– })Tis approximately pseudoinvex type I of ordermatx= with respect toη(x,x) =x–xandψ(x,x) = ( x
√
+x
, )T. Actually,
it is not difficult to getA= (–,a)T∈∂
Jf(x) ={(–,a)T: –≤a≤}. We suppose that
Aη(x,x) = (–,a)Tx= (–x,ax)T,
then we arrive at
f(x) –f(x) =
–x,max{–x, ,x– }T–(α,α)T
x
√
m
= –αψ(x,x)
m
.
This fulfills the condition of an approximately pseudoinvex type I of ordermfunction at x= with respect toηandψ.
Remark . It is obvious that iff :X→Rp is pseudoinvex atx
∈Xwith respect toη,
then it is also approximately pseudoinvex type I of ordermatxwith respect toηandψ.
But the converse does not hold. For example, considerϕ:X→R, given by
ϕ(x) =
⎧ ⎨ ⎩
–x ifx< ,
Takingη(x,x) =x–x,ψ(x,x) =x–xandm= , thenϕis approximately pseudoinvex
type I of order matx= with respect toηandψ. As for anyα> , there existsδ= min(π,α) > such thatAη(x,x)≥ for someA∈∂Jϕ(x) implies
ϕ(x) –ϕ(x)≥–αx–x, ∀x∈B(x,δ)∩X.
However,ϕis not pseudoinvex atxwith respect toη. Indeed, for everyδ> , there
ex-istsx∈B(x,δ)∩Xsuch thatAη(x,x)≥,A∈∂Jϕ(x) does not implyϕ(x)≥ϕ(x) (see
Remark in []).
We consider the followingnonsmooth multi-objective programming problem(NMP):
(NMP)
⎧ ⎨ ⎩
minimizef(x) = (f(x),f(x), . . . ,fp(x))T,
subject tox∈X,
wherefi:X→R,i∈P={, , . . . ,p}are non-differentiable functions.
In multi-objective programming problems, efficient and weakly efficient solutions are widely used. Considering the complexity of the optimization problem in reality and in order to find the optimal solution of multi-objective optimization problem in a smaller range, the notion of quasi-efficient and weakly quasi-efficient are introduced as follows (see [, , ]).
Definition . A pointx∈Xis said to be an efficient solution to the (NMP), if there
exists nox∈Xsuch that
f(x)≤f(x).
Definition . A pointx∈Xis said to be a weakly efficient solution to the (NMP), if
there exists nox∈Xsuch that
f(x) <f(x).
Definition . Letx∈X.
(i) A pointxis said to be a quasi-efficient solution to the (NMP), if there exists α∈int(Rp+)such that, for anyx∈X, the following cannot hold:
f(x)≤f(x) –αx–x.
(ii) A pointxis said to be a weakly quasi-efficient solution to the (NMP), if there exists α∈int(Rp+)such that, for anyx∈X, the following cannot hold:
f(x) <f(x) –αx–x.
Definition . Letm≥ be a positive integer. A pointx∈Xis called a quasi-efficient
solution of ordermfor (NMP) with respect toψ, if there exist a functionψ:X×X→Rn
andα∈int(Rp+) such that, for anyx∈X, the following cannot hold:
f(x)≤f(x) –αψ(x,x)m.
Definition . Letm≥ be a positive integer. A pointx∈Xis called a weakly
quasi-efficient solution of ordermfor (NMP) with respect toψ, if there exist a functionψ : X×X→Rnandα∈int(Rp+) such that, for anyx∈X, the following cannot hold:
f(x) <f(x) –αψ(x,x)
m
.
Remark . It is clear that efficient solution implies quasi-efficient solution of orderm with respect toψto the (NMP), but the converse may not be true. To illustrate this fact, we consider the following multi-objective programming problem:
⎧ ⎨ ⎩
minimizef(x) = (ln(x+ ) –x,x–x)T,
subject tox≥,
wheref :R+→R. Thenx
= is a quasi-efficient solution of ordermfor (NMP) with
respect toψ(x,x) = (x,x)Tforα= (, )Tandm= , but not an efficient solution. Remark . A quasi-efficient solution of ordermfor (NMP) with respect toψis not to be a quasi-efficient solution in the sense of Definition .. For example, letX={x∈R: ≤x≤}andf :X→Rbe defined asf(x) = (–x, –sinx)T, thenx= is not a
quasi-efficient solution in the sense of Definition ., because, for anyα= (α,α)T∈int(R +),
there exists anxsatisfyingx≥α
,
sinx
x ≥αsuch thatf(x)≤f(x) –αx–x; however,
x= is a quasi-efficient solution of orderm= for (NMP) with respect toψ(x,x) =
(x–sinx,x)Tforα= (, )T.
Associated with the problem (NMP), we consider the following generalized (weakly) vector variational-like inequalities problems:
(GVVI) Find a pointx∈Xsuch that there exists nox∈Xsuch that
Aη(x,x)≤, ∀A∈∂Jf(x).
(GWVVI) Find a pointx∈Xsuch that there exists nox∈Xsuch that
Aη(x,x) < , ∀A∈∂Jf(x).
3 Relations between (GVVI), (GWVVI) and (NMP)
Theorem . Let f:X→Rpbe approximately invex of order m at x
∈X with respect to ηandψ.If xsolves(GVVI),then xis a quasi-efficient solution of order m for(NMP)with
respect to the sameψ.
Proof Suppose thatxis not a quasi-efficient solution of ordermfor (NMP) with respect
toψ, then there existxˆ∈Xandα∈int(Rp+) such that
f(x)ˆ ≤f(x) –αψ(x,ˆ x)m. (.)
Sincef is approximately invex of ordermatxwith respect toηandψ onX, it follows
from Definition . that
f(x)ˆ f(x) +Aη(x,ˆ x) –αψ(x,ˆ x)m, ∀A∈∂Jf(x). (.)
From inequalities (.) and (.), we see that there existsxˆ∈Xsuch that
Aη(x,ˆ x)≤, ∀A∈∂Jf(x),
which is inconsistent with the fact thatxsolves (GVVI).
Theorem . Let f:X→Rpbe approximately invex of order m at x
∈X with respect to ηandψ.If xsolves(GWVVI),then xis a weakly quasi-efficient solution of order m for
(NMP)with respect to the sameψ.
Proof Assume thatxis not a weakly quasi-efficient solution of ordermfor (NMP) with
respect toψ, then there existα∈int(Rp+) andxˆ∈Xsuch that
f(x) <ˆ f(x) –αψ(x,ˆ x)
m
.
Becausef is approximately invex of ordermatxwith respect toηandψonX, therefore,
in particular forα∈int(Rp+) andx, we haveˆ
f(x)ˆ f(x) +Aη(x,ˆ x) –αψ(x,ˆ x)
m
, ∀A∈∂Jf(x).
Furthermore, we arrive at
Aη(x,ˆ x) < , ∀A∈∂Jf(x),
which contradicts the hypothesis thatxsolves (GWVVI).
Theorem . Let f :X→Rpbe approximately pseudoinvex type I of order m at x
∈X
with respect toηandψ.If xsolves(GWVVI),then xis a weakly quasi-efficient solution
of order m for(NMP)with respect to the sameψ.
Proof Supposexsolves (GWVVI) but is not a weakly quasi-efficient solution of orderm
for (NMP) with respect toψ, then there existα∈int(Rp+) andxˆ∈X, satisfying
f(x) <ˆ f(x) –αψ(x,x)
m
Noticing thatf is approximately pseudoinvex type I of ordermatxwith respect toηand ψonX, it follows from Definition . and inequality (.) that there existα∈int(Rp+) and
ˆ
xsuch that
Aη(x,ˆ x) < , ∀A∈∂Jf(x). (.)
This is obviously not in agreement with the hypothesis thatxsolves (GWVVI).
4 Characterization of generalized quasi-efficient solutions by vector critical points
This section is devoted to investigating the relations between vector critical points and weakly quasi-efficient solution of ordermfor (NMP) with respect toψunder generalized invexity (introduced in Section ) hypotheses imposed on the involved functions.
Definition .(see []) A feasible solutionx∈Xis said to be a vector critical point of
(NMP), if there exists a vetorξ∈Rpwithξ≥ such thatξTA= for someA∈∂Jf(x). Lemma .(see [] (Gordan’s theorem)) Let A be a p×n matrix.Then exactly one of the following two systems has a solution:
System : Ax< for somex∈Rn.
System : ATy= ,y≥for some nonzeroy∈Rp.
Theorem . Let x∈X be a vector critical point of(NMP)and f :X→Rp be
approx-imately pseudoinvex type I of order m at xwith respect toηandψ,then xis a weakly
quasi-efficient solution of order m for(NMP)with respect toψ.
Proof Letxbe a vector critical point of (NMP), it follows from Definition . that there
exist a vectorξ∈Rpwithξ≥ andA∈∂
Jf(x) such that ξTA= .
By contradiction, suppose thatxis not a weakly quasi-efficient solution of ordermfor
(NMP) with respect toψ, then for anyα∈int(Rp+) there exists anxˆ∈Xsatisfying
f(x) <ˆ f(x) –αψ(x,ˆ x)
m
. (.)
Noticing thatf is approximately pseudoinvex type I of ordermatxwith respect toηand ψ onX, it follows from Definition . and inequality (.) that
Aη(x,ˆ x) < , ∀A∈∂Jf(x).
Using Gordan’s theorem, the system
ξTA= , ∀A∈∂Jf(x), ξ≥, ξ∈Rp,
has no solution for ξ, which contradicts the fact that x is a vector critical point of
Theorem . Any vector critical point is a weakly quasi-efficient solution of order m for (NMP)with respect toψ,if and only if f :X→Rpis approximately pseudoinvex type I of
order m at that point with respect toηandψ.
Proof The sufficient condition is obtained by Theorem .. In the following we only need to prove the necessary condition. Letxbe a weakly quasi-efficient solution of orderm
for (NMP) with respect toψ, then there existsα∈int(Rp+) such that, for anyx∈X, the
following cannot hold:
f(x) –f(x) < –αψ(x,x)m. (.)
Noticing thatxis a vector critical point, then there existξ∈Rpwithξ≥ andA∈∂Jf(x)
such that
ξTA= .
Using Gordan’s theorem, there existsA∈∂Jf(x) such that the system μTA<
has no solutionμ∈Rp. Thus, the system
μTA< , ∀A∈∂Jf(x), (.)
has no solutionμ∈Rp. Therefore, (.) and (.) are equivalent. Hence, ifxis a weakly
quasi-efficient solution of ordermfor (NMP) with respect toψ, that is, for anyα∈int(Rp+),
there exists nox∈Xsuch that
f(x) –f(x) < –αψ(x,x)m,
thenxsolves (GWVVI), that is, there exists nox∈Xwithη(x,x)∈Rpsatisfying
Aη(x,x) < , ∀A∈∂Jf(x).
This satisfies the condition of the approximately pseudoinvexity of type I of ordermoff
atx.
5 Conclusions
obtained, that is, a vector critical point is a weakly quasi-efficient solution of higher order with respect to a function (Theorem . and Theorem .).
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grant Nos. 61650104, 11361001.
Competing interests
The authors have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approve the final text.
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Received: 16 June 2017 Accepted: 18 July 2017 References
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