Abstract and Applied Analysis

Volume 2013 (2013), Article ID 619206, 10 pages

http://dx.doi.org/10.1155/2013/619206

Research Article

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

Qinghai He¹ and Weili Kong²

¹Department of Mathematics, Yunnan University, Kunming 650091, China

²Teacher Education Institute, Qujing Normal University, Qujing 655011, China

Received 3 July 2013; Accepted 6 November 2013

Academic Editor: Marek Wisla

Copyright © 2013 Qinghai He and Weili Kong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let and be Banach spaces, a closed convex subset of , a closed convex cone of , , and a linear function. Consider the following optimization problem:

Linear multiobjective optimization problem 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 , , and . If is a polyhedron of and the Pareto solution set of is nonempty, then (i) and the weak Pareto solution set of are the unions of finitely many polyhedra of ; (ii) the Pareto optimal value set and the weak Pareto optimal value set are the unions of finitely many polyhedra of . 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:

where is a multifunction whose graph is the union of finitely many convex polyhedra and is a polyhedron of . The following results of the structure of (weak) Pareto solution sets were obtained.

Theorem Z 1 (see [13]). Let and be Banach spaces, the union of finitely many polyhedra of , and a multifunction whose graph is the union of finitely many convex polyhedra of . Suppose that the ordering cone is closed, convex, pointed, and has a weakly compact base. If is convex, then the Pareto solution set and the Pareto optimal value set of   are the unions of finitely many polyhedra of and , respectively.

Theorem Z 2 (see [13]). Let and be Banach spaces, the union of finitely many polyhedra of , and a multifunction whose graph is the union of finitely many convex polyhedra of . Suppose that the ordering cone is closed, convex, and pointed. If the interior of is nonempty, then the weak Pareto solution set and the Pareto optimal value set of are the unions of finitely many polyhedra of and , 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 of a multifunction is closed when it is the union of finitely many polyhedra, while the graph is not necessarily closed when it is the union of finitely many G-polyhedra. In the case that is single valued, the former implies that is a piecewise continuous linear function but the latter does not imply that 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 whose graph 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 () and the (weak) Pareto optimal value set () of   are pathwise connected, respectively, when , , and . Recently, Zheng and Yang [11] proved that the weak Pareto set and the weak Pareto optimal value set of   are pathwise connected, respectively, when the ordering cone has a nonempty interior and the mapping is -convex. Zheng [13] proved that the Pareto set and the Pareto optimal value set of are pathwise connected, respectively, when the ordering cone is a pointed, closed, convex cone with a weakly compact base and is a -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 set and the Pareto optimal value set of without the assumption of the ordering cone having a weakly compact base but with that of the cone being polyhedral.

2. Preliminaries

Let and be Banach spaces and let be a convex cone. We say that is pointed if . In this case, one can define a partial order in as follows: for , if and only if . Let denote the interior of . When , by , we mean that . Let denote the dual space of and the dual cone of , defined by

We denote by the set of all strictly positive continuous linear functionals; that is,

We say a convex subset of is a base of if it satisfies that

where denotes the closure. 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 if and only if has a base.

Let be a subset of and in . As usual, we denote by , , and the set of all Pareto points of , the set of all weak Pareto points of , and the set of all positively proper Pareto points of , respectively; that is,

It is clear that

For a multifunction , we denote by and the graph and -epigraph of , respectively; that is,

We say that is -convex, ifis convex. Obviously, is -convex if and only if

Recall [14] that a subset of a Banach space is a (convex) polyhedron, if there exist and such that

A subset of is said to be a G-polyhedron, if is a polyhedron, or there exist a polyhedron of and , such that

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 and are G-polyhedra of Banach space , is a G-polyhedron of , and is the union of finitely many G-polyhedra of .

In this paper, we will consider set-valued vector optimization problem . In the remainder of the paper, we always assume that the graph of the objective mapping is the union of finitely many G-polyhedra or polyhedra of .

We say that is a Pareto (resp., weak Pareto or positively proper Pareto) solution of , if there exists such that (resp., or ); in this case, we say that is a Pareto (resp., weak Pareto or positively proper Pareto) optimal value of . Let , , and , respectively, denote the sets of all Pareto, weak Pareto and positively proper Pareto solutions of . Let , , and , respectively, denote the sets of all Pareto, weak Pareto, and positively proper Pareto optimal values of . Obviously,

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 subset of is the union of finitely many G-polyhedra (resp., polyhedra) if and only if there exist closed subspaces and of , closed subspaces and of , and the union of finitely many G-polyhedra (resp., polyhedron) of such that

The proof of Lemma 1 is similar to that for polyhedron or G-polyhedron in [13] and is omitted.

If in Lemma 1, we have the following corollary.

Corollary 2. A subset of is the union of finitely many G-polyhedra (resp., polyhedra) if and only if there exist closed subspaces and of and the union of finitely many G-polyhedra (resp., polyhedra) of such that

For a subset of , let

Lemma 3 (see [13]). Let be a G-polyhedron (resp., polyhedron) of . Then is the union of finitely many G-polyhedra (resp., polyhedra) of .

Corollary 4. Let be a multifunction whose graph is a convex G-polyhedron (resp., polyhedra) of . Let and be convex G-polyhedra (resp., polyhedra) of and , respectively. Then and are the union of finitely many G-polyhedra (resp., polyhedra) of and , respectively.

Noting that

Corollary 4 is a consequence of Lemma 3.

Lemma 5. Let be a subset of with a closed convex cone . Then

Proof. Let . Then, one can easily verify that

Noting that

it follows that (15) holds. The proof is completed.

Lemma 6. Let be a subset of Banach space with the ordering cone . Then,

Proof. It is obvious that . We only need to show that . Let . Then ; that is

Since is open, one has and hence,

Thus . The proof is completed.

We will also use the following lemma [13, Proposition 2.1] in the sequel.

Lemma 7. Let and be closed subspaces of a Banach space such that

Let for some subset of . Let be the projection of the ordering cone on ; that is,

Then, the following assertions hold.

(i) if .(ii) if .(iii) if .(iv) if .(v) if and .

Proof. For the proofs of (ii)–(v), see [13, Proposition 2.1]. We only need to show (i).

For (i), we assume that . Let and . Then there exists such that

Define by

Then is well defined and it is easy to verify that . On the other hand, (23) and (24) imply that

Hence .

Remark 8. In view of Lemma 7, it is practical to investigate some topics on in a finite dimensional framework. This is our main ideal to consider 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 whose graph is the union of finitely many G-polyhedra. Throughout the remainder of this paper, we assume that , , and are nonempty.

Let be a Banach space. For and , let

It is easy to verify that

In addition, if is convex and has a nonempty interior, by [21, Corollary 5.29], one has

The following lemma is also useful in our later analysis and can be found in [13].

Lemma 9. Let be a Banach space with the ordering cone which has a weakly compact base. Let be the union of finitely many polyhedra of . Suppose that is convex and that is nonempty. Then is the union of finitely many convex polyhedra. More precisely, there exist such that

We have the following proposition.

Proposition 10. Let be a Banach space with the ordering cone and let ,. Set

Then, .

Proof. Let . Then, there exists an integer such that . For each with , noting that

one has and . It follows that

Therefore, one has which implies that .

Conversely, let . Then, for some integer in . Let . We only need to show that . By the definition of , there exists an integer in such that . Noting that , one has if . Now suppose that . Then, by the definition of , it follows that . Since , one has . It follows that . The proof is completed.

Lemma 11. Let be a Banach space, let be the union of finitely many G-polyhedra of , and let be a subset of . Then, is the union of finitely many G-polyhedra of and, more precisely, there exists a finite subset of such that

Proof. By the assumptions, there exist finitely many G-polyhedra such that . Then for each , let

Then

Let . Since and is a polyhedron, (26) implies that is a face of . Noting that every polyhedron has finitely many faces. Hence, there exists a finite subset of , such that

Now, we show that for any ,

If (37) holds, by (36), we have

From (35), it follows that

Let . Noting that , we have that (33) holds. Hence, we only need to show (37). Suppose there would exist such that , but . Let with . By (26) and the definition of , we have , which implies that . This is a contradiction. The proof is completed.

Theorem 12. Let and be Banach spaces, a G-polyhedron, and the union of finitely many G-polyhedra of and let the ordering cone have nonempty interior. Suppose that is convex. Then, and are the union of finitely many G-polyhedra of and , respectively. More precisely, there exist with for some integer such that

Proof. Since is convex, by (28), we have

Noting that , we have

Since the feasible set is a G-polyhedron of and is the union of finitely many G-polyhedra of , Corollary 4 implies that is the union of finitely many G-polyhedra of . Since , it follows from (26) and Lemma 11 that is the union of finitely many G-polyhedra of and there exist with ( for some integer ) such that (40) holds. On the other hand, from Corollary 4, one can easily see that is the union of finitely many G-polyhedra of . The proof is completed.

By Lemma 3, dropping convexity of but requiring that the ordering cone is polyhedral, we generalize Theorem  3.3 in [13] to the union of finitely many G-polyhedra setting.

Theorem 13. Let and be Banach spaces. Let the ordering cone be polyhedral and have nonempty interior. Suppose that and are the unions of finitely many G-polyhedra of and , respectively. Then, and are the unions of finitely many G-polyhedra of and , respectively.

Proof. Suppose that is the union of finitely many G-polyhedra of . Since , by Lemma 3, one has that is the union of finitely many polyhedra of . It is similar to the proof of Theorem  3.3 in [13]; we can show that is the union of finitely many polyhedra of . By Lemma 6, one has

which implies that is the union of finitely many G-polyhedra of . Noting that , Lemma 3 implies that is the union of finitely many G-polyhedra of . The proof is completed.

Dropping the assumption of the ordering cone having a weakly compact base but inquiring being a polyhedral cone, we have the following result analogous to that of Lemma 11.

Lemma 14. Let be a Banach space, let the ordering cone be pointed and polyhedral, and let be the union of finitely many polyhedra of . Suppose that is convex and that is nonempty. Then is the union of finitely many polyhedra of . More precisely, there exist such that

Proof. By Corollary 2 there exist a closed subspace , a finite dimensional subspace of , and the union of finitely many polyhedra of such that

Let be the projection of the ordering cone on ; that is,

Since is polyhedral, together with Corollary 4, we have that the cone is the union of finitely many polyhedra of . Hence it is also closed. From the assumption of , by Lemma 7(ii), we have . From this, with Lemma 7(iii), we have . By Lemma 7(i), we have

We claim that is the union of finitely many polyhedra of . For any , there exist and such that

Since is a subspace of , we have

Hence we have

Conversely, for any , there exists such that . Hence we have

Thus we have shown that

Let . Then there exist such that

Since is convex, by (53), one has

Noting that

from (45) and (55), one has

Hence is convex.

Let . Then there exist such that . Since is a convex cone, it follows that

Hence

We have

Together with (46), we have and .

We have shown that is a closed, convex, and pointed cone of . Noting that is finite dimensional, it follows that has a compact base. Applying Lemma 9, it follows that there exist in such that

Thus is the union of finitely many polyhedra of . By Lemma 7((ii), (iii)), (48), and (61), one has that

Hence,

From (45), (46), and (63), applying Corollary 2, it follows that is the union of finitely many polyhedra of .

For each , , we define by

Then, one can easily show that and

In fact, for any , there exists a pair such that . If , then and thus , a contradiction. Hence . Then we have

It follows that . For (65), let . By (26), (46), and (64), there exists such that and

We have and . The conclusion, holds obviously. Thus (65) holds. From (48) and (63)–(65), we have

Finally, we show that . Indeed, for any , (45) implies that there exists unique pair , such that . Since is finitely dimensional, one can take a constant which is independent on satisfying

From (46), we have

It follows that

and thus

On the other hand, for each sequenc with converging to some , there exists a pair such that . Together with (69), we have

and hence and converge to (since is closed) and , respectively. Thus we have and hence is closed. Together with (70), we have

From (72) and (74), we have

and together with (65), one has

By Theorem 3.2(i) in [13], it follows that

This and (65) imply that

The proof is completed.

Theorem 15. Let and be Banach spaces and let the ordering cone be pointed and polyhedral. Suppose that and are the unions of finitely many polyhedra of and , respectively, and that is convex. Then, and are the union of finitely many convex polyhedra of and , respectively, and more precisely, there exist for some integer such that

Consequently, and are the unions of finitely many polyhedra.

Proof. Suppose that and are the unions of finitely many polyhedra of and , respectively. Noting that

and by Lemma 3, one has that is the union of finitely many polyhedra of . From this and Lemma 14, it follows that there exist ( for some integer ) such that (79) holds. Hence, is the union of finitely many polyhedra of . This and Corollary 4 imply that is the union of finitely many polyhedra of . The proof is completed.

Without the convexity of in Lemma 14, we have the following lemma which will also be applied to consider .

Lemma 16. Let be a Banach space, let the ordering cone be pointed and polyhedral, and let be the union of finitely many G-polyhedra of . Suppose that is nonempty. Then is the union of finitely many G-polyhedra of .

Proof. By Corollary 2 there exist a closed subspace , a finite dimensional subspace of , and the union of finitely many G-polyhedra of , such that (45) and (46) hold.

Let be the projection of on . We first show that is the union of finitely many G-polyhedra of . Indeed, since is a polyhedron on , Lemma 3 implies that is the union of finitely many polyhedra of . We can assume

where each is a G-polyhedron of and each is a polyhedron of . It follows that

Noting that is finitely dimensional, by Proposition  2.1 in [15], we have that each is a G-polyhedron of . This with (82) implies that is the union of finitely many G-polyhedra of . Together with (45), (46), and (53), applying Corollary 2, it follows that is the union of finitely many G-polyhedra of . Let , where each is a G-polyhedron of . Noting that is a face of , by the above proof, applying Proposition  2.1 in [15] again, we have that each as well as is the union of finitely many G-polyhedra of . So is the complimentary . By Lemma 5, we have

which implies that is the union of finitely many G-polyhedra of . So is the set

Let

Then is also the union of finitely many G-polyhedra of . Applying Proposition 10, we have

Thus is the union of finitely many G-polyhedra of . The proof is completed.

Theorem 17. Let and be Banach spaces and let the ordering cone be pointed and polyhedral. Suppose that and are the unions of finitely many G-polyhedra of and , respectively. Then, and are the union of finitely many convex G-polyhedra of and , respectively.

Proof. Suppose that and are the unions of finitely many G-polyhedra of and , respectively. Noting that

and by Lemma 3, one has that is the union of finitely many polyhedra of . From this and Lemma 16, we have that is the union of finitely many G-polyhedra of . This and Corollary 4 imply that is the union of finitely many G-polyhedra of 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 cone has a weakly compact base but requiring that is pointed and polyhedral, under the -convexity assumption on the set-valued objective mapping , we will establish connectedness of Pareto solution set and Pareto optimal value set of .

Let be such that

Consider the following vector optimization problem:

Let and denote the set of all Pareto solutions of and the set of all Pareto optimal values of , respectively.

Lemma 18. Let be a polyhedron of , let be a closed, convex, pointed, and polyhedral cone of , and let be the union of finitely many polyhedra of . Suppose that is -convex. Then and .

Proof. By Corollary 4, is the union of finitely many polyhedra of . From Corollary 2, there exist a closed subspace , a finite dimensional subspace of , and the union of finitely many polyhedra of such that

By (75) in the proof of Lemma 14, we have

Since is a polyhedron of , by Lemma  2.1 in [13] and Theorem  19.6 in [14], is a polyhedron of . By Lemma 3, is the union of finitely many polyhedra of . Hence it is closed. On the other hand, From the convexity of , we have that is convex. Noting that , it follows that . Thus by (90), one has

Let be the projection of on . Analogously to the proof of Lemma 14, one can show

In order to show , we only need to show

Indeed, since is -convex, it follows that and are convex. It is the same to prove that is convex in the proof of Lemma 14; we have that both and are also convex. Noting that is finite dimensional, by Theorem 3.2(i) in [13], one has

On the other hand, (27) implies that

From (94)–(96), it follows that (93) holds. Hence, we have shown . It remains to show . Noting that , , the following assertion in [13, see the proof of Lemma 4.1]

it follows that . The proof is completed.

For our main result, we need the following lemma; see [13, Lemma 4.2].

Lemma 19. Let be a Banach space and a polyhedron of . Let be a convex subset of . Then is pathwise connected.

Theorem 20. Let and be Banach spaces, a polyhedron of a closed, convex, pointed, and polyhedral cone of , and the union of finitely many polyhedra of . Suppose that the set-valued objective function is -convex. Then, the Pareto solution set and the Pareto optimal value set of are pathwise connected.

Proof. Let be defined by (88). Then, is a polyhedron of . By (27), one has . It follows that

This and Lemma 19 imply that is pathwise connected. Hence and are pathwise connected. By Lemma 18, one sees that and are pathwise connected.

Corollary 21. Let be a Banach space with the pointed ordering being closed and polyhedral and let be the union of finitely many convex polyhedra of . Suppose that is convex. Then, is pathwise connected.

Remark 22. In [13, Theorem 4.1 and Corollary 4.1], under the assumptions that the ordering cone has a weakly compact base and the Banach space is finite dimensional, respectively, the corresponding results of connectedness of were established. In our results, Theorem 20 and Corollary 21, we drop these two assumptions, respectively.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant nos. 11261067, 11061039, XT412003) and IRTSTYN.

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