# Structure of Pareto Solutions of Generalized Polyhedral-Valued Vector Optimization Problems in Banach Spaces

Authors: Qinghai He; Weili Kong
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra or the union of finitely many generalized polyhedra. Dropping the compactness assumption, we establish some results on structure of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set, and Pareto optimal value set of (SVOP) and on connectedness of Pareto solution set and Pareto optimal value set of (SVOP). In particular, we improved and generalize, Arrow, Barankin, and Blackwell’s classical results in Euclidean spaces and Zheng and Yang’s results in general Banach spaces.


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## Body

## 1. Introduction

LetX and Y be Banach spaces, Γ a closed convex subset of X,  C a closed convex cone of Y, b∈Y, and T:X→Y a linear function. Consider the following optimization problem:
(LMOP)C-min{T(x)+b}subjecttox∈Γ.Linear multiobjective optimization problem(LMOP) has been extensively studied and applied to various decision-making problems in economics, management science, and engineering (see [1–9] and references therein). One of the important topics in vector optimization is the study of the structure of Pareto solution sets. In 1953, Arrow et al. [2] studied the structure of the Pareto solution set, weak Pareto solution set, Pareto optimal value set, and weak Pareto optimal value set for a linear vector optimization problem in Euclidean spaces. In particular, let X=ℝm, Y=ℝn, and C=ℝ+n. If Γ is a polyhedron of Rm and the Pareto solution set S of (LMOP) is nonempty, then (i) S and the weak Pareto solution set Sw of (LMOP) are the unions of finitely many polyhedra of Rm; (ii) the Pareto optimal value set V and the weak Pareto optimal value set Vw are the unions of finitely many polyhedra of Rn. This theorem is well known as ABB theorem. Since the family of all piecewise linear functions is much larger than that of all linear functions and there exists a wide class of functions that can be approximated by piecewise linear functions, it is of value to study piecewise linear multiobjective optimization (cf. [10–12]). Recently, In Banach spaces setting, Zheng and Yang [11] generalized ABB theorem to the case when the objective is a piecewise linear single-valued mapping. Note that the graph of a piecewise linear function is the union of finitely many polyhedra. Zheng [13] considered the following vector optimization problem:
(SVOP)C-minF(x)subjecttox∈Γ,
where F:X⇉Y is a multifunction whose graph is the union of finitely many convex polyhedra and Γ is a polyhedron of X. The following results of the structure of (weak) Pareto solution sets were obtained.Theorem Z 1 (see [13]).
LetX and Y be Banach spaces, Γ the union of finitely many polyhedra of  X, and F:Γ⇉Y a multifunction whose graph Gr(F) is the union of finitely many convex polyhedra of X×Y. Suppose that the ordering cone C⊂Y is closed, convex, pointed, and has a weakly compact base. If F(Γ)+C is convex, then the Pareto solution set S and the Pareto optimal value set V of  (SVOP) are the unions of finitely many polyhedra of X and Y, respectively.Theorem Z 2 (see [13]).
LetX and Y be Banach spaces, Γ the union of finitely many polyhedra of X, and F:Γ⇉Y a multifunction whose graph Gr(F) is the union of finitely many convex polyhedra of X×Y. Suppose that the ordering cone C⊂Y is closed, convex, and pointed. If the interior int(C) of C is nonempty, then the weak Pareto solution set Sw and the Pareto optimal value set Vw of (SVOP) are the unions of finitely many polyhedra of X and Y, respectively.A polyhedron is defined [12–15] as the intersection of finitely many closed half-spaces. Polyhedra exist in many contexts such as linear and quadratic programs, game theory, statistical decision theory, and mathematical biology as well. For more details on the theory of polyhedra, we refer the reader to [12, 14] and the references therein. A generalized polyhedron (G-polyhedron or semiclosed polyhedron) in [13, 15] is defined as the intersection of finitely many closed and/or open half-spaces. A G-polyhedron can be regarded as an extension of a polyhedron. It has nice properties analogous to a polyhedron; see [13, 15] and the references therein. It is necessarily noted that the graph Gr(F) of a multifunction F is closed when it is the union of finitely many polyhedra, while the graph Gr(F) is not necessarily closed when it is the union of finitely many G-polyhedra. In the case that F is single valued, the former implies that F is a piecewise continuous linear function but the latter does not imply that F is necessarily a piecewise continuous linear function. G-polyhedra are important in piecewise linear programs as well as polyhedra. For example, Fang et al. [15] proved that (weak) Pareto solution set of a piecewise linear multicriteria program with possible discontinuity is a union of finitely many G-polyhedra.One of our main aims in this work is to investigate the structure of the (weak) Pareto solution set and the (weak) Pareto optimal set of(SVOP) whose graph Gr(F) is the union of finitely many G-polyhedra.Another topic in linear optimization problems is to study the connectedness of (weak) Pareto solution sets. Many authors researched this issue; see [2, 11, 13, 16–20] and the references therein. Arrow et al. [2] proved that the (weak) Pareto solution set (Sw) S and the (weak) Pareto optimal value set (Vw) V of  (LMOP) are pathwise connected, respectively, when X=Rm, Y=Rn, and C=R+n. Recently, Zheng and Yang [11] proved that the weak Pareto set Sw and the weak Pareto optimal value set Vw of  (LMOP) are pathwise connected, respectively, when the ordering cone C has a nonempty interior and the mapping T(·)+b is C-convex. Zheng [13] proved that the Pareto set S and the Pareto optimal value set V of (SVOP) are pathwise connected, respectively, when the ordering cone C is a pointed, closed, convex cone with a weakly compact base and F is a C-convex multifunction whose graph is the union of finitely many convex polyhedra.The other of our main aims is to study the connectedness of the Pareto setS and the Pareto optimal value set V of (SVOP) without the assumption of the ordering cone C having a weakly compact base but with that of the cone C being polyhedral.


## 2. Preliminaries

LetX and Y be Banach spaces and let C⊂Y be a convex cone. We say that C is pointed if C∩-C={0}. In this case, one can define a partial order ≤C in Y as follows: for y1,y2∈Y, y1≤Cy2 if and only if y2-y1∈C. Let int(C) denote the interior of C. When int(C)≠∅, by y1<Cy2, we mean that y2-y1∈int(C). Let Y* denote the dual space of Y and C+ the dual cone of C, defined by
(1)C+∶={c*∈Y*:〈c*,c〉≥0,∀c∈C}.
We denote by C+1 the set of all strictly positive continuous linear functionals; that is,
(2)C+1∶={c*∈Y*:〈c*,c〉>0,∀c∈C∖{0}}.
We say a convex subset Θ of C is a base of C if it satisfies that
(3)C={tθ:t∈ℝ+,θ∈Θ},0∉cl(Θ),
where cl(·) denotes the closure. C is said to have a bounded (resp., weakly compact) base, if it has a base that is bounded (resp., weakly compact). It is known that C+1≠∅ if and only if C has a base.LetA be a subset of A and a in A. As usual, we denote by E(A,C), WE(A,C), and Pos(A,C) the set of all Pareto points of A, the set of all weak Pareto points of A, and the set of all positively proper Pareto points of A, respectively; that is,
(4)a∈E(A,C)⟺(a-C)∩A={a},a∈WE(A,C)⟺(a-int(C))∩A=∅,a∈Pos(A,C)⟺∃c*∈C+1satisfying〈c*,y-a〉≥0∀y∈A.
It is clear that
(5)Pos(A,C)⊂E(A,C)=E(A+C,C)⊂WE(A,C).For a multifunctionF:X⇉Y, we denote by Gr(F) and epiC(F) the graph and C-epigraph of F, respectively; that is,
(6)Gr(F)∶={(x,y):x∈X,y∈F(x)},epiC(F)∶={(x,y):x∈X,y∈F(x)+C}.
We say that F is C-convex, ifepiC(F)is convex. Obviously, F is C-convex if and only if
(7)tF(x1)+(1-t)F(x2)⊂F(tx1+(1-t)x2)+C∀x1,x2∈X,∀t∈[0,1].Recall [14] that a subset P of a Banach space Z is a (convex) polyhedron, if there exist u1*,…,um*∈Z* and s1,…,sm∈ℝ such that
(8)P={z∈Z:〈ui*,z〉≤si,i=1,…,m}.
A subset Q of Z is said to be a G-polyhedron, if Q is a polyhedron, or there exist a polyhedron P of Z and v1*,…,vm*∈Z*, r1,…,rm∈ℝ such that
(9)Q=P∩{z∈Z:〈vi*,z〉<ri,i=1,…,m};
see [11, 13, 15]. Clearly, each polyhedron is closed but a G-polyhedron is not necessarily closed. From the definitions, it is easy to verify that if Q1 and Q2 are G-polyhedra of Banach space Z, Q1∩Q2 is a G-polyhedron of Z, and Q1∖Q2 is the union of finitely many G-polyhedra of Z.In this paper, we will consider set-valued vector optimization problem(SVOP). In the remainder of the paper, we always assume that the graph Gr(F) of the objective mapping F is the union of finitely many G-polyhedra or polyhedra of X×Y.We say thatx-∈Γ is a Pareto (resp., weak Pareto or positively proper Pareto) solution of (SVOP), if there exists y-∈F(x-) such that y-∈E(F(Γ),C) (resp., y-∈WE(F(Γ),C) or y-∈Pos(F(Γ),C)); in this case, we say that y- is a Pareto (resp., weak Pareto or positively proper Pareto) optimal value of (SVOP). Let S, Sw, and Sp, respectively, denote the sets of all Pareto, weak Pareto and positively proper Pareto solutions of (SVOP). Let V, Vw, and Vp, respectively, denote the sets of all Pareto, weak Pareto, and positively proper Pareto optimal values of (SVOP). Obviously,
(10)Sp⊂S⊂Sw,Vp⊂V⊂Vw,Sp=P∩F-1(Vp),S=Γ∩F-1(V),Sw=Γ∩F-1(Vw).Next, we provide some properties on the union of finitely many G-polyhedra in general Banach spaces, which generalize the corresponding results in [13].Lemma 1.
A subsetP of X×Y is the union of finitely many G-polyhedra (resp., polyhedra) if and only if there exist closed subspaces X1 and X2 of X, closed subspaces Y1 and Y2 of Y, and the union of finitely many G-polyhedra (resp., polyhedron) P2 of X2×Y2 such that
(11)X×Y=X1×Y1+X2×Y2,(X1×Y1)∩(X2×Y2)={(0,0)},max{dim(X2),dim(Y2)}<+∞,P=X1×Y1+P2.The proof of Lemma1 is similar to that for polyhedron or G-polyhedron in [13] and is omitted.IfY={0} in Lemma 1, we have the following corollary.Corollary 2.
A subsetP of X is the union of finitely many G-polyhedra (resp., polyhedra) if and only if there exist closed subspaces X1 and X2 of X and the union of finitely many G-polyhedra (resp., polyhedra) P2 of X2 such that
(12)X=X1+X2,X1∩X2={0},dim(X2)<+∞,P=X1+P2.For a subsetA of X×Y, let
(13)AX∶={x∈X:∃y∈Ys.t.(x,y)∈A},AY∶={y∈Y:∃x∈Xs.t.(x,y)∈A}.Lemma 3 (see [13]).
LetP be a G-polyhedron (resp., polyhedron) of X×Y. Then PX is the union of finitely many G-polyhedra (resp., polyhedra) of X.Corollary 4.
LetG:X⇉Y be a multifunction whose graph is a convex G-polyhedron (resp., polyhedra) of X×Y. Let P and Q be convex G-polyhedra (resp., polyhedra) of X and Y, respectively. Then G(P) and G-1(Q) are the union of finitely many G-polyhedra (resp., polyhedra) of Y and X, respectively.Noting that(14)G(P)=(Gr(G)∩(P×Y))Y,G-1(Q)=(Gr(G)∩(X×Q))X,
Corollary 4 is a consequence of Lemma 3.Lemma 5.
LetA be a subset of Y with a closed convex cone C. Then
(15)E(A,C)=A∖(A+C∖{0}).Proof.
Leta∈A. Then, one can easily verify that
(16)(a-C∖{0})∩A=∅⟺a∈A∖(A+C∖{0}).
Noting that
(17)a∈E(A,C)⟺(a-C∖{0})∩A=∅,
it follows that (15) holds. The proof is completed.Lemma 6.
LetA be a subset of Banach space Y with the ordering cone C. Then,
(18)WE(A,C)=A∩WE(clA,C).Proof.
It is obvious thatA∩WE(clA,C)⊆WE(A,C). We only need to show that WE(A,C)⊆A∩WE(clA,C). Let x∈WE(A,C). Then (x-int(C))∩A=∅; that is
(19)A⊂Y∖(x-int(C)).
Since x-int(C) is open, one has cl(A)⊂Y∖(x-int(C)) and hence,
(20)(x-int(C))∩cl(A)=∅.
Thus x∈WE(clA,C). The proof is completed.We will also use the following lemma [13, Proposition 2.1] in the sequel.Lemma 7.
LetY1 and Y2 be closed subspaces of a Banach space Y such that
(21)Y=Y1+Y2,Y1∩Y2={0}.
Let A∶=Y1+A2 for some subset A2 of Y2. Let C2 be the projection of the ordering cone C on Y2; that is,
(22)C2={c2∈Y2:y1+c2∈Cforsomey1∈Y1}.
Then, the following assertions hold.(i)
Y1+Pos(A2,C2)⊂Pos(A,C) if Y1∩C={0}.(ii)
E(A,C)=∅ if Y1∩C≠{0}.(iii)
E(A,C)=Y1+E(A2,C2) if Y1∩C={0}.(iv)
WE(A,C)=∅ if Y1∩int(C)≠∅.(v)
WE(A,C)=Y1+WE(A2,C2) if int(C)≠∅ and Y1∩int(C)=∅.Proof.
For the proofs of (ii)–(v), see [13, Proposition 2.1]. We only need to show (i).
For (i), we assume thatY1∩C={0}. Let y1∈Y1 and y2∈Pos(A2,C2). Then there exists c~*∈C2+1 such that
(23)〈c~*,a2-y2〉≥0∀a2∈A2.
Define c* by
(24)〈c*,x1+x2〉∶=〈c~*,x2〉,∀(x1,x2)∈Y1×Y2.
Then c* is well defined and it is easy to verify that c*∈C+1. On the other hand, (23) and (24) imply that
(25)〈c*,x1+a2-(y1+y2)〉≥0∀x1∈Y1,∀a2∈A2.
Hence y1+y2∈Pos(A,C).Remark 8.
In view of Lemma7, it is practical to investigate some topics on (SVOP) in a finite dimensional framework. This is our main ideal to consider (SVOP) in general Banach spaces.


## 3. The Structure of Solution Sets and Optimal Value Sets

In this section, our aim is to investigate the structure of the (weak) Pareto solution set and the (weak) Pareto optimal set of(SVOP) whose graph Gr(F) is the union of finitely many G-polyhedra. Throughout the remainder of this paper, we assume that Sw, S, and Sp are nonempty.LetY be a Banach space. For y*∈Y* and A⊂Y, let
(26)λy*(A)∶=infa∈A〈y*,a〉,Ly*(A)∶={a∈A:〈y*,a〉=infy∈A〈y*,y〉}.
It is easy to verify that
(27)Pos(A,C)=⋃y*∈C+1‍Ly*(A).
In addition, if A+C is convex and C has a nonempty interior, by [21, Corollary 5.29], one has
(28)WE(A,C)=⋃y*∈C+∖{0}‍Ly*(A).The following lemma is also useful in our later analysis and can be found in [13].Lemma 9.
LetY be a Banach space with the ordering cone C which has a weakly compact base. Let A be the union of finitely many polyhedra of Y. Suppose that A+C is convex and that E(A,C) is nonempty. Then E(A,C) is the union of finitely many convex polyhedra. More precisely, there exist y1*,…,yp*∈C+1 such that
(29)E(A,C)=Pos(A,C)=⋃i=1p‍Lyi*(A).We have the following proposition.Proposition 10.
LetY be a Banach space with the ordering cone C and let A∶=⋃i=1kAi,Ai⊂Y. Set
(30)Ei∶=E(Ai,C)∖⋃i≤j≤k,j≠i‍[(Aj+C)∖(E(Ai,C)∩E(Aj,C))].
Then, E(A,C)=⋃i=1kEi.Proof.
Letx∈E(A,C). Then, there exists an integer i0∈[1-k] such that x∈E(Ai0,C). For each j with x∈Aj+C, noting that
(31)(Aj+C)∩E(A,C)=(Aj+C)∩E(A+C,C)⊂E(Aj+C,C)=E(Aj,C),
one has x∈E(Aj,C) and x∈E(Ai0,C)∩E(Aj,C). It follows that
(32)x∉(Aj+C)(E(Ai0,C)∩E(Aj,C))foreachintegerj∈[1-k].
Therefore, one has x∈Ei0 which implies that E(A,C)⊂⋃i=1kEi.
Conversely, letx∈⋃i=1kEi. Then, x∈Ei for some integer i in [1-k]. Let z∈A∩(x-C). We only need to show that z=x. By the definition of A, there exists an integer j in [1-k] such that z∈Aj. Noting that Ei⊂E(Ai,C), one has z=x if j=i. Now suppose that j≠i. Then, by the definition of Ei, it follows that x∉(Aj+C)∖(E(Ai,C)∩E(Aj,C)). Since x∈z+C⊂Aj+C, one has x∈E(Aj,C). It follows that x=z. The proof is completed.Lemma 11.
LetY be a Banach space, let A be the union of finitely many G-polyhedra of Y, and let Λ be a subset of Y*. Then, ⋃y*∈ΛLy*(A) is the union of finitely many G-polyhedra of Y and, more precisely, there exists a finite subset Λ0 of Λ such that
(33)⋃y*∈Λ‍Ly*(A)=⋃y*∈Λ0‍Ly*(A).Proof.
By the assumptions, there exist finitely many G-polyhedraP1,…,Pn such that A=⋃i=1nPi. Then for each i∈[1-n], let
(34)Λi∶={y*∈Λ:Pi∩Ly*(A)≠∅}.
Then
(35)⋃y*∈Λ‍Ly*(A)=⋃i=1n‍(⋃y*∈Λi‍Pi∩Ly*(A)).
Let y*∈Λi. Since λy*(A)=λy*(cl(A)) and cl(Pi) is a polyhedron, (26) implies that cl(Pi)∩Ly*(A) is a face of cl(Pi). Noting that every polyhedron has finitely many faces. Hence, there exists a finite subset Λ~i of Λi, such that
(36){cl(Pi)∩Ly*(A):y*∈Λi}={cl(Pi)∩Ly*(A):y*∈Λ~i}.
Now, we show that for any y1*,y2*∈Λi,
(37)cl(Pi)∩Ly1*(A)=cl(Pi)∩Ly2*(A)impliesPi∩Ly1*(A)=Pi∩Ly2*(A).
If (37) holds, by (36), we have
(38)⋃y*∈Λi‍Pi∩Ly*(A)=⋃y*∈Λ~i‍Pi∩Ly*(A).
From (35), it follows that
(39)⋃y*∈Λ‍Ly*(A)=⋃i=1n‍(⋃y*∈Λ~i‍Pi∩Ly*(A)).
Let Λ0∶=⋃i=1nΛ~i. Noting that Pi∩Ly*(A)⊂Ly*(A), we have that (33) holds. Hence, we only need to show (37). Suppose there would exist y1*,y2*∈Λi such that cl(Pi)∩Ly1*(A)=cl(Pi)∩Ly2*(A), but Pi∩Ly1*(A)≠Pi∩Ly2*(A). Let y∈Pi∩Ly1*(A) with y∉Pi∩Ly2*(A). By (26) and the definition of Λi, we have 〈y2*,y〉>λy*(A), which implies that y∉cl(Pi)∩λy2*(A)=cl(Pi)∩λy1*(A). This is a contradiction. The proof is completed.Theorem 12.
LetX and Y be Banach spaces, Γ a G-polyhedron, and Gr(F) the union of finitely many G-polyhedra of X×Y and let the ordering cone C have nonempty interior. Suppose that F(Γ)+C is convex. Then, Sw and Vw are the union of finitely many G-polyhedra of X and Y, respectively. More precisely, there exist yi*∈C+ with ∥yi*∥=1(i∈[i-q] for some integer q) such that
(40)Vw=⋃y*∈C+∖{0}‍Ly*(F(Γ))=⋃i=1q‍Lyi*(F(Γ)).Proof.
SinceF(Γ)+C is convex, by (28), we have
(41)WE(F(Γ)+C,C)=⋃y*∈C+∖{0}‍Ly*(F(Γ)+C).
Noting that WE(F(Γ),C)=F(Γ)∩WE(F(Γ)+C,C), we have
(42)WE(F(Γ),C)=⋃y*∈C+∖{0}‍F(Γ)∩Ly*(F(Γ)+C)=⋃y*∈C+∖{0}‍Ly*(F(Γ)).
Since the feasible set Γ is a G-polyhedron of X and Gr(F) is the union of finitely many G-polyhedra of X×Y, Corollary 4 implies that F(Γ) is the union of finitely many G-polyhedra of Y. Since Vw=WE(F(Γ),C), it follows from (26) and Lemma 11 that Vw is the union of finitely many G-polyhedra of Y and there exist yi*∈C+ with ∥yi*∥=1 (i∈[i-q] for some integer q) such that (40) holds. On the other hand, from Corollary 4, one can easily see that Sw=Γ∩F-1(Vw) is the union of finitely many G-polyhedra of X. The proof is completed.By Lemma3, dropping convexity of F(Γ)+C but requiring that the ordering cone C is polyhedral, we generalize Theorem  3.3 in [13] to the union of finitely many G-polyhedra setting.Theorem 13.
LetX and Y be Banach spaces. Let the ordering cone C be polyhedral and have nonempty interior. Suppose that Γ and Gr(F) are the unions of finitely many G-polyhedra of X and X×Y, respectively. Then, Sw and Vw are the unions of finitely many G-polyhedra of X and Y, respectively.Proof.
Suppose thatGr(F) is the union of finitely many G-polyhedra of X×Y. Since F(Γ)=(Gr(F)∩(Γ×Y))Y, by Lemma 3, one has that cl(F(Γ)) is the union of finitely many polyhedra of Y. It is similar to the proof of Theorem  3.3 in [13]; we can show that WE(cl(F(Γ))) is the union of finitely many polyhedra of Y. By Lemma 6, one has
(43)Vw=WE(F(Γ),C)=F(Γ)∩WE(cl(F(Γ)),C)
which implies that Vw is the union of finitely many G-polyhedra of Y. Noting that Sw=(Gr(F)∩(Γ)×Vw)X, Lemma 3 implies that Sw is the union of finitely many G-polyhedra of X. The proof is completed.Dropping the assumption of the ordering coneC having a weakly compact base but inquiring C being a polyhedral cone, we have the following result analogous to that of Lemma 11.Lemma 14.
LetY be a Banach space, let the ordering cone C be pointed and polyhedral, and let A be the union of finitely many polyhedra of Y. Suppose that A+C is convex and that E(A,C) is nonempty. Then E(A,C) is the union of finitely many polyhedra of Y. More precisely, there exist c1*,…,cp*∈C+1 such that
(44)E(A,C)=Pos(A,C)=⋃i=1p‍Lci*(A),Lci*(clcoA)=Lci*(A).Proof.
By Corollary2 there exist a closed subspace Y1, a finite dimensional subspace Y2 of Y, and the union A2 of finitely many polyhedra of Y2 such that
(45)Y=Y1+Y2,Y1∩Y2={0},(46)A=Y1+A2.
Let C2 be the projection of the ordering cone C on Y2; that is,
(47)C2={c2∈Y2:c1+c2∈Cforsomec1∈Y1}.
Since C is polyhedral, together with Corollary 4, we have that the cone C2 is the union of finitely many polyhedra of Y2. Hence it is also closed. From the assumption of E(A,C)≠∅, by Lemma 7(ii), we have Y1∩C={0}. From this, with Lemma 7(iii), we have E(A2,C2)≠∅. By Lemma 7(i), we have
(48)Pos(A,C)⊃Y1+Pos(A2,C2).
We claim that Pos(A2,C2) is the union of finitely many polyhedra of Y2. For any (x,c)∈A×C, there exist (x1,x2)∈Y1×A2 and (c1,c2)∈Y1×C2 such that
(49)x=x1+x2,c=c1+c2.
Since Y1 is a subspace of Y, we have
(50)x+c=(x1+c1)+x2+c2∈Y1+A2+C2.
Hence we have
(51)A+C⊂Y1+A2+C2.
Conversely, for any (x1,a2,c2)∈Y1×A2×C2, there exists c1∈Y1 such that c1+c2∈C. Hence we have
(52)y1+a2+c2=(y1-c1)+a2+(c1+c2)∈Y1+A2+C=A+C.
Thus we have shown that
(53)A+C=Y1+A2+C2.
Let x2,y2∈A2+C2. Then there exist x1,y1∈Y1 such that
(54)x1+x2,y1+y2∈A+C.
Since A+C is convex, by (53), one has
(55)λ(x1+x2)+(1-λ)(y1+y2)∈Y1+(A2+C2)∀λ∈[0,1].
Noting that
(56)λx1+(1-λ)y1∈Y1,λx2+(1-λ)y2∈Y2∀λ∈[0,1],
from (45) and (55), one has
(57)λx2+(1-λ)y2∈A2+C2∀λ∈[0,1].
Hence A2+C2 is convex.
Letx2∈C2∩-C2. Then there exist x1′,x1′′∈Y1 such that x1′+x2,x1′′-x2∈C. Since C is a convex cone, it follows that
(58)x1′+x1′′=(x1′+x2)+(x1′′-x2)∈C.
Hence
(59)x1′+x1′′∈Y1∩C={0}.
We have
(60)x1′′=-x1′,x1′+x2∈C∩-C={0}.
Together with (46), we have x2=0 and C2∩-C2={0}.
We have shown thatC2 is a closed, convex, and pointed cone of Y2. Noting that Y2 is finite dimensional, it follows that C2 has a compact base. Applying Lemma 9, it follows that there exist c~1*,…,c~p* in C2+1 such that
(61)Pos(A2,C2)=E(A2,C2)=⋃i=1p‍Lc~i*(A2).
Thus Pos(A2,C2) is the union of finitely many polyhedra of Y2. By Lemma 7((ii), (iii)), (48), and (61), one has that
(62)Pos(A,C)⊃Y1+Pos(A2,C2)=Y1+E(A2,C2)=E(A,C).
Hence,
(63)E(A,C)=Pos(A,C)=Y1+Pos(A2,C2).
From (45), (46), and (63), applying Corollary 2, it follows that Pos(A,C) is the union of finitely many polyhedra of Y.
For eachc~i*, i∈[1-p], we define c* by
(64)〈ci*,y1+y2〉∶=〈c~i*,y2〉,∀(y1,y2)∈Y1×Y2.
Then, one can easily show that ci*∈C+1 and
(65)Lci*(A)=Y1+Lc~i*(A2).
In fact, for any y∈C∖{0}, there exists a pair (y1,y2)∈Y1×C2 such that y=y1+y2. If y2=0, then y=y1∈Y1 and thus y∈Y1∩C={0}, a contradiction. Hence y2≠0. Then we have
(66)〈ci*,y〉=〈ci*,y1+y2〉=〈c~i*,y2〉>0.
It follows that ci*∈C+1. For (65), let a∈Lci*(A). By (26), (46), and (64), there exists (a1,a2)∈Y1×A2 such that a=a1+a2 and
(67)〈c~i*,a2〉=〈ci*,a1+a2〉=infy1∈Y1,y2∈A2〈ci*,y1+y2〉=infy2∈A2〈c~i*,y2〉.
We have a2∈Lc~i*(A2) and a∈Y1+Lc~i*(A2). The conclusion, Y1+Lc~i*(A2)⊂A holds obviously. Thus (65) holds. From (48) and (63)–(65), we have
(68)E(A,C)=Pos(A,C)=Y1+⋃i=1p‍Lc~i*(A2)=⋃i=1p‍(Y1+Lc~i*(A2))=⋃i=1p‍Lci*(A).
Finally, we show that Lci*(clcoA)=Lci*(A). Indeed, for any y∈Y, (45) implies that there exists unique pair (y1,y2)∈Y1×Y1, such that y=y1+y2. Since Y2 is finitely dimensional, one can take a constant M>0 which is independent on y satisfying
(69)∥y1∥+∥y2∥≤M∥y∥.
From (46), we have
(70)coA=Y1+coA2⊂Y1+clcoA2.
It follows that
(71)y1+clcoA2⊂clcoA∀y1∈Y1,
and thus
(72)Y1+clcoA2⊂clcoA.
On the other hand, for each sequenc {(y1,n,a2,n)}⊂Y1×clcoA2 with yn∶=y1,n+a2,n converging to some y∈Y, there exists a pair (y1,y2)∈Y1×Y2 such that y=y1+y2. Together with (69), we have
(73)∥y1,n-y1∥+∥y2,n-y2∥≤M∥yn-y∥,
and hence {y1,n} and {y2,n} converge to y1∈Y1 (since Y1 is closed) and y2∈clcoA2, respectively. Thus we have y∈Y1+clcoA2 and hence Y1+clcoA2 is closed. Together with (70), we have
(74)clcoA⊂Y1+clcoA2.
From (72) and (74), we have
(75)clcoA=Y1+clcoA2,
and together with (65), one has
(76)Lci*(clcoA)=Y1+Lc~i*(clcoA2).
By Theorem 3.2(i) in [13], it follows that
(77)Lc~i*(clcoA2)=Lc~i*(A2).
This and (65) imply that
(78)Lci*(clcoA)=Y1+Lc~i*(clcoA2)=Y1+Lc~i*(A2)=Lci*(A).
The proof is completed.Theorem 15.
LetX and Y be Banach spaces and let the ordering cone C be pointed and polyhedral. Suppose that Γ and Gr(F) are the unions of finitely many polyhedra of X and X×Y, respectively, and that F(Γ)+C is convex. Then, S and V are the union of finitely many convex polyhedra of X and Y, respectively, and more precisely, there exist ci*∈C+1(i∈[1-q] for some integer q) such that
(79)V=Vp=⋃i=1q‍Lci*(F(Γ)).
Consequently, S=Sp and V are the unions of finitely many polyhedra.Proof.
Suppose thatΓ and Gr(F) are the unions of finitely many polyhedra of X and X×Y, respectively. Noting that
(80)F(Γ)=(Gr(F)∩(Γ×Y))Y,
and by Lemma 3, one has that F(Γ) is the union of finitely many polyhedra of Y. From this and Lemma 14, it follows that there exist ci*∈C+1 (i∈[1-q] for some integer q) such that (79) holds. Hence, V=Vp is the union of finitely many polyhedra of Y. This and Corollary 4 imply that S=Sp=Γ∩F-1(V) is the union of finitely many polyhedra of X. The proof is completed.Without the convexity ofA+C in Lemma 14, we have the following lemma which will also be applied to consider (SVOP).Lemma 16.
LetY be a Banach space, let the ordering cone C be pointed and polyhedral, and let A be the union of finitely many G-polyhedra of Y. Suppose that E(A,C) is nonempty. Then E(A,C) is the union of finitely many G-polyhedra of Y.Proof.
By Corollary2 there exist a closed subspace Y1, a finite dimensional subspace Y2 of Y, and the union A2 of finitely many G-polyhedra of Y2, such that (45) and (46) hold.
LetC be the projection of C on Y2. We first show that A2+C2 is the union of finitely many G-polyhedra of Y2. Indeed, since C is a polyhedron on Y1×Y2, Lemma 3 implies that C2 is the union of finitely many polyhedra of Y2. We can assume
(81)A2=⋃i=1p‍A2,i,C2=⋃j=1q‍C2,j,
where each A2,i is a G-polyhedron of Y2 and each C2,j is a polyhedron of Y2. It follows that
(82)A2+C2=⋃i=1p‍⋃j=1q‍(A2,i+C2,j).
Noting that Y2 is finitely dimensional, by Proposition  2.1 in [15], we have that each A2,i+C2,j is a G-polyhedron of Y2. This with (82) implies that A2+C2 is the union of finitely many G-polyhedra of Y2. Together with (45), (46), and (53), applying Corollary 2, it follows that A+C is the union of finitely many G-polyhedra of Y. Let A=⋃i=1k‍Ai, where each Ai is a G-polyhedron of Y. Noting that {0} is a face of C, by the above proof, applying Proposition  2.1 in [15] again, we have that each Ai+C∖{0} as well as Ai+C is the union of finitely many G-polyhedra of Y. So is the complimentary (Ai+C∖{0})c. By Lemma 5, we have
(83)(Ai,C)=Ai∖(Ai+C∖{0})=Ai∩(Ai+C∖{0})c,
which implies that (Ai,C) is the union of finitely many G-polyhedra of Y. So is the set
(84)⋃i≤j≤k,j≠i‍[(Aj+C)∖(E(Ai,C)∩E(Aj,C))].
Let
(85)Ei∶=E(Ai,C)∖⋃i≤j≤k,j≠i‍[(Aj+C)∖(E(Ai,C)∩E(Aj,C))].
Then Ei is also the union of finitely many G-polyhedra of Y. Applying Proposition 10, we have
(86)E(A,C)=⋃i=1k‍Ei.
Thus E(A,C) is the union of finitely many G-polyhedra of Y. The proof is completed.Theorem 17.
LetX and Y be Banach spaces and let the ordering cone C be pointed and polyhedral. Suppose that Γ and Gr(F) are the unions of finitely many G-polyhedra of X and X×Y, respectively. Then, S and V are the union of finitely many convex G-polyhedra of X and Y, respectively.Proof.
Suppose thatΓ and Gr(F) are the unions of finitely many G-polyhedra of X and X×Y, respectively. Noting that
(87)F(Γ)=(Gr(F)∩(Γ×Y))Y,
and by Lemma 3, one has that F(Γ) is the union of finitely many polyhedra of Y. From this and Lemma 16, we have that V is the union of finitely many G-polyhedra of Y. This and Corollary 4 imply that S=Γ∩F-1(V) is the union of finitely many G-polyhedra of X too. The proof is completed.


## 4. Connectedness of Pareto Solution Sets and Pareto Optimal Value Sets

For various applications, it is of special interest to move continuously from an optimal solution to another along optimal alternatives. In order to do this movement, one needs that the optimal solution set is pathwise connected or at least connected. Many authors studied connectedness of optimal solution sets in multiobjective optimization under some restrictive assumptions (cf. [2, 16–20, 22, 23]).In this section, dropping the assumption that the ordering coneC has a weakly compact base but requiring that C is pointed and polyhedral, under the C-convexity assumption on the set-valued objective mapping F, we will establish connectedness of Pareto solution set S and Pareto optimal value set V of (SVOP).LetF′:X⇉Y be such that
(88)Gr(F′)=clco(Gr(F)∩(Γ×Y)).
Consider the following vector optimization problem:
(SVOP′)C-minF′(x)subjecttox∈Γ.LetS′ and V′ denote the set of all Pareto solutions of (SVOP′) and the set of all Pareto optimal values of (SVOP′), respectively.Lemma 18.
LetΓ be a polyhedron of X, let C be a closed, convex, pointed, and polyhedral cone of Y, and let GrF(Γ) be the union of finitely many polyhedra of X×Y. Suppose that F is C-convex. Then S′=S and V′=V.Proof.
By Corollary4, F(Γ) is the union of finitely many polyhedra of Y. From Corollary 2, there exist a closed subspace Y1, a finite dimensional subspace of Y, and the union of finitely many polyhedra F(Γ)2 of Y2 such that
(89)Y=Y1+Y2,Y1∩Y2={0},F(Γ)=Y1+F(Γ)2.
By (75) in the proof of Lemma 14, we have
(90)clcoF(Γ)=Y1+clcoF(Γ)2.
Since Γ is a polyhedron of X, by Lemma  2.1 in [13] and Theorem  19.6 in [14], Gr(F′) is a polyhedron of X×Y. By Lemma 3, F′(Γ) is the union of finitely many polyhedra of X. Hence it is closed. On the other hand, From the convexity of Gr(F′), we have that F′(Γ) is convex. Noting that F(Γ)⊂F′(Γ)⊂clcoF(Γ), it follows that F′(Γ)=clcoF(Γ). Thus by (90), one has
(91)F′(Γ)=Y1+clcoF(Γ)2.
Let C2 be the projection of C on Y2. Analogously to the proof of Lemma 14, one can show
(92)V′=E(F′(Γ),C)=Pos(F′(Γ),C)=Y1+Pos(clcoF(Γ)2,C2),V=E(F(Γ),C)=Pos(F(Γ),C)=Y1+Pos(F(Γ)2,C2).
In order to show V′=V, we only need to show
(93)Pos(clcoF(Γ)2,C2)=Pos(F(Γ)2,C2).
Indeed, since F is C-convex, it follows that F′(Γ)+C and F(Γ)+C are convex. It is the same to prove that A2+C2 is convex in the proof of Lemma 14; we have that both F′(Γ)2+C2 and F(Γ)2+C2 are also convex. Noting that Y2 is finite dimensional, by Theorem 3.2(i) in [13], one has
(94)Lc~*(clcoF(Γ))=Lc~*(F(Γ))∀c~*∈C2+1.
On the other hand, (27) implies that
(95)Pos(clcoF(Γ)2,C2)=⋃c~*∈C2+1‍Lc~*(clcoF(Γ)2),(96)Pos(F(Γ)2,C2)=⋃c~*∈C2+1‍Lc~*(F(Γ)2).
From (94)–(96), it follows that (93) holds. Hence, we have shown V′=V. It remains to show S′=S. Noting that S′=Γ∩F′-1(V′), S=Γ∩F-1(V), the following assertion in [13, see the proof of Lemma 4.1]
(97)Γ∩F-1(Lc*(F(Γ)))=Γ∩F′-1(Lc*(F′(Γ)))∀c*∈C+,V′=⋃c*∈C+1‍Lc*(F′(Γ)),V=⋃c*∈C+1‍Lc*(F(Γ)),
it follows that S′=S. The proof is completed.For our main result, we need the following lemma; see [13, Lemma 4.2].Lemma 19.
LetY be a Banach space and A a polyhedron of Y. Let Λ be a convex subset of Y*. Then ⋃y*∈ΛLy*(A) is pathwise connected.Theorem 20.
LetX and Y be Banach spaces, Γ a polyhedron of X,C a closed, convex, pointed, and polyhedral cone of Y, and GrF(Γ) the union of finitely many polyhedra of X×Y. Suppose that the set-valued objective function F is C-convex. Then, the Pareto solution set S and the Pareto optimal value set V of (SVOP) are pathwise connected.Proof.
LetF′:X⇉Y be defined by (88). Then, Gr(F′) is a polyhedron of X×Y. By (27), one has V′=⋂y*∈C+1Ly*(F′(Γ)). It follows that
(98)Gr(F′)∩(Γ×V′)=⋃y*∈C+1‍L(0,y*)(Gr(F′)∩(Γ×Y)).
This and Lemma 19 imply that Gr(F′)∩(Γ×V′) is pathwise connected. Hence S′=(Gr(F′)∩(Γ×V′))X and V′=(Gr(F′)∩(Γ×V′))Y are pathwise connected. By Lemma 18, one sees that S and V are pathwise connected.Corollary 21.
LetY be a Banach space with the pointed ordering C being closed and polyhedral and let A be the union of finitely many convex polyhedra of Y. Suppose that A+C is convex. Then, E(A,C) is pathwise connected.Remark 22.
In [13, Theorem 4.1 and Corollary 4.1], under the assumptions that the ordering cone C has a weakly compact base and the Banach space Y is finite dimensional, respectively, the corresponding results of connectedness of (SVOP) were established. In our results, Theorem 20 and Corollary 21, we drop these two assumptions, respectively.



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