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

Theorem Z 1 (see [13].)Let X 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].)Let X 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 S_w and the Pareto optimal value set V_w of (SVOP) are the unions of finitely many polyhedra of X and Y, respectively.

Lemma 1. A subset P 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

Corollary 2. A subset P 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

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

Corollary 4. Let G : 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.

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

Proof. Let a ∈ A. Then, one can easily verify that

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

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

Proof. It is obvious that A∩WE(cl A, C)⊆WE(A, C). We only need to show that WE(A, C)⊆A∩WE(cl A, C). Let x ∈ WE(A, C). Then (x − int (C))∩A = ∅; that is

Since x − int (C) is open, one has cl (A) ⊂ Y∖(x − int (C)) and hence, Thus x ∈ WE(cl A, C). The proof is completed.

Lemma 7. Let Y₁ and Y₂ be closed subspaces of a Banach space Y such that

Let A∶ = Y₁ + A₂ for some subset A₂ of Y₂. Let C₂ be the projection of the ordering cone C on Y₂; that is, 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 that Y₁∩C = {0}. Let y₁ ∈ Y₁ and y₂ ∈ Pos(A₂, C₂). Then there exists $$ such that

Define c^* by Then c^* is well defined and it is easy to verify that c^* ∈ C⁺¹. On the other hand, (23) and (24) imply that Hence y₁ + y₂ ∈ Pos(A, C).

Remark 8. In view of Lemma 7, 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.

Lemma 9. Let Y 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 $$ such that

Proposition 10. Let Y be a Banach space with the ordering cone C and let $$,  A_i ⊂ Y. Set

Then, $$.

Proof. Let x ∈ E(A, C). Then, there exists an integer i₀ ∈ [1 − k] such that $$. For each j with x ∈ A_j + C, noting that

one has x ∈ E(A_j, C) and $$. It follows that Therefore, one has $$ which implies that $$.

Conversely, let $$. Then, x ∈ E_i 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 ∈ A_j. Noting that E_i ⊂ E(A_i, C), one has z = x if j = i. Now suppose that j ≠ i. Then, by the definition of E_i, it follows that x ∉ (A_j + C)∖(E(A_i, C)∩E(A_j, C)). Since x ∈ z + C ⊂ A_j + C, one has x ∈ E(A_j, C). It follows that x = z. The proof is completed.

Lemma 11. Let Y 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, $$ is the union of finitely many G-polyhedra of Y and, more precisely, there exists a finite subset Λ₀ of Λ such that

Proof. By the assumptions, there exist finitely many G-polyhedra P₁, …, P_n such that $$. Then for each i ∈ [1 − n], let

Then Let y^* ∈ Λ_i. Since $$ and cl (P_i) is a polyhedron, (26) implies that $$ is a face of cl (P_i). Noting that every polyhedron has finitely many faces. Hence, there exists a finite subset $$ of Λ_i, such that Now, we show that for any $$, 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 $$. By (26) and the definition of Λ_i, we have $$, which implies that $$. This is a contradiction. The proof is completed.

Theorem 12. Let X 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, S_w and V_w are the union of finitely many G-polyhedra of X and Y, respectively. More precisely, there exist $$ with $$ for some integer q) such that

Proof. Since F(Γ) + C is convex, by (28), we have

Noting that WE(F(Γ), C) = F(Γ)∩WE(F(Γ) + C, C), we have 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 V_w = WE(F(Γ), C), it follows from (26) and Lemma 11 that V_w is the union of finitely many G-polyhedra of Y and there exist $$ with $$ (i ∈ [i − q] for some integer q) such that (40) holds. On the other hand, from Corollary 4, one can easily see that S_w = Γ∩F⁻¹(V_w) is the union of finitely many G-polyhedra of X. The proof is completed.

Theorem 13. Let X 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, S_w and V_w are the unions of finitely many G-polyhedra of X and Y, respectively.

Proof. Suppose that Gr(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

which implies that V_w is the union of finitely many G-polyhedra of Y. Noting that S_w = (Gr(F)∩(Γ) × V_w) _X, Lemma 3 implies that S_w is the union of finitely many G-polyhedra of X. The proof is completed.

Lemma 14. Let Y 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 $$ such that

Proof. By Corollary 2 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

Let C₂ be the projection of the ordering cone C on Y₂; that is, 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 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 Since Y₁ is a subspace of Y, we have Hence we have Conversely, for any (x₁, a₂, c₂) ∈ Y₁ × A₂ × C₂, there exists c₁ ∈ Y₁ such that c₁ + c₂ ∈ C. Hence we have Thus we have shown that Let x₂, y₂ ∈ A₂ + C₂. Then there exist x₁, y₁ ∈ Y₁ such that Since A + C is convex, by (53), one has Noting that from (45) and (55), one has Hence A₂ + C₂ is convex.

Let x₂ ∈ C₂∩−C₂. Then there exist $$ such that $$. Since C is a convex cone, it follows that

Hence We have Together with (46), we have x₂ = 0 and C₂∩−C₂ = {0}.

We have shown that C₂ 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 $$ in $$ such that

Thus Pos(A₂, C₂) is the union of finitely many polyhedra of Y₂. By Lemma 7((ii), (iii)), (48), and (61), one has that Hence, From (45), (46), and (63), applying Corollary 2, it follows that Pos(A, C) is the union of finitely many polyhedra of Y.

For each $$, i ∈ [1 − p], we define c^* by

Then, one can easily show that $$ and In fact, for any y ∈ C∖{0}, there exists a pair (y₁, y₂) ∈ Y₁ × C₂ such that y = y₁ + y₂. If y₂ = 0, then y = y₁ ∈ Y₁ and thus y ∈ Y₁∩C = {0}, a contradiction. Hence y₂ ≠ 0. Then we have It follows that $$. For (65), let $$. By (26), (46), and (64), there exists (a₁, a₂) ∈ Y₁ × A₂ such that a = a₁ + a₂ and We have $$ and $$. The conclusion, $$ holds obviously. Thus (65) holds. From (48) and (63)–(65), we have Finally, we show that $$. Indeed, for any 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 From (46), we have It follows that and thus On the other hand, for each sequenc {(y_1,n, a_2,n)} ⊂ Y₁ × cl  co A₂ with y_n∶ = y_1,n + a_2,n converging to some y ∈ Y, there exists a pair (y₁, y₂) ∈ Y₁ × Y₂ such that y = y₁ + y₂. Together with (69), we have and hence {y_1,n} and {y_2,n} converge to y₁ ∈ Y₁ (since Y₁ is closed) and y₂ ∈ cl  co A₂, respectively. Thus we have y ∈ Y₁ + cl  co A₂ and hence Y₁ + cl  co A₂ 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 X 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 $$ for some integer q) such that

Consequently, S = S_p and V are the unions of finitely many polyhedra.

Proof. Suppose that Γ and Gr(F) are the unions of finitely many polyhedra of X and X × Y, respectively. Noting that

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 $$ (i ∈ [1 − q] for some integer q) such that (79) holds. Hence, V = V_p is the union of finitely many polyhedra of Y. This and Corollary 4 imply that S = S_p = Γ∩F⁻¹(V) is the union of finitely many polyhedra of X. The proof is completed.

Lemma 16. Let Y 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 Corollary 2 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.

Let C 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

where each A_2,i is a G-polyhedron of Y₂ and each C_2,j is a polyhedron of Y₂. It follows that Noting that Y₂ is finitely dimensional, by Proposition  2.1 in [15], we have that each A_2,i + C_2,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 $$, where each A_i 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 A_i + C∖{0} as well as A_i + C is the union of finitely many G-polyhedra of Y. So is the complimentary (A_i + C∖{0}) ^c. By Lemma 5, we have which implies that (A_i, C) is the union of finitely many G-polyhedra of Y. So is the set Let Then E_i is also the union of finitely many G-polyhedra of Y. Applying Proposition 10, we have Thus E(A, C) is the union of finitely many G-polyhedra of Y. The proof is completed.

Theorem 17. Let X 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 that Γ and Gr(F) are the unions of finitely many G-polyhedra of X and X × Y, respectively. Noting that

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.

Lemma 18. Let Γ be a polyhedron of X, let C be a closed, convex, pointed, and polyhedral cone of Y, and let Gr F(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 Corollary 4, 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

By (75) in the proof of Lemma 14, we have 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) ⊂ cl  co F(Γ), it follows that F^′(Γ) = cl  co F(Γ). Thus by (90), one has Let C₂ be the projection of C on Y₂. Analogously to the proof of Lemma 14, one can show In order to show V^′ = V, we only need to show 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 On the other hand, (27) implies that 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] it follows that S^′ = S. The proof is completed.

Lemma 19. Let Y be a Banach space and A a polyhedron of Y. Let Λ be a convex subset of Y^*. Then $$ is pathwise connected.

Theorem 20. Let X and Y be Banach spaces, Γ a polyhedron of X, C a closed, convex, pointed, and polyhedral cone of Y, and Gr F(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. Let F^′ : X⇉Y be defined by (88). Then, Gr(F^′) is a polyhedron of X × Y. By (27), one has $$. It follows that

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. Let Y 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.