Optimality conditions for vector optimization problems with generalized order

The aim of this paper is to present new optimality conditions for vector optimization problems with generalized order by using the extremal principle.The aim of this paper is to present new optimality conditions for vector optimization problems with generalized order by using the extremal principle.

Noname manuscript No (will be inserted by the editor) Optimality conditions for vector optimization problems with generalized order N V Tuyen Received: date / Accepted: date Abstract The aim of this paper is to present new optimality conditions for vector optimization problems with generalized order by using the extremal principle Keywords Vector optimization · Generalized order · Optimality conditions Mathematics Subject Classification (2000) 90C29 · 90C46 · 49J53 Introduction Kruger and Mordukhovich [16, Definition 5.53] have introduced the new concept of the locally (f ; Θ)-optimal solution, where f is a single-valued mapping between Banach spaces and the ordering set Θ (may not be convex and/or conic) containing the origin This notion is directly induced by the concept of local extremal points for systems of sets and covers all the traditional notions of optimality in vector optimization To the best of our knowledge, there are a few works studying the necessary and sufficient optimality conditions for optimality solutions to vector optimization problems with generalized order (see, e.g., [8, 16, 17]) In [16, Theorem 5.59], Mordukhovich established some preliminary necessary conditions to vector optimization problems with geometric contraints Bao [8] used subdifferentials of set-valued mapping to establish This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No 101.01-2014.39 A part of this work was done when the author was working at the Vietnam Institute for Advanced Study in Mathematics (VIASM) He would like to thank the VIASM for providing a fruitful research environment and working condition N V Tuyen Department of Mathematics, Hanoi Pedagogical Institute No 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam E-mail: tuyensp2@yahoo.com 2 N V Tuyen some necessary conditions to set-valued optimization problems with equilibrium contraints In [17], Tuyen and Yen established some sufficient conditions for a point satisfying the necessary optimality condition [16, Theorem 5.59] to be a generalized order solution of the vector optimization problem under convexity assumptions Recently, Bao and Mordukhovich [3, 4, 7] gave some new sufficient conditions for global weak Pareto (global Pareto) solutions to setvalued optimization problems However, we are not familiar with any sufficient optimality conditions for nonconvex vector optimization problems with generalized order This motivates us to study the sufficient optimality conditions for vector optimization problems with respect to generalized order optimality The rest of this paper is organized as follows Section investigates the notion of generalized order optimality and preliminaries from variational analysis In Section 3, we establish some necessary and sufficient conditions for global generalized solutions in vector optimization Preliminaries Let Z be a Banach space and Θ ⊂ Z be a set containing origin The topological interior, topological closure, the relative interior and the affine hull of Θ are denoted respectively by intΘ, cl Θ (or A), ri Θ, and aff Θ The dual space of Z is denoted by Z ∗ The weak∗ topology in Z ∗ is denoted by w∗ The closed unit ball in Z is abbreviated to B Let A ⊂ Z be given arbitrarily A point z is said to be a boundary point of A if every neighborhood U of z, we have U ∩ A = ∅ and U ∩ AC = ∅, with AC := Z\A The set of all the boundary points of A denoted by bd (A) Definition A point z¯ ∈ A is said to be a generalized efficient point of A with respect to Θ, if there is a sequence {zk } ⊂ Z with zk → as k → ∞ such that A ∩ (Θ + z¯ − zk ) = ∅, ∀k ∈ N (1) The set of all the generalized efficient points of A with respect to Θ is denoted by GE(A | Θ) In Definition we don’t assume that Θ is a convex cone, and we also don’t require that the interior of Θ is nonempty If Θ is a convex cone with riΘ = ∅, then the above optimality concept covers the conventional concept of optimality Definition Suppose that Θ is a convex cone with riΘ = ∅ A point z¯ ∈ A is said to be a (Slater) relative efficient point of A with respect to the order generated by cone Θ, if A ∩ (¯ z + ri Θ) = ∅ (2) The set of all the relative efficient points of A is denoted by RE(A | Θ) Optimality conditions for vector optimization problems with generalized order Let Θ be a convex cone in Z The cone Θ induces a partial order in Z as follows: z1 , z2 ∈ Z, z1 ≤ z2 if z1 − z2 ∈ Θ Definition Let A be a nonempty subset in Z (a) Suppose that intΘ = ∅ A point z¯ ∈ A is said to be a weak efficient point of A with respect to Θ, if A ∩ (¯ z + intΘ) = ∅ The set of weak efficient points of A is denoted by W E(A | Θ) (b) A point z¯ ∈ A is said to be an efficient point of A with respect to Θ, if (¯ z ≥ y, for some y ∈ A) ⇒ (y ≥ z¯) The set of efficient points of A is denoted by E(A | Θ) Proposition (see [17, Proposition 2.11]) If Θ is a convex cone, then the following holds: (i) If intΘ = ∅, then GE(A | Θ) ⊂ W E(A | Θ) (ii) If intΘ = ∅, then W E(A | Θ) ⊂ RE(A | Θ) (iii) If riΘ = ∅, then RE(A | Θ) ⊂ GE(A | Θ) Thus, if Θ is a convex cone with nonempty interior, then W E(A | Θ) = RE(A | Θ) = GE(A | Θ) (3) Proposition (see [11, Proposition 2]) Suppose that Θ is a convex cone with Θ\l(Θ) = ∅ If z¯ is an efficient point of A with respect to Θ, then z¯ is a generalized efficient point of A with respect to Θ, or E(A | Θ) ⊂ GE(A | Θ) (4) Proposition (see [11, Theorem 2.3]) Let Z be a Banach space, A be a nonempty set in Z, and ∈ Θ ⊂ Z Then GE(A | Θ) = A ∩ bd (A − Θ) (5) Proof Let z¯ ∈ GE(A | Θ) Then, there exists (zk ) ⊂ Z with zk → as k → ∞ such that z¯ − zk ∈ / (A − Θ) for all k ∈ N Thus z¯ − zk ∈ (A − Θ)C for all k ∈ N Let U be an arbitrary neighborhood of z¯ From z¯ ∈ A and ∈ Θ we have z¯ ∈ (A − Θ) Thus U ∩ (A − Θ) = ∅ Since lim (¯ z − zk ) = z¯, we k→∞ have z¯ − zk ∈ U for large enough k Thus z¯ − zk ∈ U ∩ (A − Θ)C for large enough k It follows that U ∩ (A − Θ)C = ∅ Therefore z¯ ∈ bd (A − Θ) This shows that GE(A | Θ) ⊂ A ∩ bd (A − Θ) To prove the converse inclusion, let z¯ ∈ A ∩ bd (A − Θ) Since z¯ ∈ bd (A − Θ), we have B z¯, k ∩ (A − Θ)C = ∅ ∀k ∈ N, where B z¯, k1 := z ∈ Z | z − z¯ ≤ k For each k ∈ N, let xk ∈ B z¯, k1 ∩ N V Tuyen (A − Θ)C We have lim xk = z¯ and {xk } ⊂ (A − Θ)C For each k ∈ N, put k→∞ zk = z¯ − xk , then lim zk = and z¯ − zk = xk ∈ (A − Θ)C ∀k ∈ N, k→∞ or z¯ − zk ∈ / (A − Θ) ∀k ∈ N This shows that z¯ is a generalized efficient point of A with respect to Θ Therefore A ∩ bd (A − Θ) ⊂ GE(A | Θ) The proof is complete ✷ Next, we recall some basic notions that will be used later Let F : Z ⇒ Z ∗ be a multifunction from Z to Z ∗ The Painlev´e-Kuratowski upper limit at z¯ of F with respect to the norm topology of Z and the weak∗ topology of Z ∗ is defined by w∗ Lim sup F (z) := {z ∗ ∈ Z ∗ | ∃ zk → z¯, zk∗ −→ z ∗ , zk∗ ∈ F (zk ) ∀k ∈ N} z→¯ z Definition (see [15, Difinition 1.1]) Let Ω ⊂ Z, z¯ ∈ Ω, and (i) The set of -normals of Ω at z¯ is defined by ˆ (¯ N z ; Ω) := z ∗ ∈ Z ∗ | lim sup Ω z →¯ z z ∗ , z − z¯ z − z¯ , Ω where the notation z −→ z¯ means that z → z¯ and z ∈ Ω We call the closed ˆ (¯ ˆ0 (¯ convex cone N z ; Ω) := N z ; Ω) the Fr´echet normal cone of Ω at z¯ (ii) The Mordukhovich normal cone or the limiting normal cone of Ω at z¯ is the set N (¯ z , Ω) = Lim sup N (x, Ω), z→¯ z ↓0 that is, N (¯ z ; Ω) = z ∗ ∈ Z ∗ | ∃ ∗ k Ω w ˆ (zk ; Ω) ∀k → 0+ , zk → z¯, zk∗ → z ∗ , zk∗ ∈ N k (6) Definition (see [15, Difinition 1.20]) A set Ω ⊂ Z is said to be sequentially Ω normally compact (SNC) at z¯ if for any sequences k ↓ 0, zk −→ z¯ and zk∗ ∈ ˆ (zk , Ω) it holds N k w∗ zk∗ −−→ as k → =⇒ where k zk∗ → as k → , can be omitted if Z is Asplund and Ω is locally closed around z¯ Recall that a Banach space is Asplund if every convex continuous function ϕ : U → R defined on an open convex subset U of Z is Fr´echet differentiable on a dense subset of U The class of Asplund spaces is quite broad including every reflexive Banach space and every Banach space with a separable dual are Asplund spaces It is known from [15, Theorem 1.21] that a convex set Optimality conditions for vector optimization problems with generalized order is fulfilled the SNC condition if it has a nonempty relative interior and the closure of affine hull of Θ has a finite-codimensions The codimension of aff Θ is defined as the dimension of the quotient space X/(aff Θ −θ), for some θ ∈ Θ, and is denoted by codim aff Θ Definition (see [15, Definition 2.1]) Let Ω1 , Ω2 be nonempty subsets of a Banach space Z and z¯ ∈ Ω1 ∩ Ω2 We say that z¯ is a global extremal point of the system {Ω1 , Ω2 } in Z if there exists a sequence (ak ) such that ak → as k → ∞ and Ω1 ∩ (Ω2 − ak ) = ∅ ∀k ∈ N (7) In this case {Ω1 , Ω2 , z¯} is said to be an extremal system in Z Theorem (The Extremal Principle, see [15, Theorem 2.20]) If z¯ is an extremal point of the closed set system {Ω1 , Ω2 } in the Asplund space Z, then it satisfies the following relationships: for every > 0, there are xi ∈ Z and x∗i ∈ Z ∗ satisfying ˆ (xi , Ωi ) f or i = 1, 2, xi ∈ Ωi ∩ (¯ z + B), x∗i ∈ N x∗1 + x∗2 ≤ , and − ≤ x∗1 + x∗2 ≤ + Main results Theorem (Necessary condition) Let Z be an Asplund space, and let ∅ = A ⊂ Z, ∈ Θ ⊂ Z be closed subsets If z¯ ∈ GE(A | Θ), then there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A) ∩ (−N (0; Θ)) (8) provided that either A is SNC at z¯ or Θ is SNC at Proof Put Ω1 := Z × (A − Θ), Ω2 := A × {¯ z } Obviously, (¯ z , z¯) ∈ Ω1 ∩ Ω2 We claim that {Ω1 , Ω2 , (¯ z , z¯)} is an extremal system Indeed, from z¯ ∈ GE(A | Θ) it follows that there exists a sequence (zk ) ⊂ Z such that zk → 0, and A ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N, or, equivalent to (A − Θ) ∩ (¯ z − zk ) = ∅ ∀k ∈ N (9) Put ak := (0, zk ) We have ak → as k → ∞ and Ω1 ∩ (Ω2 − ak ) = ∅ ∀k ∈ N (10) Arguing by contradiction, suppose that (10) does not hold for some k0 ∈ N This mean that Ω1 ∩ (Ω2 − ak0 ) = ∅ 6 N V Tuyen Then, there exist a ∈ A and θ ∈ Θ such that a − θ = z¯ − zk0 , contrary to (9) Thus {Ω1 , Ω2 , (¯ z , z¯)} is an extremal system Employing the extremal principle from Theorem to the system {Ω1 , Ω2 , (¯ z , z¯)} shows that for ∗ each k ∈ N, there are elements (uik , zik ) and (u∗ik , −zik ) for i = 1, satisfying the relationships  z , z¯),  (uik , zik ) ∈ Ωi with (uik , zik ) → (¯   and (u∗ , −z ∗ ) ∈ N ˆ ((uik , zik ), Ωi ) , i = 1, 2, ik ik (11) ∗ ∗  with (u∗1k , z1k ) + (u∗2k , z2k ) → 1,    ∗ ∗ ) + (u∗2k , z2k ) → and (u∗1k , z1k From ∗ ˆ ((u2k , z2k ), Ω2 ) (u∗2k , −z2k )∈N ˆ ((u2k , z2k ), A × {¯ ∈N z }) ˆ (u2k , A) × N ˆ (¯ =N z , {z}) ∗ ˆ = N (u2k , A) × Z (12) ˆ (u2k , A) Since it follows that u∗2k ∈ N ∗ ˆ ((u1k , z1k ), Ω1 ) (u∗1k , −z1k )∈N ˆ ((u1k , z1k ), Z × (A − Θ)), =N (13) we have ∗ ∗ ˆ ((u1k , 0), Z × (−Θ)) (u∗1k − z1k , −z1k )∈N ˆ (u1k , Z) × N ˆ (0, −Θ) =N ˆ (0, Θ)) = {0} × (−N (14) ∗ ∗ ˆ (0, Θ) The Asplund property of Hence, we get u∗1k − z1k = and z1k ∈ N ∗ ∗ ∗ , u∗2k , z2k ) in (11) allow us to Z and the boundedness of the sequence (u1k , z1k ∗ ∗ ∗ ∗ find a quadruple (u1 , z1 , u2 , z2 ) such that w∗ ∗ (u∗ik , −zik ) → (u∗i , −zi∗ ) for i = 1, along some subsequences Employing (11)–(14)gives us (u∗1 , −z1∗ ) = (−u∗2 , z2∗ ) = (z ∗ , −z ∗ ), −z ∗ ∈ N (¯ z , A) and z ∗ ∈ N (0, Θ) To complete the proof of the theorem, it remains to show that z ∗ = in (8) under assumeed SNC Arguing by contradiction, suppose that z ∗ = The imposed SNC assumptions give us two cases: Optimality conditions for vector optimization problems with generalized order ∗ ˆ (u2k , A), u2k → z¯ and u∗ w Case1 : A is SNC at z¯ From u∗2k ∈ N 2k → it follows that u∗2k → Since u∗1k ≤ u∗1k + u∗2k + u∗2k ∗ ∗ ≤ (u∗1k , z1k ) + (u∗2k , z2k ) + u∗2k for all k ∈ N, (15) (16) ∗ ∗ we have u∗1k → as k → ∞ Thus z1k → and z2k → This contradicts the nontriviality in the third line of (11) ∗ ∗ ˆ (0, Θ) for all k ∈ N, Case2 : Θ is SNC at From u∗1k = z1k , z1k ∈ N w∗ ∗ ∗ and z1k → imply that z1k → and u∗1k ∗ ∗ (u2k , z2k ) → 0, a contrary again → Since (11), we have ✷ Corollary Let Z be an Asplund space, and let ∅ = A ⊂ Z be a closed subset Suppose that Θ is a closed convex cone and ri Θ = ∅ If z¯ ∈ RE(A | Θ) then there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A) ∩ (−N (0; Θ)) (17) provided that either A is SNC at z¯ or codim aff Θ < ∞ Corollary Let Z be an Asplund space, and let ∅ = A ⊂ Z be a closed subset Suppose that Θ is a closed convex cone and intΘ = ∅ If z¯ ∈ W E(A | Θ) then there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A) ∩ (−N (0; Θ)) (18) provided that either A is SNC at z¯ or codim aff Θ < ∞ Corollary Let Z be an Asplund space, and let ∅ = A ⊂ Z be a closed subset Suppose that Θ is a closed convex cone and Θ\l(Θ) = ∅ If z¯ ∈ E(A | Θ) then there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A) ∩ (−N (0; Θ)) (19) provided that either A is SNC at z¯ or Θ is SNC at Proof We have E(A | Θ) ⊂ GE(A | Θ) by Proposition Assertion (19) is immediate from Theorem ✷ Corollary Let Z be an Asplund space and ∅ = A ⊂ Z, ∈ Θ ⊂ Z, z¯ ∈ GE(A | Θ) Suppose that A − Θ and Θ are closed subsets and the following condition Θ+Θ =Θ (20) holds true Then, there exists z ∗ ∈ Z ∗ such that = −z ∗ ∈ N (¯ z , A − Θ) ∩ (−N (0, Θ)) provided that either A − Θ is SNC at z¯ or Θ is SNC at (21) N V Tuyen Proof We first show that if the condition (20) is satisfied, then GE(A | Θ) ⊂ GE(A − Θ | Θ) (22) Indeed, suppose that z¯ ∈ GE(A | Θ) Then, there exists a sequence (zk ) ∈ Z such that zk → as k → ∞ and A ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N From this and (20) imply that A ∩ (Θ + Θ + z¯ − zk ) = ∅ ∀k ∈ N, or, equivalent to (A − Θ) ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N Thus z¯ ∈ GE(A − Θ | Θ) By Theorem 2, there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A − Θ) ∩ (−N (0, Θ)) ✷ The proof is complete Theorem (Necessary and sufficient condition) Let Z be an Asplund space, ∅ = A ⊂ Z, ∈ Θ ⊂ Z and z¯ ∈ A Assume that A − Θ is a closed subset in Z and either A − Θ is SNC at z¯ or dim Z < ∞ Then, z¯ ∈ GE(A | Θ) if and only if there exists z ∗ ∈ Z ∗ satisfying = −z ∗ ∈ N (¯ z , A − Θ) ∩ (−N (0, Θ)) (23) Proof (⇒): Suppose that z¯ ∈ GE(A | Θ) Then, there exists a sequence zk → as k → ∞ satisfying A ∩ (Θ + z¯ − zk ) = ∅ ∀k ∈ N, or (A − Θ) ∩ (¯ z − zk ) = ∅ ∀k ∈ N (24) The equation (24) implies that {A − Θ, {¯ z }, z¯} is an extremal system in Z By [15, Theorem 1.21], {¯ z } is SNC if and only if dim Z < ∞ Thus A − Θ or {¯ z } is SNC at z¯ Clearly, the singleton set {¯ z } is a closed subset in Z Theorem 2.22 [15] now shows that the exact extremal principle holds for {A−Θ, {¯ z }, z¯} Thus there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A − Θ) ∩ N (¯ z , {¯ z }), (25) −z ∗ ∈ N (¯ z , A − Θ) (26) or, equivalent to As in the proof of [5, Theorem 3.1], equation (26) gives −z ∗ ∈ (−N (0, Θ)) Optimality conditions for vector optimization problems with generalized order Hence, there exists z ∗ ∈ Z ∗ such that = −z ∗ ∈ N (¯ z , A − Θ) ∩ (−N (0, Θ)) (⇐): Arguing by contradiction, assume that there is z ∗ ∈ Z ∗ such that = −z ∗ ∈ N (¯ z , A − Θ) ∩ (−N (0, Θ)) , (27) but z¯ ∈ / GE(A | Θ) Therefore z¯ ∈ / bd (A − Θ) by Lemma From this and z¯ ∈ (A − Θ) imply that z¯ ∈ int(A − Θ) Thus N (¯ z , A − Θ) = {0}, contrary to (27) The proof is complete ✷ Example Let Z = R2 , A = {z = (z1 , z2 ) ∈ R2 | z2 = −z1 , ≤ z1 ≤ 1}, Θ = {z = (z1 , z2 ) ∈ R2 | z2 = −z1 , z1 ≤ 0} ∪ {z = (z1 , z2 ) ∈ R2 | z2 ≤ − |z1 | , −1 ≤ z1 ≤ 1}, and z¯ = (0, 0) ∈ A It is easy to see that Θ is neither convex nor conic We have A − Θ = {z ∈ R2 | z2 ≥ |z1 |, −1 ≤ z1 ≤ 1} ∪ {z ∈ R2 | z1 − ≤ z2 ≤ z1 , z1 ≤ 1} ∪ {z ∈ R2 | z1 ≤ −|z2 | + 2, ≤ z1 ≤ 2} From this we obtain N (¯ z ; A − Θ) = {z = (z1 , z2 ) ∈ R2 | z1 = −|z2 |}, and N (¯ z ; Θ) = {z = (z1 , z2 ) ∈ R2 | z2 = z1 } ∪ {z = (z1 , z2 ) ∈ R2 | z2 = −z1 , z1 ≤ 0} We have N (¯ z , A − Θ) ∩ (−N (0, Θ)) = {z = (z1 , z2 ) ∈ R2 | z1 = −|z2 |} Thus z¯ ∈ GE(A | Θ) by Theorem Now we compare Theorem with [16, Theorem 89], which characterizes the linear suboptimality of set systems via the relations of the exact extremal principle Given two subsets Ω1 and Ω2 of a Banach space Z Put ϑ(Ω1 , Ω2 ) := sup{υ ≥ | υB ⊂ Ω1 − Ω2 } (28) The constant ϑ(Ω1 , Ω2 ) describing the measure of overlapping for these sets Ω1 and Ω2 Note that one has ϑ(Ω1 , Ω2 ) = −∞ if Ω1 ∩ Ω2 = ∅ It is easy to observe that a point z¯ ∈ Ω1 ∩ Ω2 is locally extremal for the set system {Ω1 , Ω2 } if and only if ϑ(Ω1 ∩ Br (¯ z ), Ω2 ∩ Br (¯ z )) = for some r > (29) 10 N V Tuyen Definition (see [16, Definition 5.87]) Given Ω1 , Ω2 ⊂ Z and z¯ ∈ Ω1 ∩ Ω2 We say that the set system {Ω1 , Ω2 } is linearly subextremal around the point z¯ if ϑlin (Ω1 , Ω2 , x ¯) = 0, where ϑlin (Ω1 , Ω2 , z¯) := lim inf Ω x1 −→¯ z ϑ ([Ω1 − x1 ] ∩ rB, [Ω2 − x2 ] ∩ rB) , r (30) Ω x2 −→¯ z r↓0 where the measure of overlapping ϑ(·, ·) is defined in (28) The next result characterizes the linear suboptimality of set systems Theorem (see [16, Theorem 89]) Let Ω1 and Ω2 be two subsets in a Asplund space Z Assume that {Ω1 , Ω2 } ⊂ Z is a linearly subextremal around z¯ ∈ Ω1 ∩ Ω2 , that the sets Ω1 , Ω2 are locally closed around z¯, and that one of them is SNC at this point Then there is z ∗ ∈ Z ∗ satisfying = z ∗ ∈ N (¯ z ; Ω1 ) ∩ (−N (¯ z ; Ω2 )) (31) Furthermore, condition (31) is necessary and sufficient for the linear subextremality of {Ω1 , Ω2 } around z¯ if dim Z < ∞ The following examples show that the sufficient condition in Theorem is not sufficient for an extremal system even in the finite dimensional case Example Let X = R2 , A = (1, 0) + B, Θ = (−2, 0) + 2B, z¯ = (0, 0) Clearly, Θ is not a cone in R2 An easy computation shows that N ((0, 0); A) = R− × {0}, N ((0, 0); Θ) = R+ × {0} This implies that N ((0, 0); A) ∩ − N ((0, 0); Θ) = R− × {0} = {0} Thus the system {A, Θ} is linearly subextremal around the point z¯ by Theorem However, z¯ is not an extremal point of the system {A, Θ} Indeed, we have A − Θ = (1, 0) + 3B Thus N (¯ z , A − Θ) = {0} Theorem now shows that z¯ is not an extremal point of the system {A, Θ} Example Let Z = R2 , A = {z = (z1 , z2 ) ∈ R2 | z1 = 0, −1 ≤ z2 ≤ 0}, and Θ = {z = (z1 , z2 ) ∈ R2 | z2 = z1 } ∪ {z = (z1 , z2 ) ∈ R2 | z2 = −z1 } ∪ {z = (z1 , z2 ) ∈ R2 | z2 < − |z1 |} It is easy to see that Θ is a nonconvex cone, and z¯ = (0, 0) ∈ A ∩ Θ We have N (¯ z ; A) = {z = (z1 , z2 ) ∈ R2 | z2 ≥ 0}, and N (¯ z ; Θ) = {z = (z1 , z2 ) ∈ R2 | z2 = z1 } ∪ {z = (z1 , z2 ) ∈ R2 | z2 = −z1 } Optimality conditions for vector optimization problems with generalized order 11 Thus N (¯ z ; A) ∩ (−N (¯ z ; Θ)) = {z = (z1 , z2 ) ∈ R2 | z2 = |z1 |} By Theorem 4, {A, Θ} is linearly subextremal around the point z¯ However, z¯ is not an extremal point of the system {A, Θ} Indeed, we have A−Θ = {z ∈ R2 | z2 = z1 −1}∪{z ∈ R2 | z2 = −z1 −1}∪{z ∈ R2 | z2 > |z1 |−1} It is easy to check that z¯ ∈ int(A − Θ) This implies that N (¯ z , A − Θ) = {0} Thus z¯ ∈ / GE(A | Θ) by Theorem Theorem (Sufficient condition under convexity assumption, see [17, Theorem 4.3]) Let Z be a Banach space, ∅ = A ⊂ Z, ∈ Θ ⊂ Z Suppose that Θ, A − Θ are convex subsets and intΘ = ∅ If there exists = z ∗ ∈ Z ∗ such that −z ∗ ∈ N (¯ z , A − Θ), (32) then z¯ ∈ GE(A | Θ) Proof Since A − Θ is a convex set in Z, the Mordukhovich normal cone of A − Θ at z¯ coincides with the normal cone of A − Θ in the sense of convex analysis; that is, −z ∗ ∈ N (¯ z , A − Θ) if and only if z ∗ , z − z¯ ≥ ∀z ∈ A − Θ (33) From definition of the Mordukhovich normal cone and (33) imply that z ∗ ∈ N (0; Θ) We have to prove that z¯ is a generalized efficient point of A with respect to Θ Arguing by contradiction, assume that z¯ ∈ / GE(A | Θ) Since z0 for all k ∈ N Since zk → as intΘ = ∅, let z0 ∈ intΘ and put zk := − k k → ∞ and z¯ ∈ / GE(A | Θ) imply that there exists k0 ∈ N such that A ∩ (Θ + z¯ − zk0 ) = ∅ Thus there are z ∈ A and θ ∈ Θ satisfying a = θ + z¯ − zk0 Put z := a − θ, we have z ∈ A − Θ and z0 (34) z − z¯ = k0 Substituting (34) into the left side of (33) we obtain z ∗ , z0 ≥ (35) Since z0 ∈ intΘ, it follows that there exists > such that B(z0 , ) ⊂ Θ, where B(z0 , ) := {z ∈ Z | z − z0 ≤ } 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optimization problems with generalized order This motivates... sufficient optimality conditions for vector optimization problems with respect to generalized order optimality The rest of this paper is organized as follows Section investigates the notion of generalized. .. Banach space with a separable dual are Asplund spaces It is known from [15, Theorem 1.21] that a convex set Optimality conditions for vector optimization problems with generalized order is fulfilled