R E S E A R C H
Open Access
Generic stability of the solution mapping
for set-valued optimization problems
Xian-Jun Long
*, Ying-Quan Huang and Li-Ping Tang
*Correspondence:
[email protected] College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, P.R. China
Abstract
In this paper, we consider the generic stability of the weakly efficient solution mapping for set-valued optimization problems. Firstly, we obtain the upper semicontinuity of the weakly efficient solution mapping for set-valued optimization problems. Secondly, we show that, in the sense of Baire category, most set-valued optimization problems are stable. Finally, we give sufficient conditions ensuring the existence of essential. Our results extend and improve the corresponding results of Songet al.(J. Optim. Theory Appl. 156:591-599, 2013).
MSC: 49J53; 90C29; 90C31
Keywords: generic stability; set-valued optimization problem; weakly efficient solution; essential; semicontinuity
1 Introduction
Set-valued optimization is a vibrant and expanding branch of applied mathematics that deals with optimization problems where the objective function is a set-valued map acting between abstract spaces. Set-valued optimization provides an important generalization and unification of scalar as well as vector optimization problems. Therefore, this relatively new discipline has justifiably attracted a great deal of attention in recent years (see [–]). Stability is very interesting and important in optimization theory and applications. It may be understood as the solution set having some topological properties such as semi-continuity, well-posedness, essential stability and so on. Essential stability was firstly in-troduced by Fort [] for the study of fixed points of a continuous mapping. Since then, essentiality was applied in many nonlinear problems such as KKM points, vector equilib-rium problems and Nash equilibequilib-rium problems (see [–]). Recently, Xiang and Zhou [] obtained the essential stability of efficient solution sets for continuous vector op-timization problems. Very recently, Songet al.[] generalized the results obtained by Xiang and Zhou [] to a set-valued case. They obtained the essential stability of efficient solution sets for set-valued optimization problems with the only perturbation of the ob-jective function in compact metric spaces.
In this paper, we consider the stability of a weakly efficient solution mapping for set-valued optimization problems with the perturbation of both the objective function and the constraint set in noncompact Banach spaces. In Section we recall some basic definitions and some known results. In Section we obtain the upper semicontinuity of the weakly
efficient solution mapping for set-valued optimization problems. Moreover, we show that, in the sense of Baire category, most set-valued optimization problems are stable. Finally, we give sufficient conditions ensuring the existence of essential. Our results extend and improve the corresponding results of Songet al.[].
2 Preliminaries
LetXandYbe two topological vector spaces. LetC⊂Ybe a closed convex pointed cone withintC=∅, whereintCdenotes the interior ofC. LetA⊂Ybe a nonempty subset. We denote by
WMinA:=y∈A: (A–y)∩–intC=∅ the set of weakly efficient elements ofAand by
MinA:=y∈A: (A–y)∩–C={} the set of efficient elements ofA.
LetF:X→Y be a set-valued map,K ⊆X be a nonempty subset. We consider the following set-valued optimization problem (in short, SOP):
min
C F(x) subject tox∈K.
We denote
F(K) =
x∈K
F(x).
Definition . A pointx∈Kis said to be a weakly efficient (resp. an efficient) solution of
problem (SOP) iff there existsy∈F(x) such thaty∈WMinF(K) (resp.y∈MinF(K)).
Definition .[] LetG:X→Y be a set-valued map.Tis said to be
() upper semicontinuous atx∈Xif, for any open setVcontainingG(x), there exists
a neighborhoodU(x)ofxsuch thatG(x)⊂Vfor allx∈U(x);Gis said to be
upper semicontinuous onXif it is upper semicontinuous at eachx∈X; () lower semicontinuous atx∈Xif, for any open setVwithG(x)∩V=∅, there
exists a neighborhoodU(x)ofxsuch thatG(x)∩V=∅for allx∈U(x);Gis said
to be lower semicontinuous onXif it is lower semicontinuous at eachx∈X; () continuous onXif it is both upper semicontinuous and lower semicontinuous onX; () closed ifGraph(G) :={(x,y) :x∈X,y∈G(x)}is a closed set inX×Y.
Lemma .[] Let G:X→Y be a set-valued map.If G is upper semicontinuous and for
any x∈X,G(x)is a closed set,then G is closed.
Definition . Let (X,d) be a metric space and letA,Bbe nonempty subsets ofX. The Hausdorff distanceH(·,·) betweenAandBis defined by
where e(A,B) =supa∈Ad(a,B) with d(a,B) =infb∈B a–b . Let {An} be a sequence of
nonempty subsets of X. We say that An converges toAin the sense of Hausdorff
dis-tance (denoted byAn→A) ifH(An,A)→. It is easy to see thate(An,A)→ if and only
ifd(an,A)→ for all selectionan∈An. For more details on this topic, we refer the readers
to [, ].
Lemma .[] Let A and An(n= , , . . .)all be nonempty compact subsets of the
Haus-dorff topological space X with An→A.Then the following statements hold:
(i) +n∞=An∪Ais also a nonempty compact subset ofX.
(ii) Ifxn∈Anconverging tox,thenx∈A.
A topological spaceXis said to be a Baire space if the following condition holds: given any countable collection{An}+n∞=of the closed subsets ofXeach of which has empty
inte-rior inX, their union∪Analso has empty interior inX. A subsetGofXis called residual
if it contains a countable intersection of open dense subsets ofX.
Lemma .(Baire category theorem) If X is a compact Hausdorff space or a complete metric space,then X is a Baire space.
Lemma .([], Theorem ) Let X be a Baire space,Y be a metric space and G:X→Y
be upper semicontinuous with compact values.Then there exists a dense residual subset Q of X such that G is lower semicontinuous at each x∈Q.
For convenience in the later presentation, denote byK(X) andK(Y) all nonempty com-pact subsets ofXandY, respectively.
Lemma .[] Let(X,d)be a metric space and H be Hausdorff distance on X.Then
(K(X),H)is complete if and only if(X,d)is complete. The next lemma is a special case of Lemma . in [].
Lemma . Let K be a nonempty compact subset of X and G:K→Y be a set-valued
map with nonempty compact values.Then G is continuous if and only if for any x∈K,
x→ximplies G(x)→G(x).
Lemma .[] Let Fn→F,n= , , . . . ,where Fn,F:X→Y are continuous on X and
have nonempty compact values.If yn∈Fn(xn),xn→x∗and yn→y∗,then y∗∈F(x∗).
Lemma . Let Fn→F,n= , , . . . ,where Fn,F:X→Y are continuous on X and have
nonempty compact values.Then,for any x∈X,y∈F(x),xn→x,there exists yn∈Fn(xn)
such that yn→y.
Proof SinceFn→F,H(Fn(x),F(x))→ for anyx∈X. Note that
HFn(xn),F(x)
≤HFn(xn),F(xn)
+HF(xn),F(x)
.
By the continuity ofFand Lemma .,H(Fn(xn),F(x))→. Therefore, for anyy∈F(x),
3 Main results
Throughout this section, letXandYbe two real Banach spaces,Kbe a nonempty subset ofX,C⊂Ybe a closed convex pointed cone withintC=∅.
The spaceMof the problem (SOP) is defined by
M:=
u= (F,K) :F:K→
Yis continuous and has nonempty compact values,
Kis a nonempty compact subset ofX. . For anyu= (F,K),u= (F,K)∈M, we define the metricρas follows:
ρ(u,u) :=sup
x∈K
HF
F(x),F(x)
+HK(K,K),
whereHF,HKare two Hausdorff distances onYandX, respectively.
Lemma . (M,ρ)is a complete metric space.
Proof Clearly, (M,ρ) is a metric space. We only need to show that (M,ρ) is complete. Let
{un}be a Cauchy sequence ofM, whereun= (Fn,Kn). Then, for anyε> , there exists a
positive integerNsuch that
ρ(un,um) <
ε
for allm,n≥N. It follows that for anyx∈K,
HF
Fn(x),Fm(x)
<ε
and HK(Kn,Km) <
ε
. ()
This implies that {Fn(x)}is a Cauchy sequence inK(Y) and{Kn}is a Cauchy sequence
inK(X). By the assumption and Lemma ., (K(Y),HF) and (K(X),HK) are complete. It
follows that there existF(x)∈K(Y) andK∈K(X) such that
Fn(x)→F(x) and Kn→K. ()
For fixedn≥Nand anyx∈K, letm→+∞in (), we have
HF
Fn(x),F(x)
<ε
and HK(Kn,K) <
ε
. ()
We now show thatFis continuous. In fact, by the continuity ofFnand Lemma ., there
exist a neighborhoodU(x) ofxand a positive integerNsuch that
HF
Fn(x),Fn(x)
<ε
for allx∈U(x)∩K, for anyn≥N. () LetN=max{N,N}. Combining with (), () and () yields
HF
F(x),F(x)
≤HF
F(x),Fn(x)
+HF
Fn(x),Fn(x)
+HF
Fn(x),F(x)
for allx∈U(x)∩Kand for anyn≥N. By Lemma .,Fis continuous onK. Setu= (F,K)
and sou∈M. It follows that
ρ(un,u) =sup x∈K
HF
Fn(x),F(x)
+HK(Kn,K) <ε,
which impliesun
ρ
−
→u. Therefore, (M,ρ) is a complete metric space. The proof is
com-plete.
For anyu= (F,K)∈M, we denote byS(u) andSw(u) the efficient solution set and the
weakly efficient solution set of problem (SOP), respectively. ThenS andSw define two
set-valued maps from Mto X. By the compactness of K and the continuity ofF, the setMin(F(X)) is nonempty, and soS(u) is nonempty for anyu∈M. Moreover,Sw(u) is
nonempty sinceS(u)⊂Sw(u).
Theorem . The set-valued map Sw:M→X is upper semicontinuous with compact
values.
Proof For anyu= (F,K)∈M, we prove that the set
Sw(u) =
x∈K:F(K) –y∩–intC=∅,∃y∈F(x)
is compact. In fact, let{xn} ⊆Sw(u) withxn→x. Thenxn∈Kand there existsyn∈F(xn)
such that
F(K) –yn
∩–intC=∅. ()
Note thatKis a compact set. It follows thatx∈K. SinceF(K)⊃F(xn) is compact, there
exists a subsequence of{yn} which converges toy. Without loss of generality, we may
assume thatyn→y. By the continuity ofF,y∈F(x). This fact together with () yields
x∈Sw(u). It follows thatSw(u) is closed. Therefore,Sw(u) is compact sinceKis compact.
Next, we prove thatSwis upper semicontinuous onM. Suppose by contradiction that
there existsu= (F,K)∈Msuch thatSwis not upper semicontinuous atu. Then there exists
an open neighborhoodUinXwithU⊃Sw(u) such that, for eachn= , , . . . and each open
neighborhoodVn:={u= (F,K)∈M:ρ(u,u) <n}ofu, there existun= (Fn,Kn)∈Vnand
xn∈Sw(un) butxn∈/U.
Fromun= (Fn,Kn)∈Vnfor eachn= , , . . . , we haveρ(un,u) <n →. This implies
Fn→F and Kn→K.
Asxn∈Sw(un), we havexn∈Knand there existsyn∈Fn(xn) such that
Fn(Kn) –yn
∩–intC=∅.
By the compactness ofK andKnand Lemma .(i),
+∞
n=Kn∪Kis compact. Note that
{xn} ⊆
+∞
n=Kn∪K. Then{xn}has a convergent subsequence. Without loss of generality,
of{xn},xn→x∗∈K. Sincexn∈/UandUis open,x∗∈/ U. FromSw(u)⊂U, we havex∗∈/
Sw(u). It follows that
F(K) –y∩–intC=∅, ∀y∈Fx∗. ()
On the other hand, sinceyn∈Fn(xn) andFn(xn) is compact for anyn, there existsy
such that yn→y. By Lemma .,y∈F(x∗). Note thatKn→K. Then, for any z∈K,
there exists a sequence{zn}such thatzn∈Knandzn→z. By Lemma ., for anyw∈F(z),
there existswn∈Fn(zn) such thatwn→w. Since (Fn(Kn) –yn)∩–intC=∅, one has
wn–yn∈/–intC.
It follows that
w–y∈/–intC.
This contradicts (). Therefore,Sw is upper semicontinuous onM. The proof is
com-plete.
From the proof of Theorem ., we obtain that for anyu∈M, the weakly efficient solu-tion setSw(u) is closed. By Lemma ., we have the following corollary.
Corollary . The set-valued map Sw:M→Xis closed.
Remark . Corollary . generalizes and improves the corresponding result of Songet al.[], Theorem ., in the following four aspects:
() the assumption that the metric space is compact is removed; () the setting of Euclidean spaces is generalized to Banach spaces; () the order coneRn
+is generalized to any closed convex pointed cone;
() we not only consider the perturbation of the set-valued map, but also consider the perturbation of the feasible set; while Songet al.[] only considered the former.
Definition . Letu∈M. The weakly efficient solution setSw(u) is called stable if the
set-valued mapSwis continuous atu.
Remark . The following example shows that there existsu∈Msuch thatSw(u) is not
stable.
Example . LetX=R,Y=R,C=R
+,K= [, ] andKn= [n, ]. Define set-valued
map-pingsF,Fn:X→R
such that for anyx∈X,
F(x) = [, ]×[x, ] and Fn(x) =
x n,
×[x, ].
ThenFn→FandKn→Kwhenn→+∞. By a simple computation,
Sw(un) =
n, un= (Fn,Kn).
It is easy to see thatSwis upper semicontinuous atu. However,Swis not lower
semicontin-uous atu. In fact, letx= ∈Sw(u), one can easily find that for small enough neighborhood
U(x) ofxand large enoughn,Sw(un)∩U(x) =∅. Therefore,Swis not stable atu.
Definition . Foru∈M, a pointx∈Sw(u) is said to be essential if, for any open
neighbor-hoodUofxinX, there exists an open neighborhoodVofuinMsuch thatSw(u)∩U=∅
for allu∈V.uis said to be essential if everyx∈Sw(u) is essential.
From Definition ., it is easy to see that the following lemma holds, so we omit its proof.
Lemma . The set-valued map Sw is lower semicontinuous at u∈M if and only if u is
essential.
We now give a generic stability result for set-valued optimization problems.
Theorem . There exists a dense residual subset Q of M such that,for every u∈Q,u is essential.
Proof By Lemmas . and .,Mis a Baire space. By Theorem ., the set-valued mapSw:
M→Xis upper semicontinuous with compact values. By Lemma ., there exists a dense
residual subsetQofMsuch thatSwis lower semicontinuous at eachu∈Q. Therefore, the
conclusion holds by Lemma ..
Remark . Example . shows that there existsu∈Msuch thatuis not essential. The following theorem gives a sufficient condition thatu∈Mis essential.
Theorem . If u∈M and Sw(u)is a singleton set,then u is essential.
Proof Suppose thatSw(u) ={x}. LetUbe any open set inXsuch thatSw(u)∩U=∅. Then
x∈UandSw(u)⊂U. By Theorem .,Swis upper semicontinuous atu∈M. It follows
that there exists an open neighborhoodVofuinMsuch thatSw(u)⊂Ufor eachu∈V.
This implies thatSw(u)∩U=∅for eachu∈V. Thus,Swis lower semicontinuous atu.
By Lemma .,uis essential.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
Acknowledgements
The first author was supported by the National Natural Science Foundation of China (11001287, 11471059), the Chongqing Research Program of Basic Research and Frontier Technology (cstc2014jcyjA00037), the Education Committee Project Research Foundation of Chongqing (KJ1400618) and the Program for Core Young Teacher of the Municipal Higher Education of Chongqing ([2014]47).
References
1. Ansari, QH, Jahn, J:T-Epiderivative of set-valued maps and its application to set optimization and generalized variational inequalities. Taiwan. J. Math.14, 2447-2468 (2010)
2. Aubin, JP, Frankowska, H: Set-Valued Analysis. Birkhäuser, Boston (1990)
3. Chen, GY, Jahn, J: Optimality conditions for set-valued optimization problems. Math. Methods Oper. Res.48, 187-200 (1998)
4. Chen, GY, Huang, XX, Yang, XQ: Vector Optimization: Set-Valued and Variational Analysis. Springer, Berlin (2005) 5. Durea, M, Strugariu, R: Calculus of tangent sets and derivatives of set-valued maps under metric subregularity
conditions. J. Glob. Optim.56, 587-603 (2013)
6. Flores-Bazán, F: Radial epiderivatives and asymptotic functions in nonconvex vector optimization. SIAM J. Optim.14, 284-305 (2003)
7. Gong, XH, Dong, HB, Wang, SY: Optimality conditions for proper efficient solutions of vector set-valued optimization. J. Math. Anal. Appl.284, 332-350 (2003)
8. Hernández, E, Rodríguez-Marín, L, Sama, M: On solutions of set-valued optimization problems. Comput. Math. Appl.
60, 1401-1408 (2010)
9. Jahn, J: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)
10. Kuroiwa, D: On derivatives of set-valued maps and optimality conditions for set optimization. J. Nonlinear Convex Anal.10, 41-50 (2009)
11. Khan, AA, Tammer, C, Zalinescu, C: Set-Valued Optimization: An Introduction with Applications. Springer, Berlin (2015)
12. Lalitha, CS, Arora, R: Weak Clarke epiderivative in set-valued optimization. J. Math. Anal. Appl.342, 704-714 (2008) 13. Luc, DT: Theory of Vector Optimization. Lecture Notes in Economics and Mathematics Systems, vol. 319. Springer,
New York (1989)
14. Long, XJ, Li, XB, Zeng, J: Lagrangian conditions for approximate solutions on nonconvex set-valued optimization problems. Optim. Lett.7, 1847-1856 (2013)
15. Long, XJ, Peng, JW: Lagrangian duality for vector optimization problems with set-valued mappings. Taiwan. J. Math.
17, 287-297 (2013)
16. Long, XJ, Peng, JW: GeneralizedB-well-posedness for set optimization problems. J. Optim. Theory Appl.157, 612-623 (2013)
17. Long, XJ, Peng, JW, Li, XB: Weak subdifferentials for set-valued mappings. J. Optim. Theory Appl.162, 1-12 (2014) 18. Long, XJ, Peng, JW, Peng, ZY: Scalarization and pointwise well-posedness for set optimization problems. J. Glob.
Optim.62, 763-773 (2015)
19. Tammer, C, Zalinescu, C: Vector variational principles for set-valued functions. In: Recent Developments in Vector Optimization, pp. 367-415. Springer, Berlin (2012)
20. Qiu, QS, Yang, XM: Some properties of approximate solutions for vector optimization problem with set-valued functions. J. Glob. Optim.47, 1-12 (2010)
21. Zhou, ZA, Yang, XM: Scalarization ofε-super efficient solutions of set-valued optimization problems in real ordered linear spaces. J. Optim. Theory Appl.162, 680-693 (2014)
22. Fort, MK: Essential and nonessential fixed points. Am. J. Math.72, 315-322 (1950) 23. Yu, J, Xiang, SW: The stability of the set of KKM points. Nonlinear Anal.54, 839-844 (2003)
24. Chen, JC, Gong, XH: The stability of set of solutions for symmetric vector quasi-equilibrium problems. J. Optim. Theory Appl.136, 359-374 (2008)
25. Yang, H, Yu, J: Essential components of the set of weakly Pareto-Nash equilibrium points. Appl. Math. Lett.15, 553-560 (2002)
26. Lin, Z: Essential components of the set of weakly Pareto-Nash equilibrium points for multiobjective generalized games in different topological spaces. J. Optim. Theory Appl.124, 387-450 (2005)
27. Luo, Q: Essential component and essential optimum solution of optimization problems. J. Optim. Theory Appl.102, 433-438 (1999)
28. Xiang, SW, Zhou, YH: On essential sets and essential components of efficient solutions for vector optimization problems. J. Math. Anal. Appl.315, 317-326 (2006)
29. Song, QQ, Tang, GQ, Wang, LS: On essential stable sets of solutions in set optimization problems. J. Optim. Theory Appl.156, 591-599 (2013)
30. Klein, E, Thompson, AC: Theory of Correspondences. Wiley, New York (1968) 31. Kuratowski, K: Topology, Vols. 1 and 2. Academic Press, New York (1968)
