Optimality conditions of set-valued optimization problem involving relative algebraic interior in ordered linear spaces Zhi-Ang Zhoua , Xin-Min Yangb and Jian-Wen Pengc1 a
Department of Applied Mathematics, Chongqing University of Technology, Chongqing 400054, P.R. China; b School of Mathematics, Chongqing Normal University, Chongqing 400047, P.R. China; c School of Mathematics, Chongqing Normal University, Chongqing 400047, P.R. China
In this paper, firstly, a generalized subconvexlike set-valued map involving the relative algebraic interior is introduced in ordered linear spaces. Secondly, some properties of a generalized subconvexlike set-valued map are investigated. Finally, the optimality conditions of set-valued optimization problem are established. Keywords Relative algebraic interior · Generalized cone subconvexlikeness · Set-valued map· Separation property· Optimality condition AMS 2010 Subject Classifications: 90C26, 90C29, 90C30
1
Introduction
Recently, many authors have been interested in vector optimization problems with set-valued maps. For example, Rong and Wu [11] gave characterizations of super efficiency for vector 1
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1
optimization with cone convexlike set-valued maps. Li [2] established an alternative theorem for cone subconvexlike set-valued maps and Kuhn-Tucker conditions for vector optimization problems with cone subconvexlike set-valued maps. Yang et al. [12] established an alternative theorem for the generalized cone subconvexlike set-valued maps. Yang et al. [13] established an alternative theorem for the nearly cone subconvexlike set-valued maps. We note that the results in [2] and [11-13] are established in linear topological spaces. It is well-known that linear spaces are wider than linear topological spaces. Hence, it is natural to consider the following interesting and meaningful problem: How to generalize those results in [2, 12, 13] from linear topological spaces to linear spaces. Li [1] generalized those results in [2] from linear topological spaces to linear spaces. Huang and Li [3] also generalized the results obtained in [1] from cone subconvexlikeness case to generalized cone-subconvexlikeness case. Note that the results in [1] and [3] were established under the condition that the algebraic interior of ordered cone C denoted by cor(C) is nonempty. However, in some optimization problems, it is possible that cor(C) = ∅. In order to overcome this flaw, the authors in [15-16] introduced the notion of relative algebraic interiors in linear spaces. It is worth noting that the algebraic interior of a set C is the subset of the relative algebraic interior of C. Thus, the notion of the relative algebraic interior generalizes that of the algebraic interior. Ad´an and Novo [58] investigated weak or proper efficiency of vector optimization problems with generalized convex set-valued maps involving relative algebraic interior and vector closure of C in linear spaces. Hern´andez et al. [9] introduced a cone subconvexlike set-valued map and established optimality conditions and Lagrangian multiplier rule involving relative algebraic interior of C in linear spaces. This paper is organized as follows. In section 2, we give some preliminaries. In section 3, the new notion of generalized cone subconvexlike set-valued map involving the relative 2
algebraic interior of C is introduced, and some properties of the generalized cone subconvexlike set-valued maps are investigated. In section 4, a separation property is obtained, and the optimality conditions of set-valued optimization problem are established. The results in this paper generalize some known results in some literature.
2
Preliminaries
Let X be a nonempty set, and let Y and Z be two ordered linear spaces. Let 0 stand for the zero element of every space. Let K be a nonempty subset in Y. The generated cone of K is defined as coneK = {λa|a ∈ K, λ ≥ 0}. A cone K ⊆ Y is said to be pointed if K ∩ (−K) = {0}. A cone K ⊆ Y is said to be nontrivial if K ̸= {0} and K ̸= Y. Let Y ∗ and Z ∗ stand for algebraic dual spaces in Y and Z, respectively. From now on, let C and D be nontrivial pointed convex cones in Y and Z, respectively. The algebraic dual cone C + and strictly algebraic dual cone C +i of C are defined as C + = {y ∗ ∈ Y ∗ |⟨y, y ∗ ⟩ > 0, ∀y ∈ C} and C +i = {y ∗ ∈ Y ∗ |⟨y, y ∗ ⟩ > 0, ∀y ∈ C \ {0}}, where ⟨y, y ∗ ⟩ denotes the value of the linear functional y ∗ at the point y. The meanings of D+ and D+i are similar. Let K be a nonempty subset in Y . We denote by af f (K), span(K) and L(K) = span(K − K) the affine hull, linear hull and generated linear subspace of K, respectively. Definition 2.1 [14] Let K be a nonempty subset in Y . The algebraic interior of K is the set cor(K) = {k ∈ K|∀v ∈ Y, ∃λ0 > 0, ∀λ ∈ [0, λ0 ], k + λv ∈ K}. 3
Definition 2.2 [15,16] Let K be a nonempty subset in Y . The relative algebraic interior of K is the set icr(K) = {k ∈ K|∀v ∈ L(K), ∃λ0 > 0, ∀λ ∈ [0, λ0 ], k + λv ∈ K}. LEMMA 2.1 Let K be a nonempty subset in Y . Then af f (K) = x + L(K), ∀x ∈ K.
(1)
Proof Firstly, we will show that af f (K) ⊆ x + L(K), ∀x ∈ K. Let y ∈ af f (K). Then, there exist ki ∈ K, αi ∈ R with
n ∑
(2)
αi = 1 such that
i=1
y=
n ∑
αi ki .
i=1
Thus, we obtain y = x+y−x = x+
n ∑
αi ki −
i=1
n ∑
αi x = x +
i=1
n ∑
αi (ki − x) ∈ x + span(K − K) = x + L(K).
i=1
Therefore, (2) holds. Finally, we will show that x + L(K) ⊆ af f (K), ∀x ∈ K.
(3)
Let z ∈ x + L(K). Then, there exist xi , yi ∈ K and λi ∈ R(i = 1, · · · , n) such that z =x+
n ∑
λi (xi − yi ) = x +
i=1
n ∑
λi x i −
i=1
n ∑
λi y i .
(4)
i=1
Clearly, 1+
n ∑
λi −
i=1
n ∑ i=1
4
λi = 1.
(5)
It follows from (4) and (5) that z ∈ af f (K). Therefore, (3) holds. It follows from (2) and (3)
that (1) holds. Remark 2.1 It follows from Lemma 2.1 that icr(K) = {k ∈ K|∀v ∈ af f (K) − k, ∃λ0 > 0, ∀λ ∈ [0, λ0 ], k + λv ∈ K}. Remark 2.2 It follows from Lemma 2.1 that if 0 ∈ K ⊆ Y, then icr(K) = {k ∈ K|∀v ∈ af f (K), ∃λ0 > 0, ∀λ ∈ [0, λ0 ], k + λv ∈ K}.
Remark 2.3 If K be a nontrivial pointed cone in Y then 0 ∈ / icr(K). In fact, if 0 ∈ icr(K), then it follows from Remark 2.2 that af f (K) = K.
(6)
af f (K) = span(K).
(7)
span(K) = K.
(8)
Since 0 ∈ K, it is clear that
By (6) and (7), we have
Since K is nontrivial, by (8), there exists a nonzero k ∈ K such that −k ∈ K, which contradicts that K is pointed. Remark 2.4 It is easy to check that icr(K) is a convex set and icr(K) ∪ {0} is a convex cone if K is a convex cone. LEMMA 2.2 Let K be a convex cone in Y. Then K + icr(K) = icr(K). Proof Clearly, (9) holds if icr(K) = ∅. 5
(9)
Now, let k1 ∈ K and k2 ∈ icr(K). k2 ∈ icr(K) implies that k2 ∈ K, ∀v ∈ L(K), ∃λ0 > 0, ∀λ ∈ [0, λ0 ], we have k2 + λv ∈ K. Since K is a convex cone, we obtain (k1 + k2 ) + λv = k1 + (k2 + λv) ∈ K + K ⊆ K, which implies k1 + k2 ∈ icr(K). Therefore, K + icr(K) ⊆ icr(K).
(10)
icr(K) ⊆ K + icr(K).
(11)
Since 0 ∈ K, we have
According to (10) and (11), (9) holds. LEMMA 2.3 Let K1 and K2 be two nonempty subsets in Y . Then af f (K1 + K2 ) = af f (K1 ) + af f (K2 ).
(12)
af f (K1 + K2 ) ⊆ af f (K1 ) + af f (K2 ).
(13)
af f (K1 ) + af f (K2 ) ⊆ af f (K1 + K2 ).
(14)
Proof Clearly,
Now, we will show that
Let y ∈ af f (K1 ) + af f (K2 ). Then, there exist k1i ∈ K1 , k2j ∈ K2 , λi1 ∈ R, λj2 ∈ R(i = m1 m2 ∑ ∑ 1, · · · m1 ; j = 1, · · · m2 ) with λi1 = 1 and λj2 = 1 such that i=1
y=
m1 ∑ i=1
j=1
λi1 k1i +
m2 ∑
λj2 k2j =
j=1
m1 ∑ m2 ∑ i=1 j=1
where λij = λi1 λj2 (i = 1, · · · , m1 ; j = 1, · · · , m2 ).
6
λij (k1i + k2j ),
It is easy to check that
m1 ∑ m2 ∑
λij = 1 and k1i + k2j ∈ K1 + K2 (i = 1, · · · , m1 ; j = 1, · · · , m2 ).
i=1 j=1
Hence, y ∈ af f (K1 +K2 ). Thus, (14) holds. By (13) and (14), (12) holds.
LEMMA 2.4 Let K be a nonempty subset in Y . Then (a) af f (K) = k + af f (K − K), ∀k ∈ K. If K is convex and icr(K) ̸= ∅, then (b) icr(icr(K)) = icr(K); (c) af f (icr(K)) = af f (K). Proof (b) and (c) can be found in [5] and [9], respectively. We will only prove (a). Clearly, it follows from that Lemma 2.3 that af f (K) − k ⊆ af f (K) − af f (K) ⊆ af f (K − K), ∀k ∈ K, i.e., af f (K) ⊆ k + af f (K − K), ∀k ∈ K.
(15)
Finally, we will show that k + af f (K − K) ⊆ af f (K), ∀k ∈ K.
(16)
By Lemma 2.3, we only need to show that k + af f (K) − af f (K) ⊆ af f (K), ∀k ∈ K.
(17)
Let y ∈ k + af f (K) − af f (K). Then, there exist k1i ∈ K, k2j ∈ K, λi1 ∈ R, λj2 ∈ R(i = m1 m2 ∑ ∑ 1, · · · , m1 ; j = 1, · · · , m2 ) with λi1 = 1 and λj2 = 1 such that i=1
y=k+
j=1 m1 ∑
λi1 k1i
i=1
−
m2 ∑ j=1
7
λj2 k2j .
(18)
Clearly, 1+
m1 ∑
λi1
−
i=1
m2 ∑
λj2 = 1.
(19)
j=1
By (18) and (19), y ∈ af f (K). Thus, (17) holds. Hence, (16) holds. By (15) and (16), we obtain af f (K) = k + af f (K − K), ∀k ∈ K.
LEMMA 2.5 Let K1 and K2 be two nonempty subsets in Y . Then af f (K1 × K2 ) = af f (K1 ) × af f (K2 ).
(20)
af f (K1 × K2 ) = af f (K1 × {0} + {0} × K2 ).
(21)
Proof Clearly,
It follows from Lemma 2.3 that af f (K1 × {0} + {0} × K2 ) = af f (K1 × {0}) + af f ({0} × K2 ) = af f (K1 ) × {0} + {0} × af f (K2 ) = af f (K1 ) × af f (K2 )
(22)
By (21) and (22), (20) holds.
LEMMA 2.6 Let K1 and K2 be two nontrivial convex cone in Y and Z. If icr(K1 ) ̸= ∅ and icr(K2 ) ̸= ∅, then icr(K1 × K2 ) = icr(K1 ) × icr(K2 ).
(23)
Proof Firstly, we will show that icr(K1 ) × icr(K2 ) ⊆ icr(K1 × K2 ).
(24)
Let (k1 , k2 ) ∈ icr(K1 ) × icr(K2 ). Clearly, (k1 , k2 ) ∈ K1 × K2 . Let (v1 , v2 ) ∈ af f (K1 × K2 ) − (k1 , k2 ). By Lemma 2.5, (v1 , v2 ) ∈ (af f (K1 ) − k1 ) × (af f (K2 ) − k2 ). Since k1 ∈ icr(K1 ), for 8
v1 ∈ af f (K1 ) − k1 , there exists ε10 > 0 such that k1 + εv1 ∈ K1 , ∀ε ∈ [0, ε10 ].
(25)
Since k2 ∈ icr(K2 ), for v2 ∈ af f (K2 ) − k2 , there exists ε20 > 0 such that k2 + εv2 ∈ K2 , , ∀ε ∈ [0, ε20 ].
(26)
Choose ε30 = min{ε10 , ε20 }. It follows from (25) and (26) that there exists ε30 > 0 such that (k1 , k2 ) + ε(v1 , v2 ) ∈ K1 × K2 , ∀ε ∈ [0, ε30 ], which implies (k1 , k2 ) ∈ icr(K1 × K2 ). Thus, (24) holds. Finally, we will show that icr(K1 × K2 ) ⊆ icr(K1 ) × icr(K2 ).
(27)
Since icr(K1 ) ̸= ∅ and icr(K2 ) ̸= ∅, it follows from (24) that icr(K1 × K2 ) ̸= ∅. Let (k1 , k2 ) ∈ icr(K1 × K2 ). Clearly, (k1 , k2 ) ∈ K1 × K2 . Let (v1 , v2 ) ∈ (af f (K1 ) − k1 ) × (af f (K2 ) − k2 ). Clearly, (v1 , v2 ) ∈ af f (K1 × K2 ) − (k1 , k2 ). Therefore, there exists ε0 > 0 such that (k1 , k2 ) + ε(v1 , v2 ) ⊆ K1 × K2 , ∀ε ∈ [0, ε0 ], which implies k1 + εv1 ⊆ K1
(28)
k2 + εv2 ⊆ K2 .
(29)
and
9
By (28) and (29), we have (k1 , k2 ) ∈ icr(K1 ) × icr(K2 ). Therefore, (27) holds. By (24) and (27), (23) holds.
LEMMA 2.7 [15,16] Let K be a convex set with icr(K) ̸= ∅ in Y. If 0 ∈ / icr(K), then there exists y ∗ ∈ Y ∗ \ {0} such that ⟨k, y ∗ ⟩ ≥ 0, ∀k ∈ K.
3
Generalized subconvexlike set-valued map
In this section, firstly, we will introduce several classes of generalized convex set-valued maps. Secondly, we will discuss their relationships. Finally, we will obtain some properties of the generalized cone subconvexlike set-valued maps. From now on, let A be a nonempty subset in X, and let F : A → 2Y be a set-valued map. ∪ Write F (A) = F (x). We suppose that icr(C) × icr(D) ̸= ∅. x∈A
Definition 3.1 [11] A set-valued map F : A → 2Y is called C-convexlike if, ∀x1 , x2 ∈ A and ∀λ ∈ (0, 1), λF (x1 ) + (1 − λ)F (x2 ) ⊆ F (A) + C, Remark 3.1 It follows from [11] that F : A → 2Y is C-convexlike if and only if F (A) + C is a convex set. Definition 3.2 [9] A set-valued map F : A → 2Y is called C-subconvexlike if, ∃c ∈ icr(C) such that, ∀x1 , x2 ∈ A, ∀λ ∈ (0, 1), ∀ε > 0, εc + λF (x1 ) + (1 − λ)F (x2 ) ⊆ F (A) + C. Remark 3.2 Proposition 3.2 of [9] shows that F : A → 2Y is C-subconvexlike if and only if 10
F (A) + icr(C) is a convex set. Definition 3.3
A set-valued map F : A → 2Y is called generalized C-subconvexlike if
cone(F (A)) + icr(C) is a convex set. THEOREM 3.1 If the set-valued map F : A → 2Y is C-convexlike, then F is C-subconvexlike. Proof It follows from the C-convexlikeness of F that F (A) + C is a convex set. We need to prove that F (A) + icr(C) is a convex set. Indeed, let mi ∈ F (A) + icr(C)(i = 1, 2), λ ∈ (0, 1). Then, there exist xi ∈ A, ci ∈ icr(C)(i = 1, 2) such that mi = yi + ci , yi ∈ F (xi ), i = 1, 2. By Remark 3.1 and Lemma 2.2, λm1 + (1 − λ)m2 = λ(y1 + c1 ) + (1 − λ)(y2 + c2 ) = [λy1 + (1 − λ)y2 ] + [λc1 + (1 − λ)c2 ] ∈ (F (A) + C) + icr(C) = F (A) + (C + icr(C)) = F (A) + icr(C)
Remark 3.3 The following example shows that a C-subconvexlike set-valued map may not be C-convexlike. Thus, the C-subconvexlikeness is a generalization of C-convexlikeness. Example 3.1 Let X = Y = R2 , C = {(y1 , 0)|y1 ≥ 0} and A = {(1, 0), (0, 2)}. The set-valued map F : A → 2Y is defined as follows: F (1, 0) = {(y1 , y2 )|1 < y1 ≤ 2, 0 ≤ y2 ≤ 1} ∪ {(1, 0), (1, 1)}; F (0, 2) = {(y1 , y2 )|1 < y1 ≤ 2, 1 ≤ y2 ≤ 2} ∪ {(1, 2), (1, 1)}. Clearly, F (A) + icr(C) is a convex set, and F (A) + C is not a convex set. Therefore, F is C-subconvexlike. However, F is not C-convexlike. THEOREM 3.2 If the set-valued map F : A → 2Y is C-subconvexlike, then F is generalized C-subconvexlike. Proof It follows from the C-subconvexlikeness of F that F (A)+icr(C) is a convex set. We need 11
to prove that cone(F (A)) + icr(C) is a convex set. Indeed, let mi ∈ cone(F (A)) + icr(C)(i = 1, 2), λ ∈ (0, 1). Then, there exist ρi ≥ 0, xi ∈ A, ci ∈ icr(C)(i = 1, 2) such that mi = ρi yi + ci , yi ∈ F (xi ), i = 1, 2. Case 1. ρ1 = 0 or ρ2 = 0. Clearly, λm1 + (1 − λ)m2 ∈ cone(F (A)) + icr(C). Case 2. ρ1 > 0 and ρ2 > 0. Since F (A) + icr(C) is a convex set, we have λm1 + (1 − λ)m2 = λ(ρ1 y1 + c1 ) + (1 − λ)(ρ2 y2 + c2 ) = λρ1 (y1 + λρ1 = [λρ1 + (1 − λ)ρ2 ]{ λρ1 +(1−λ)ρ (y1 + 2
1 c) ρ1 1
+
(1−λ)ρ2 (y2 λρ1 +(1−λ)ρ2
+
1 c) ρ1 1
+ (1 − λ)ρ2 (y2 +
1 c) ρ2 2
1 c )} ρ2 2
∈ [λρ1 + (1 − λ)ρ2 ](F (A) + icr(C)) ⊆ cone(F (A)) + icr(C). Case 1 and Case 2 show that cone(F (A)) + icr(C) is a convex set.
Remark 3.4 The following example shows that a generalized C-subconvexlike set-valued map may be not C-subconvexlike. Thus, generalized C-subconvexlikeness is a generalization of C-subconvexlikeness. Example 3.2 Let X = Y = R2 , C = {(y1 , 0)|y1 ≥ 0} and A = {(1, 0), (0, 2)}. The set-valued map F : A → 2Y is defined as follows: F (1, 0) = {(y1 , y2 )|y2 ≤ −y1 + 2, y1 ≥ 0, y2 ≥ 1}; F (0, 2) = {(y1 , y2 )|y2 ≤ −y1 + 2, y1 ≥ 1, y2 ≥ 0}. Clearly, cone(F (A))+icr(C) is a convex set, and F (A)+icr(C) is not a convex set. Therefore, F is generalized C-subconvexlike. However, F is not C-subconvexlike. Next, we give some equivalent properties of generalized C-subconvexlike set-valued maps. THEOREM 3.3 A set-valued map F : A → 2Y is generalized C-subconvexlike if and only if, ∀c ∈ icr(C), ∀x1 , x2 ∈ A, ∀λ ∈ (0, 1), c + λF (x1 ) + (1 − λ)F (x2 ) ⊆ cone(F (A)) + icr(C).
12
(30)
Proof Necessity. Let c ∈ icr(C), x1 , x2 ∈ A, y1 ∈ F (x1 ), y2 ∈ F (x2 ) and λ ∈ (0, 1). Clearly, y1 + c ∈ cone(F (A)) + icr(C)
(31)
y2 + c ∈ cone(F (A)) + icr(C).
(32)
and
Since F is generalized C-subconvexlike, it follows from (31) and (32) that c + λy1 + (1 − λ)y2 = λ(y1 + c) + (1 − λ)(y2 + c) ∈ cone(F (A)) + icr(C). Therefore, (30) holds. Sufficiency. Let mi ∈ cone(F (A)) + icr(C)(i = 1, 2), λ ∈ (0, 1). Then, there exist ρi ≥ 0, xi ∈ A, ci ∈ icr(C)(i = 1, 2) such that mi = ρi yi + ci , yi ∈ F (xi ), i = 1, 2. Case 1. ρ1 = 0 or ρ2 = 0. Clearly, λm1 + (1 − λ)m2 ∈ cone(F (A)) + icr(C). Case 2. ρ1 > 0 and ρ2 > 0. Since F (A) + icr(C) is a convex set, we have λm1 + (1 − λ)m2 = λ(ρ1 y1 + c1 ) + (1 − λ)(ρ2 y2 + c2 ) = [λc1 + (1 − λ)c2 ] + [λρ1 y1 + (1 − λ)ρ2 y2 ] = [λρ1 + (1 − λ)ρ2 ]{
1 λρ1 (1 − λ)ρ2 [λc1 + (1 − λ)c2 ] + y1 + y2 }. λρ1 + (1 − λ)ρ2 λρ1 + (1 − λ)ρ2 λρ1 + (1 − λ)ρ2 (33)
Clearly, 1 [λc1 + (1 − λ)c2 ] ∈ icr(C), λρ1 + (1 − λ)ρ2 λρ1 ∈ (0, 1), λρ1 + (1 − λ)ρ2 (1 − λ)ρ2 ∈ (0, 1) λρ1 + (1 − λ)ρ2 13
and λρ1 (1 − λ)ρ2 + = 1. λρ1 + (1 − λ)ρ2 λρ1 + (1 − λ)ρ2 By (30) and (33), we obtain λm1 + (1 − λ)m2 ∈ [λρ1 + (1 − λ)ρ2 ](cone(F (A)) + icr(C)) ⊆ cone(F (A)) + icr(C). Case 1 and Case 2 shows that cone(F (A))+icr(C) is a convex set. Therefore, F is generalized
C-subconvexlike. THEOREM 3.4 The following statements are equivalent: (a) F : A → 2Y is generalized C-subconvexlike; (b) ∀c ∈ icr(C), ∀x1 , x2 ∈ A, ∀λ ∈ (0, 1), c + λF (x1 ) + (1 − λ)F (x2 ) ⊆ cone(F (A)) + icr(C); (c) ∃c′ ∈ icr(C), ∀x1 , x2 ∈ A, ∀λ ∈ (0, 1), ∀ε > 0, εc′ + λF (x1 ) + (1 − λ)F (x2 ) ⊆ cone(F (A)) + C; (d) ∀x1 , x2 ∈ A, ∀λ ∈ (0, 1), ∃c′′ ∈ C, ∀ε > 0, εc′′ + λF (x1 ) + (1 − λ)F (x2 ) ⊆ cone(F (A)) + C.
Proof By Theorem 3.3, (a) ⇔ (b). The implications (b) ⇒ (c) ⇒ (d) are clear. Therefore, we need to prove that (d) ⇒ (b). Let c ∈ icr(C), x1 , x2 ∈ A and λ ∈ (0, 1). Then, ∃c′′ ∈ C, ∀ε > 0, εc′′ + λF (x1 ) + (1 − λ)F (x2 ) ⊆ cone(F (A)) + C.
(34)
Since c ∈ icr(C) = icr(icr(C)), by Lemma 2.4 (a) and (c), for −c′′ = 0 − c′′ ∈ C − C ⊆ af f (C − C) = af f (C) − c = af f (icr(C)) − c, there exists λ0 > 0 such that c + λ0 (−c′′ ) ∈ icr(C). 14
Write c = c + λ0 (−c′′ ). Clearly, c = c + λ0 c′′ . By (34) and Lemma 2.2, we obtain c+λF (x1 )+(1−λ)F (x2 ) = (c+λ0 c′′ )+λF (x1 )+(1−λ)F (x2 ) = [λ0 c′′ +λF (x1 )+(1−λ)F (x2 )]+c ⊆ cone(F (A)) + C + c ⊆ cone(F (A)) + C + icr(C) = cone(F (A)) + icr(C).
4
Optimality conditions
In order to establish optimality conditions for vector optimization problems with set-valued maps, we need to present a separation property for a generalized C-subconvexlike set-valued map. Now, we consider the following two systems: System 1: There exists x0 ∈ A such that F (x0 ) ∩ (−icr(C)) ̸= ∅; System 2: There exists y ∗ ∈ C + \ {0} such that ⟨y, y ∗ ⟩ ≥ 0, ∀y ∈ F (A). THEOREM 4.1 (i) Suppose that F : A → 2Y is generalized C-subconvexlike and icr(coneF (A)+ icr(C)) ̸= ∅. If System 1 has no solutions, then System 2 has a solution. (ii) If y ∗ ∈ C +i is a solution of System 2, then System 1 has no solutions. Proof (i) Firstly, we assert that 0 ∈ / cone(F (A))+icr(C). Otherwise, there exist x0 ∈ A, α ≥ 0 such that 0 ∈ αF (x0 ) + icr(C). Case 1. If α = 0, then 0 ∈ icr(C), which contradicts 0 ∈ / icr(C) (see Remark 2.3). Case 2. If α > 0, there exists y0 ∈ F (x0 ) such that −y0 ∈ α1 icr(C) ⊆ icr(C), which contradicts that System 1 has no solutions. Thus, we obtain 0∈ / cone(F (A)) + icr(C).
15
Since F is generalized C-subconvexlike, cone(F (A)) + icr(C) is a convex set. By Lemma 2.7, there exists y ∗ ∈ Y ∗ \ {0} such that ⟨y, y ∗ ⟩ ≥ 0, ∀y ∈ cone(F (A)) + icr(C). So, ⟨αF (x) + c, y ∗ ⟩ ≥ 0, ∀x ∈ A, ∀c ∈ icr(C), ∀α ≥ 0.
(35)
Letting α = 0 in (35), we obtain ⟨c, y ∗ ⟩ ≥ 0, ∀c ∈ icr(C).
(36)
We will show that y ∗ ∈ C + . Otherwise, there exists c′ ∈ C such that ⟨c′ , y ∗ ⟩ < 0. Hence, ⟨θc′ , y ∗ ⟩ < 0, ∀θ > 0. By Lemma 2.2, we have θc′ + c ∈ icr(C), ∀θ > 0, ∀c ∈ icr(C).
(37)
It follows from (36) and (37) that θ⟨c′ , y ∗ ⟩ + ⟨c, y ∗ ⟩ ≥ 0, ∀θ > 0, ∀c ∈ icr(C).
(38)
∗
⟩ ∗ + We note that (38) does not hold when θ > − ⟨c⟨c,y ′ ,y ∗ ⟩ ≥ 0. Therefore, y ∈ C .
Letting α = 1 in (35), we have ⟨F (x) + c, y ∗ ⟩ ≥ 0, ∀x ∈ A, ∀c ∈ icr(C). Letting c0 ∈ icr(C) and λn > 0 with lim λn = 0, we have n→∞
⟨F (x) + λn c0 , y ∗ ⟩ ≥ 0, ∀x ∈ A, ∀n ∈ N. It follows from (39) that ⟨F (x), y ∗ ⟩ ≥ 0, ∀x ∈ A. 16
(39)
(ii) Since y ∗ ∈ C +i is a solution of System 2, we have ⟨y, y ∗ ⟩ ≥ 0, ∀y ∈ F (A).
(40)
Now, we suppose that System 1 has a solution. Then, there exists x0 ∈ A such that F (x0 ) ∩ (−icr(C)) ̸= ∅. Thus, there exists y0 ∈ F (x0 ) such that −y0 ∈ icr(C). Clearly, −y0 ̸= 0. So, we have ⟨y0 , y ∗ ⟩ < 0, which contradicts (40).
Remark 4.1 In (i), the condition icr(cone(F (A)) + icr(C)) ̸= ∅ can be replaced by the condition af f (C)) = af f (cone(F (A)) + icr(C)). In fact, when af f (C) = af f (cone(F (A)) + icr(C)), icr(C) ⊆ icr(cone(F (A)) + icr(C)). Since icr(C) ̸= ∅, icr(cone(F (A)) + icr(C)) ̸= ∅. However, the following example shows that the condition icr(cone(F (A)) + icr(C)) ̸= ∅ is weaker than the condition af f (C) = af f (cone(F (A)) + icr(C)). Example 4.1 In Example 3.2, it is clear that icr(cone(F (A)) + icr(C)) = {(y1 , y2 )|y1 > 0, y2 ≥ 0} ̸= ∅. However, af f (C) = {(y1 , y2 )|y1 ∈ R, y2 = 0} ̸= R2 = af f (cone(F (A)) + icr(C)). Remark 4.2 Since cone(F (A)) + icr(C) is a nonempty convex subset in Y, The condition icr(cone(F (A)) + icr(C)) ̸= ∅ can be deleted if Y is a finite-dimensional space. Remark 4.3 Theorem 4.1 improves Theorem 2.1 of [1], Theorem 3.5 of [5] and Theorem 3.9 of [9]. Let F : A → 2Y and G : A → 2Z be two set-valued maps. Now, we consider the following vector optimization problem with set-valued maps: min F (x) (VP) s.t. − G(x) ∩ D ̸= ∅. The feasible set of (VP) is denoted by S = {x ∈ A| − G(x) ∩ D ̸= ∅}. Now, we define W M in(F (A), C) = {y0 ∈ F (A)|(y0 − F (A)) ∩ icr(C) = ∅}. 17
Definition 4.1 A point x0 is called a weakly efficient solution of (VP) if there exists x0 ∈ S such that F (x0 ) ∩ W M in(F (S), C) ̸= ∅. A point pair (x0 , y0 ) is called a weak minimizer of (VP) if y0 ∈ F (x0 ) ∩ W M in(F (S), C). Let I(x) = F (x) × G(x), ∀x ∈ A. It is clear that I is a set-valued map from A to Y × Z, where Y × Z is a linear space with nontrivial pointed convex cone C × D. The algebraic dual space of Y × Z is Y ∗ × Z ∗ , and the algebraic dual cone of C × D is C + × D+ . By Definition 3.3, we say that the set-valued map I : A → 2Y ×Z is generalized C × Dsubconvexlike if cone(I(A)) + icr(C × D) is a convex set in Y × Z. Now we present a necessary optimality condition for (VP) as follows: THEOREM 4.2 Suppose that the following conditions hold: (i) (x0 , y0 ) is a weak minimizer of (VP); (ii) I1 : A → 2Y ×Z is generalized C × D-subconvexlike, where I1 (x) = (F (x) − y0 ) × G(x), ∀x ∈ A; (iii) icr(cone(I1 (A)) + icr(C × D)) ̸= ∅. Then, there exists (y ∗ , z ∗ ) ∈ C + × D+ with (y ∗ , z ∗ ) ̸= (0, 0) such that inf (⟨F (x), y ∗ ⟩ + ⟨G(x), z ∗ ⟩) = ⟨y0 , y ∗ ⟩,
x∈A
inf ⟨G(x0 ), z ∗ ⟩ = 0. Proof According to Definition 4.1, we have (y0 − F (S)) ∩ icr(C) = ∅.
(41)
Clearly, I1 (x) = I(x) − (y0 , 0), ∀x ∈ A. We assert that −I1 (x) ∩ icr(C × D) = ∅, ∀x ∈ A.
18
(42)
Otherwise, there exists x ∈ A such that −I1 (x) ∩ icr(C × D) ̸= ∅. By Lemma 2.6, icr(C × D) = icr(C) × icr(D). Therefore, −I1 (x) ∩ (icr(C) × icr(D)) ̸= ∅.
(43)
(y0 − F (x)) ∩ icr(C) ̸= ∅
(44)
−G(x) ∩ icr(D) ̸= ∅.
(45)
By (43), we obtain
and
It follows from (45) that x ∈ S. Thus, by (44), we have (y0 − F (S)) ∩ icr(C) ̸= ∅, which contradicts (41). Therefore, (42) holds. By Theorem 4.1, there exists (y ∗ , z ∗ ) ∈ C + × D+ with (y ∗ , z ∗ ) ̸= (0, 0) such that ⟨I1 (x), (y ∗ , z ∗ )⟩ ≥ 0, ∀x ∈ A, i.e., ⟨F (x), y ∗ ⟩ + ⟨G(x), z ∗ ⟩ ≥ ⟨y0 , y ∗ ⟩, ∀x ∈ A.
(46)
Since x0 ∈ S, there exists p ∈ G(x0 ) such that −p ∈ D. Because z ∗ ∈ D+ , we obtain ⟨p, z ∗ ⟩ ≤ 0. Taking x = x0 in (46), we get ⟨y0 , y ∗ ⟩ + ⟨p, z ∗ ⟩ ≥ ⟨y0 , y ∗ ⟩ It follows that ⟨p, z ∗ ⟩ ≥ 0. So, ⟨p, z ∗ ⟩ = 0. Thus, we have ⟨y0 , y ∗ ⟩ ∈ ⟨F (x0 ), y ∗ ⟩ + ⟨G(x0 ), z ∗ ⟩. 19
(47)
Therefore, it follows from (46) and (47) that inf (⟨F (x), y ∗ ⟩ + ⟨G(x), z ∗ ⟩) = ⟨y0 , y ∗ ⟩.
x∈A
Taking again x = x0 in (46), we obtain ⟨y0 , y ∗ ⟩ + ⟨G(x0 ), z ∗ ⟩ ≥ ⟨y0 , y ∗ ⟩. So, ⟨G(x0 ), z ∗ ⟩ ≥ 0. We have shown that there exists p ∈ G(x0 ) such that ⟨p, z ∗ ⟩ = 0. Thus, we have inf ⟨G(x0 ), z ∗ ⟩ = 0.
The following example shows that the conditions of Theorem 4.2 can be satisfied. Example 4.2 Let X = Y = Z = R2 , C = D = {(y1 , 0)|y1 ≥ 0} and A = {(1, 0), (1, 2)}. The set-valued map F : X → 2Y on A is defined as follows: F (1, 0) = {(y1 , y2 )|y1 = 1, 0 ≤ y2 ≤ 1}, F (1, 2) = {(y1 , y2 )|y1 > 1, 0 ≤ y2 ≤ −y1 + 2}. The set-valued map G : X → 2Y on A is defined as follows: G(1, 0) = {(y1 , y2 )|y1 ≤ 0, 0 ≤ y2 ≤ y1 + 1}, G(1, 2) = {(y1 , y2 )|y1 ≥ −1, y1 + 1 ≤ y2 ≤ 1}. Let x0 = (1, 0) and y0 = (1, 0) ∈ F (x0 ). Clearly, all conditions of Theorem 4.2 are satisfied. Therefore, there exist y ∗ : ⟨(y1 , y2 ), y ∗ ⟩ = y1 + y2 and z ∗ : ⟨(y1 , y2 ), z ∗ ⟩ = −y1 + y2 such that inf (⟨F (x), y ∗ ⟩ + ⟨G(x), z ∗ ⟩) = ⟨y0 , y ∗ ⟩
x∈A
and inf ⟨G(x0 ), z ∗ ⟩ = 0. Remark 4.4 Theorem 4.2 improves Theorem 3.2 of [1], Theorem 3.1 of [3] and Theorem 4.2 of [9]. 20
We also show a sufficient optimality condition for (VP) as follows: THEOREM 4.3 Suppose that the following conditions hold: (i) x0 ∈ S; (ii) there exist y0 ∈ F (x0 ) and (y ∗ , z ∗ ) ∈ C +i × D+ such that inf (⟨F (x), y ∗ ⟩ + ⟨G(x), z ∗ ⟩) ≥ ⟨y0 , y ∗ ⟩.
x∈A
Then (x0 , y0 ) is a weak minimizer of (VP). Proof By condition (ii), we have ⟨F (x) − y0 , y ∗ ⟩ + ⟨G(x), z ∗ ⟩ ≥ 0, ∀x ∈ A.
(48)
We assert (x0 , y0 ) is a weak minimizer of (VP). Otherwise, there exists x ∈ S such that (y0 −F (x))∩icr(C) ̸= ∅. Therefore, there exists y ∈ F (x) such that y0 −y ∈ icr(C) ⊆ C \{0}. Thus, we obtain ⟨y − y0 , y ∗ ⟩ < 0.
(49)
Since x ∈ S, there exists q ∈ G(x) such that −q ∈ D. Hence, ⟨q, z ∗ ⟩ ≤ 0.
(50)
Adding (49) to (50), we have ⟨y − y0 , y ∗ ⟩ + ⟨q, z ∗ ⟩ < 0, which contradicts (48). Therefore, (x0 , y0 ) is a weak minimizer of (VP).
The following example shows that the conditions of Theorem 4.3 can be satisfied. Example 4.3 Let X = Y = Z = R2 , C = D = {(y1 , 0)|y1 ≥ 0} and A = {(1, 0), (1, 2)}. The set-valued map F : X → 2Y on A is defined as follows: F (1, 0) = {(y1 , y2 )|y1 ≥ 1, y1 ≤ y2 ≤ 2}, F (1, 2) = {(y1 , y2 )|y1 ≤ 2, 1 ≤ y2 ≤ y1 }. 21
The set-valued map G : X → 2Y on A is defined as follows: G(1, 0) = {(y1 , y2 )| − 1 ≤ y1 ≤ 0, y2 = 0}, G(1, 2) = {(y1 , y2 )| − 1 ≤ y1 ≤ 0, 0 ≤ y2 ≤ 1}. Let x0 = (1, 0), y0 = (1, 1) ∈ F (x0 ), ⟨(y1 , y2 ), y ∗ ⟩ = y1 + y2 and ⟨(y1 , y2 ), z ∗ ⟩ = −y1 . Clearly, all conditions of Theorem 4.3 are satisfied. Therefore, (x0 , y0 ) is a weak minimizer of (VP). Acknowledgements This work was supported by the National Nature Science Foundation of China (Grant No. 10831009) and the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240).
References [1] Z.M. Li, The optimality conditions for vector optimization of set-valued maps, J. Math. Anal. Appl. 237 (1999), pp. 413-424. [2] Z.M. Li, A theorem of the alternative and its Application to the optimization of set-valued maps, J. Optim. Theory Appl. 100 (1999), pp. 365-375. [3] Y.W. Huang and Z.M. Li, Optimality condition and Lagrangian multipliers of vector optimization with set-valued maps in linear spaces, Oper. Res. Tran. 5 (2001), pp. 63-69. [4] Y.W. Huang, Generalized cone-subconvexlike set-valued maps and applications to vector optimization, J. Chongqing. University (Eng.Ed). 1 (2002), pp. 67-71. [5] M. Ad´an and V. Novo, Partial and generalized subconvexity in vector optimization problems, J. Convex. Anal. 8 (2001), pp. 583-594. [6] M. Ad´an and V. Novo, Efficient and weak efficient points in vector optimization with generalized cone convexity, Appl. Math. Lett. 16 (2003), pp. 221-225. 22
[7] M. Ad´an and V. Novo, Weak efficiency in vector optimization using a closure of algebraic type under cone-convexlikeness, Eur. J. Oper. Res. 149 (2003), pp. 641-653. [8] M. Ad´an and V. Novo, Proper efficiency in vector optimization on real linear spaces, J. Optim. Theory Appl. 121 (2004), pp. 515-540. [9] E. Hern´andez, B. Jim´enez and V. Novo, Weak and proper efficiency in set-valued optimization on real linear spaces, J. Convex. Anal. 14 (2007), pp. 275-296. [10] J.B.G. Frenk and G. Kassay, Lagrangian duality and cone convexlike functions, J. Optim. Theory. Appl. 134 (2007), pp. 207-222. [11] W.D. Rong and Y.N. Wu, Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps, Math. Method. Oper. Res. 48 (1998), pp. 247-258. [12] X.M. Yang, X.Q. Yang and G.Y. Chen, Theorems of the alternative and optimization with set-valued maps, J. Optim. Theory Appl. 107 (2000), pp. 627-640. [13] X.M. Yang, D. Li and S.Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions, J. Optim. Theory Appl. 110 (2001), pp. 413-427. [14] J.V. Tiel, Convex analysis: An Introductory Text, John Wiley and Sons Inc, New York, 1984. [15] S.Z. Shi, Convex analysis, Shanghai scientical and technical press, Shanghai, 1990 (in Chinese). [16] Y.D. Hu and Z.Q. Meng, Convex analysis and nonsmooth analysis, Shanghai scientical and technical press, Shanghai, 2000 (in Chinese).
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