A new existence result of ε-vector equilibrium problem is first obtained. Then, by using the existence theorem of ε-vector equilibrium problem, a weakly ε-cone saddle point theorem is also obtained for vector-valued mappings.
1. Introduction
Saddle point problems are important in the areas of optimization theory and game theory. As for optimization theory, the main motivation of studying saddle point has been their connection with characterized solutions to minimax dual problems. Also, as for game theory, the main motivation has been the determination of two-person zero-sum games based on the minimax principle.
In recent years, based on the development of vector optimization, a great deal of papers have been devoted to the study of cone saddle points problems for vector-valued mappings and set-valued mappings, such as [1–8]. Nieuwenhuis [5] introduced the notion of cone saddle points for vector-valued functions in finite-dimensional spaces and obtained a cone saddle point theorem for general vector-valued mappings. Gong [2] established a strong cone saddle point theorem of vector-valued functions. Li et al. [4] obtained an existence theorem of lexicographic saddle point for vector-valued mappings. Bigi et al. [1] obtained a cone saddle point theorem by using an existence theorem of a vector equilibrium problem. Zhang et al. [9] established a general cone loose saddle point for set-valued mappings. Zhang et al. [8] obtained a minimax theorem and an existence theorem of cone saddle points for set-valued mappings by using Fan-Browder fixed point theorem. Some other types of existence results can be found in [3, 10–18].
On the other hand, in some situations, it may not be possible to find an exact solution for an optimization problem, or such an exact solution simply does not exist, for example, if the feasible set is not compact. Thus, it is meaningful to look for an approximate solution instead. There are also many papers to investigate the approximate solution problem, such as [19–21]. Kimura et al. [20] obtained several existence results for ε-vector equilibrium problem and the lower semicontinuity of the solution mapping of ε-vector equilibrium problem. Anh and Khanh [19] have considered two kinds of solution sets to parametric generalized ε-vector quasiequilibrium problems and established the sufficient conditions for the Hausdorff semicontinuity (or Berge semicontinuity) of these solution mappings. X. B. Li and S. J. Li [21] established some semicontinuity results on ε-vector equilibrium problem.
The aim of this paper is to characterize the ɛ-cone saddle point of vector-valued mappings. For this purpose, we first establish an existence theorem for ε-vector equilibrium problem. Then, by this existence result, we obtain an existence theorem for ε-cone saddle point of vector-valued mappings.
2. Preliminaries
Let X be a real Hausdorff topological vector space and let V be a real local convex Hausdorff topological vector space. Assume that S is a pointed closed convex cone in V with nonempty interior intS≠∅. Let V* be the topological dual space of V. Denote the dual cone of S by S*:
(1)S*={s*∈V*:s*(s)≥0,∀s∈S}.
Note that from Lemma 3.21 in [22] we have
(2)z∈S⟺{〈z*,z〉≥0,∀z*∈S*},z∈intS⟺{〈z*,z〉>0,∀z*∈S*∖{0}}.
Definition 1 (see [7, 23]).
Let f:X→V be a vector-valued mapping. f is said to be S-upper semicontinuous on X if and only if, for each x∈X and any s∈intS, there exists an open neighborhood Ux of x such that
(3)f(u)∈f(x)+s-intS,∀u∈Ux.fis said to be S-lower semicontinuous on X if and only if -f is S-upper semicontinuous on X.
Lemma 2 (see [17]).
Let f:X×X→V be a vector-valued mapping and s*∈S*∖{0}. If f is S-lower semicontinuous, then s*∘f is lower semicontinuous.
Definition 3 (see [24]).
Let A and B be nonempty subsets of X and f:A×B→V be a vector-valued mapping.
fis said to be S-concavelike in its first variable on A if and only if, for all x1,x2∈A and l∈[0,1], there exists x¯∈A such that
(4)f(x¯,y)∈lf(x1,y)+(1-l)f(x2,y)+S,∀y∈B.
f is said to be S-convexlike in its second variable on B if and only if, for all y1,y2∈B and l∈[0,1], there exists y¯∈B such that
(5)f(x,y¯)∈lf(x,y1)+(1-l)f(x,y2)-S,∀x∈A.
f is said to be S-concavelike-convexlike on A×B if and only if f is S-concavelike in its first variable and S-convexlike in its second variable.
Definition 4.
Let A⊂V be a nonempty subset and ε∈intS.
A point z∈A is said to be a weak ε-minimal point of A if and only if A∩(z-ε-intS)=∅ and MinεA denotes the set of all weak ε-minimal points of A.
A point z∈A is said to be a weak ε-maximal point of A if and only if A∩(z+ɛ+intS)=∅ and MaxεA denotes the set of all weak ε-maximal points of A.
Definition 5.
Let f:A×B→V be a vector-valued mapping and ε∈intS. A point (a,b)∈A×B is said to be a weak ε-S-saddle point of f on A×B if
(6)f(a,b)∈Maxεf(A,b)⋂Minεf(a,B).
3. Existence of ε-Vector Equilibrium Problem
In this section, we deal with the following ɛ-vector equilibrium problem (for short VAEP). Find x¯∈E such that
(7)f(x,y)+ε∉-intS,∀y∈E,
where f:X×X→V is a vector-valued mapping, E is a nonempty subset of X, and ε∈intS.
If f(x,y)=g(y)-g(x), x,y∈E, and if x¯∈E is a solution of VAEP, then x¯∈E is a solution of ε-vector optimization of g, where g is a vector-valued mapping.
Denote the ε-solution set of (VAEP) by
(8)S(ɛ)∶={x¯∈E:f(x,y)+ε∉-intS,∀y∈E}.
Lemma 6 (see [20]).
Let E be a nonempty subset of X. Suppose that f:X×X→V is a vector-valued mapping and the following conditions are satisfied:
clE is a compact set;
{x∈clE:f(x,y)∉-intS,∀y∈clE}≠∅;
f is S-lower semicontinuous on clE×clE.
Then, for each ε∈intS, S(ɛ)≠∅.
Next, we give a sufficient condition for the condition (ii) in Lemma 6.
Lemma 7.
Let E be a nonempty subset of X. Suppose that f:X×X→V is a vector-valued mapping with f(x,x)=0 for all x∈X and the following conditions are satisfied:
clE is a compact set;
f is S-concavelike-convexlike on clE×clE;
for each x∈clE, f(x,·) is S-lower semicontinuous on clE.
Then, there exists x¯∈clE such that
(9)f(x¯,y)∉-intS,∀y∈clE.
Proof.
For any t<0 and s*∈S*∖{0}, we define a multifunction G:clE→2clE by
(10)G(x)={y∈clE:s*(f(x,y))≤t},∀x∈clE.
First, by assumptions, we must have
(11)⋂x∈clEG(x)=∅.
In fact, if there exists y¯∈clE such that y¯∈G(x), for all x∈clE, then
(12)s*(f(x,y¯))≤t,∀x∈clE.
Particularly, taking x=y¯, we have 0=s*(f(y¯,y¯))≤t, which contradicts the assumption about t.
Then, by Lemma 2, G(x) is a closed set, for each x∈clE. By (11), for any y∈clE, we have
(13)y∈V∖⋂x∈clEG(x)=⋃x∈clEV∖G(x).
Since clE is compact, there exists a finite point set {x1,x2,…,xn} in clE such that
(14)clE⊂⋃1≤i≤nV∖G(xi).
Namely, for each y∈clE, there exists i∈{1,2,…,n} such that
(15)s*(f(xi,y))>t.
Now, we consider the set
(16)M∶={(z1,z2,…,zn,r)∈Rn+1∣∃y∈clE,es*(f(xi,y))≤r+zi,∀i=1,2,…,n(z1,z2,…,zn,r)∈Rn+1∣∃y∈clE,}.
Obviously, by the condition (ii), M is a convex set. By (15), we have the fact that (0Rn,t)∉M.
By the separation theorem of convex sets, there exists (λ1,λ2,…,λn,r¯)≠0Rn such that
(17)∑i=1nλizi+r¯r≥r¯t,∀(z1,z2,…,zn,r)∈M.
Since M+Rn+1⊂M, we can get λi≥0 and r¯≥0, for all i=1,2,…,n. By the definition of M, for each y∈clE,
(18)(0Rn,1+max1≤i≤ns*(f(xi,y)))∈intM,(19)(s*(f(x1,y))-r,ws*(f(x2,y))-r,…,s*(f(xn,y))-r,r)∈M.
By (18), r¯>0. Then, by (17) and (19),
(20)∑i=1nλir¯s*(f(xi,y))+r(1-∑i=1nλir¯)≥t.
By (20), ∑i=1n(λi/r¯)=1. Thus, by the condition (ii), for each y∈clE, there exists x¯∈clE such that
(21)s*(f(x¯,y))≥∑i=1nλir¯s*(f(xi,y))≥t.
By the assumption about t and s*, there exists x¯∈clE such that
(22)f(x¯,y)∉-intS,∀y∈clE.
This completes the proof.
By Lemmas 6 and 7, we can get the following result.
Theorem 8.
Let E be a nonempty subset of X. Suppose that f:X×X→V is a vector-valued mapping with f(x,x)=0 for all x∈X and the following conditions are satisfied:
clE is a compact set;
f is S-concavelike-convexlike on clE×clE;
f is S-lower semicontinuous on clE×clE.
Then, for each ε∈intS, S(ε)≠∅.
Remark 9.
Note that the condition (i) does not require the fact that clE is a convex set. So Theorem 8 is different from Theorem 3.2 in [20]. The following example explains this case.
Example 10.
Let X=R, V=R2, and E=[0,1/3]∪[2/3,1],
(23)f(x,y)={(xy,xz)∈R2∣z=1-y2},x,y∈X,S={(x,y)∈R2∣x≥0,y≥0}.
Obviously, clE is a compact set. However, clE is not a convex set. So, Theorem 3.2 in [20] is not applicable. By the definition of f, f is S-concavelike-convexlike on clE×clE and S-lower semicontinuous on clE×clE. Thus, all conditions of Theorem 8 hold. Indeed, for each ɛ=(ε1,ε2)∈intS,
(24)f(0,y)+ε=(ε1,ε2)∉-intS,∀y∈E.
Namely, 0∈S(ε).
4. Existence of ε-Cone Saddle PointsLemma 11.
Let E be a nonempty subset of X and E=A×B. Let ε∈intS and let f:X×X→V be a vector-valued mapping with f(x,y)=g(a,v)-g(u,b), where x=(a,b), y=(u,v), a,u∈A, and v,b∈B. If there exists x¯=(a¯,b¯)∈E such that
(25)f(x¯,y)+ε∉-intS,∀y∈E,
then (a¯,b¯)∈A×B is a weak ε-S-saddle point of g on A×B.
Proof.
By assumptions, we have
(26)f(x¯,y)+ɛ∉-intS,∀y∈E.
Then,
(27)g(a¯,v)-g(u,b¯)+ε∉-intS,∀(u,v)∈A×B.
By (27), taking u=a¯,
(28)g(a¯,v)-g(a¯,b¯)+ε∉-intS,∀v∈B,
which implies g(a¯,b¯)∈Minεg(a¯,B). Then, by (27), taking v=b¯,
(29)g(a¯,b¯)-g(u,b¯)+ε∉-intS,∀u∈A,
which implies g(a¯,b¯)∈Maxεg(A,b¯). Thus, (a¯,b¯)∈A×B is a weak ε-S-saddle point of g on A×B. This completes the proof.
Theorem 12.
Let A and B be nonempty sets and ε∈intS. Suppose that g is a vector-valued mapping and the following conditions are satisfied:
clA and clB are compact sets;
g is S-concavelike-convexlike on clA×clB;
g is S-upper semicontinuous on clA×clB;
g is S-lower semicontinuous on clA×clB.
Then, g has a weak ε-S-saddle point on A×B.
Proof.
Let A×B=E and f:clE×clE→V be a vector-valued mappings by
(30)f(x,y)=g(a,v)-g(u,b),∀x=(a,b)∈clE,y=(u,v)∈clE.
Next, we show that all assumptions of Theorem 8 are satisfied by g.
Clearly, by the condition (i), clE is compact. Then, by the condition (ii), we have the fact that, for each a1,a2∈clA and l∈[0,1], there exists a3∈clA such that
(31)g(a3,b)∈lg(a1,b)+(1-l)g(a2,b)+S,∀b∈clB
and, for each b1,b2∈clB and l∈[0,1], there exists b3∈clB(32)g(a,b3)∈lg(a,b1)+(1-l)g(a,b2)-S,∀a∈clA.
By (31) and (32), for each (a1,b1),(a2,b2)∈clE and l∈[0,1], there exists (a3,b3)∈clE such that
(33)g(a3,b)-g(a,b3)∈l(g(a1,b)-g(a,b1))(g(a1,b)-g(a,b1))+(1-l)(g(a2,b)-g(a,b2))+S,(g(a1,b)-g(a,b1))(g(ia1,b)-g(,b1))∀(a,b)∈clE,g(a,b3)-g(a3,b)∈l(g(a,b1)-g(a1,b))(g(a,b1)-g(a1,b))+(1-l)(g(a,b2)-g(a2,b))-S,(g(a,b1)-g(a1,b))(g(,ib1)-g(a1,b))∀(a,b)∈clE.
Namely, f is S-concavelike-convexlike on clE×clE.
Now, we show that f is S-lower semicontinuous on clE×clE. By the condition (iii), for each (a,v)∈clA×clB and s∈intS, there exists an open neighborhood Ua of a and Uv of v such that
(34)g(ua,uv)∈g(a,v)-s2+intS,∀ua∈Ua,uv∈Uv,
and, for each (u,b)∈clA×clB and s∈intS, there exists an open neighborhood Uu of u and Ub of b such that
(35)g(uu,ub)∈g(u,b)+s2-intS,∀uu∈Uu,ub∈Ub.
By (34) and (35), we have the fact that, for any ((a,b),(u,v))∈clE×clE,
(36)g(ua,uv)-g(uu,ub)∈g(a,v)-g(u,b)-s+intS,∀((ua,ub),(uu,uv))∈Ua×Ub×Uu×Uv.
Namely, f is S-lower semicontinuous on clE×clE. Therefore, by Lemma 11, g has a weak ε-S-saddle point on A×B. This completes the proof.
Remark 13.
The conditions (iii) and (iv) of Theorem 12 do not imply that g is continuous (see [23]).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper. This paper is dedicated to Professor Miodrag Mateljevi’c on the occasion of his 65th birthday.
BigiG.CapătăA.KassayG.Existence results for strong vector equilibrium problems and their applications201261556758310.1080/02331934.2010.528761MR2903505ZBL1244.49002GongX.-H.The strong minimax theorem and strong saddle points of vector-valued functions20086882228224110.1016/j.na.2007.01.056MR2398645ZBL1213.49007GongX.Strong vector equilibrium problems200636333934910.1007/s10898-006-9012-5MR2263170ZBL1120.90055LiX. B.LiS. J.FangZ. M.A minimax theorem for vector-valued functions in lexicographic order20107341101110810.1016/j.na.2010.04.047MR2653777ZBL1204.49021NieuwenhuisJ. W.Some minimax theorems in vector-valued functions198340346347510.1007/BF00933511MR715825ZBL0494.90073TanakaT.A characterization of generalized saddle points for vector-valued functions via scalarization199012209227MR1090782ZBL0956.90507TanakaT.Generalized semicontinuity and existence theorems for cone saddle points199736331332210.1007/s002459900065MR1457873ZBL0894.90132ZhangY.LiS. J.ZhuS. K.Minimax problems for set-valued mappings201233223925310.1080/01630563.2011.610915MR2876780ZBL1237.49010ZhangQ.-b.LiuM.-j.ChengC.-z.Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces2009711-221221810.1016/j.na.2008.10.040MR2518027ZBL1163.49020AnsariQ. H.KonnovI. V.YaoJ. C.Existence of a solution and variational principles for vector equilibrium problems2001110348149210.1023/A:1017581009670MR1854012ZBL0988.49004AnsariQ. H.KonnovI. V.YaoJ. C.Characterizations of solutions for vector equilibrium problems2002113343544710.1023/A:1015366419163MR1904233ZBL1012.90055AnsariQ. H.YangX. Q.YaoJ.-C.Existence and duality of implicit vector variational problems2001227-881582910.1081/NFA-100108310MR1871862ZBL1039.49003ChenJ. W.WanZ.ChoY. J.The existence of solutions and well-posedness for bilevel mixed equilibrium problems in Banach spaces201317272574810.11650/tjm.17.2013.2337MR3044531ZBL06249080ChoY. J.ChangS. S.JungJ. S.KangS. M.WuX.Minimax theorems in probabilistic metric spaces199551110311910.1017/S0004972700013939MR1313117ZBL0833.49007ChoY. J.DelavarM. R.MohammadzadehS. A.RoohiM.Coincidence theorems and minimax inequalities in abstract convex spaces20112011article 12610.1186/1029-242X-2011-126FangY.-P.HuangN.-J.Vector equilibrium type problems with (S)+-conditions200453326927910.1080/02331930410001712652MR2072921KienB. T.WongN. C.YaoJ.-C.Generalized vector variational inequalities with star-pseudomonotone and discontinuous operators20086892859287110.1016/j.na.2007.02.032MR2397768ZBL05264947LiX. B.LiS. J.Existence of solutions for generalized vector quasi-equilibrium problems201041172810.1007/s11590-009-0142-9MR2565243ZBL1183.49006AnhL. Q.KhanhP. Q.Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems2008291-2244210.1080/01630560701873068MR2387836ZBL1211.90243KimuraK.LiouY. C.Yao:J. C.ZhangJ. L.ZhangW.XiaX. P.LiaoJ. Q.Semicontinuty of the solution mapping of ε-vector equilibrium problem2007Beijing, ChinaScience Press103113LiX. B.LiS. J.Continuity of approximate solution mappings for parametric equilibrium problems201151354154810.1007/s10898-010-9641-6MR2837102ZBL1229.90235JahnJ.2004Berlin, GermanySpringerxiv+465MR2058695LucD. H.1989319Berlin, GermanySpringerviii+173Lecture Notes in Economics and Mathematical SystemsMR1116766ZhangY.LiS. J.Minimax theorems for scalar set-valued mappings with nonconvex domains and applications20135741359137310.1007/s10898-012-9992-2MR3121799ZBL06236763