On the convergence of relative Pareto efficient sets and the lower semicontinuity of relative Pareto efficient multifunctions

T. Q. Bao and B. S. Mordukhovich 2,3 have introduced some new concepts of extended Pareto efficient by using some kinds of relative interior replace for interior of ordering cones. The aim of this paper is to present new results on the convergence of relative Pareto efficient sets and the lower semicontinuity of relative Pareto efficient multifunctions under perturbations. Our results extend the results of Bednarczuk 4, Chuong and Yen 12, and Luc 18. Examples are given to illustrate the results obtained

Noname manuscript No. (will be inserted by the editor) On the convergence of relative Pareto efficient sets and the lower semicontinuity of relative Pareto efficient multifunctions N. V. Tuyen Received: date / Accepted: date Abstract T. Q. Bao and B. S. Mordukhovich [2,3] have introduced some new concepts of extended Pareto efficient by using some kinds of relative interior replace for interior of ordering cones. The aim of this paper is to present new results on the convergence of relative Pareto efficient sets and the lower semicontinuity of relative Pareto efficient multifunctions under perturbations. Our results extend the results of Bednarczuk [4], Chuong and Yen [12], and Luc [18]. Examples are given to illustrate the results obtained. Keywords Stability · Relative Pareto efficient · Kuratowski-Painlev´e convergence · Relative containment property · Lower semicontinuity. Mathematics Subject Classification (2000) 49K40 · 90C29 · 90C31. 1 Introduction To widen the applicability of the traditional solution concepts of scalar optimization and of vector optimization, Kruger and Mordukhovich (see [23, Subsection 5.5.18] and the related references) have introduced the notion of the locally (f ; Θ)-optimal solution, where f is a single-valued mapping between Banach spaces and Θ is a set (may not be convex and/or conic) containing the zero vector. In [29], Tuyen and Yen gave a detailed analysis of the notion of generalized order optimality and compared it with the traditional notions of 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 Pareto efficiency, weakly Pareto efficiency and Slater efficiency. Recently, Bao and Mordukhovich have introduced the concept of relative Pareto efficient to extended the concept of weak Pareto efficient (see [2, 3]) by using some kinds of relative interior replace for interior of ordering cones. This notions is weaker than the classical minimality with respect to a cone and is actually a generalization of weak minimality, which can be taken into consideration only if the interior of the ordering cone is nonempty. Recall that the relative interior of a convex set C in a Banach space Z, denoted ri C, is the interior of C relative to the closed affine hull of C. It is well known that ri C is nonempty for every nonempty convex set C in finite dimensions. However, it is not the case in many infinite-dimensional settings. To improve this situation, Borwein and Lewis [10] have introduced the notion of quasi relative interior of C ⊂ Z, denoted qri C, which is the set of all z ∈ C such that the closed conic hull of (C − z) is a linear subspace of Z. In [10, Theorem 2.19], the authors proved that qri C is nonempty for any nonempty closed and convex set C in a separable Banach space. Further properties of quasi relative interiors of convex sets in Banach spaces can be found in [10, 11] and the references therein. Stability analysis is one of the most important and interesting subjects and its role has been widely recognized in the theory of optimization. In the literature, two classical approaches can be found to study stability in vector optimization. One is to investigate the set-convergence of efficient sets of perturbed sets converging to a given set. Another is to study continuity properties of the optimal multifunctions. For instance, the lower (upper) semicontinuity of the optimal multifunctions have been examined by Penot [24]. Luc, Lucchetti and Malivert [18] investigated the stability of vector optimization in terms of the convergence of the efficient sets. Miglierina and Molho [20, 21] obtained some results on stability of convex vector optimization problems by considering the convergence of efficient sets. For more results concerning use of convexity in stability analysis, we refer readers to [17,19]. Various stability results on the optimal multifunctions were presented in the monographs [17, 25] and papers (see e.g. [4–8, 12, 13, 24]). Using the so-called domination property, containment property and dual containment property Bednarczuk [4–8] studied the Hausdorff upper semicontinuity, the C-Hausdorff upper semicontinuity and the lower (upper) semicontinuity of the efficient solution map and the efficient point multifunctions. Recently, by using the approach of Bednarczuk [4,6] and introducing the new concepts of local containment property, K-local domination property and uniformly local closedness of a multifunction around a given point, Chuong and Yen [12] obtain further results on the lower semicontinuity of efficient point multifunctions taking values in Hausdorff topological vector spaces. In this paper, following the ideas of [4, 12, 18] we study the stability of a vector optimization problem using the notion of relative Pareto efficiency. The rest of paper is organized as follows. Section 2 presents the notion of relative Pareto efficient and the relationships between this notions and the traditional solution concepts. In Section 3, we establish the upper part of convergence in the sense of Kuratowski-Painlev´e of relative Pareto efficient sets. In Section 4, Stability of relative Pareto efficient sets 3 we study the lower part of convergence in the sense of Kuratowski-Painlev´e of relative Pareto efficient sets. In Section 5, we propose new concepts called relative containment property, relative lower semicontinuous and relative upper Hausdorff semicontinuous of a multifunctions around a given point. Then, we derive some sufficient conditions for the lower semicontinuity of efficient point multifunctions under perturbations for cones with possibly empty interior. The new theorems extend the corresponding ones in [4, 12, 18]. Examples are given to illustrate the results obtained. 2 On the concept of extended Pareto efficient Let Z be a Banach space. For a set A ⊂ Z, the following notations will be used throughout: int A, cl A (or A), bd A, ri A, qri A and aff (A) stay for the interior, closure, boundary, relative interior, quasi relative interior and affine hull of A in Z. We denote by N (z) the set of all neighborhoods of z ∈ Z. By 0Z we denote the zero vector of Z. The closed unit ball in Z is abbreviated to B. The closed ball with center z and radius ρ denoted by B(x, ρ). Definition 1 (see [3, Definition 13.3]) Let Z be a Banach space with an ordering Θ containing the origin, A be a nonempty subset in Z and z¯ ∈ A. We say that (i) z¯ is a local extended Pareto efficient of A with respect to Θ, or it is a local Θ-efficient point of A if there exist a neighborhood V of z¯ such that A ∩ (¯ z − Θ) ∩ V = {¯ z }. (1) (ii) z¯ is a global Θ-efficient (or Θ-efficient for brevity) of A if we can choose V = Z in (1). Remark 1 (a) (see [3, Remark 13.1 (d)]) The notion Θ-efficient unifies all known kinds of Pareto-type optimality. Let A be a subset in a Banach space Z ordered by a closed and convex cone C ⊂ Z satisfying C\(−C) = ∅, i.e., C is not a linear subspace of Z, and let z¯ ∈ A. • z¯ is a Pareto efficient point of A (with respect to C) if A ∩ (¯ z − C) = {¯ z } or A ∩ (¯ z − C\{0}) = ∅. (2) • z¯ is a weak Pareto efficient point of A if A ∩ (¯ z − intC) = ∅ provided that intC = ∅. (3) • z¯ is an ideal Pareto efficient point of A if A ⊂ z¯ + C or A ∩ (¯ z − (Z\(−C))) = ∅. (4) • z¯ is a relative efficient point (or Slater efficient point) of A if A ∩ (¯ z − riC) = ∅ provided that riC = ∅, (5) 4 N. V. Tuyen where riC is the collection of interior points of C with respect to the closed affine hull of C. • z¯ is a quasi relative efficient point of A if A ∩ (¯ z − qriC) = ∅ provided that qriC = ∅, (6) where qriC is the collections of those points z ∈ C for which the set cl (cone (C − z)) is a linear subspace of Z. Obviously, a Pareto, weak Pareto, ideal, relative Pareto, and quasi relative efficient point in (3)–(6) can be unified by a Θ-efficient point, where Θ is, for each kind of efficient points, defined by • Pareto: Θ = C; • weak Pareto: Θ = int C ∪ {0}; • ideal Pareto: Θ = (Z\(−C)) ∪ {0}; • relative Pareto: Θ = ri C ∪ {0}; • quasi relative Pareto: Θ = qri C ∪ {0}; (b) Observe that each weak efficient point is a relative efficient point, since the condition int C = ∅ yeilds ri C = int C. Furthermore, each relative efficient point is a quasi relative efficient point, since the ri C = ∅ yields qri C = ri C. (c) The condition C\(−C) = ∅ is equivalent to 0 ∈ / ri C. (d) The set of all the relative Pareto efficient (weak Pareto efficient) points of A with respect to C is denoted by ReMin (A | C) (WMin (A | C)). The set of all the Pareto efficient poitnts of A with respect to C is denoted by Min (A | C). It is easy to see that Min (A | C) ⊂ ReMin (A | C). The opposite inclusion does not hold in general. However, if A is a rotund set, then we have Min (A | C) = ReMin (A | C). Definition 2 (see, e.g., [15, 21]) A nonempty convex set A ⊂ Z is said to be rotund when the boundary of A does not contain line segments. Proposition 1 Let Z be a Banach space and C be a convex cone with ri C = ∅ and C\(−C) = ∅. If A is a rotund set, then ReMin (A | C) = Min (A | C). (7) Proof By Remark 1(d), it suffices to prove that ReMin (A | C) ⊂ Min (A | C). Suppose to the contrary that there is an element z¯ ∈ ReMin (A | C)\Min (A | C). Then there exist z ∈ A such that z − z¯ ∈ (−C\(−ri C ∪ {0})). From z, z¯ ∈ A and rotundity of A imply that (¯ z , z] := {x ∈ Z | x = z¯ + α(z − z¯), α ∈ (0, 1]} Stability of relative Pareto efficient sets 5 does not lie entirely in the boundary of bd (A). Hence there exist α ¯ ∈ (0, 1] such that x ¯ := z¯ + α ¯ (z − z¯) ∈ int A. From α ¯ (z − z¯) ∈ (−C\(−ri C ∪ {0})), implies that x ¯ − z¯ ∈ (−C\(−ri C ∪ {0})). Since x ¯ ∈ int A, we can choose e ∈ −ri C such that x ¯ + e ∈ A. Then we have x ¯ + e − z¯ = (¯ x − z¯) + e ⊂ (−C\(−ri C ∪ {0})) − ri C ⊂ −C − ri C ⊂ −ri C, or x ¯ + e ∈ (¯ z − ri C). Thus (¯ x + e) ∈ A ∩ (¯ z − ri C), contrary to z¯ ∈ ReMin (A | C)\Min (A | C). The proof is complete. ✷ Corollary 1 (see [21, Proposition 4.3]) Let Z be a Banach space and C be a convex pointed cone with int C = ∅. If A is a rotund set, then WMin (A | C) = Min (A | C). In [16, 29], we provided an extended version of the notion of generalized order optimality in [23, Definition 5.53] and gave a detailed analysis of the notion of generalized order optimality and compared it with the traditional notions of Pareto efficiency, weakly Pareto efficiency and Slater efficiency. Definition 3 Let Z be a Banach space, A be a nonempty set in Z, and C ⊂ Z be a set containing 0Z . A point z¯ ∈ A is said to be a generalized efficient point of A with respect to C, if there is a sequence {zk } ⊂ Z with zk → 0 as k → ∞ such that A ∩ (¯ z − C − zk ) = ∅ ∀k ∈ N. (8) The set of all the generalized efficient points of A with respect to C is denoted by GE(A | C). Proposition 2 Suppose that C is a convex cone with qri C = ∅. If z¯ ∈ A is a quasi relative efficient point of A, then z¯ is a generalized efficient point of A with respect to C. Proof Suppose that z¯ is a quasi relative efficient point of A with respect to C. Since qri C = ∅ we can select an element z0 ∈ qri C. For each k ∈ N, put zk = (k + 1)−1 z0 . From the convexity of C and z0 ∈ qri C imply that C+ k z0 z0 = C+ ⊂ qri C ∀k ∈ N. k+1 k+1 k+1 6 N. V. Tuyen Since z¯ is a quasi relative efficient point of A with respect to C, we have A ∩ (¯ z − qri C) = ∅. Thus A ∩ (¯ z−C − z0 ) = ∅ ∀k ∈ N. k+1 (9) Obviously, zk → 0 as k → ∞. This and (9) imply that z¯ is a generalized efficient point of A with respect to C. ✷ Corollary 2 (see [29, Proposition 2.9]) Suppose that C is a convex cone with ri C = ∅. If z¯ ∈ A is a relative efficient point of A, then z¯ is a generalized efficient point of A with respect to C. Proposition 3 (see [29, Proposition 2.11]) If C is a convex cone with int C = ∅, then WMin (A | C) = GE(A | C). (10) 3 Upper convergence of relative Pareto efficient sets Let Z be a Banach space, (An ) be a sequence of subsets in Z and A ⊂ Z be a nonempty subset. We recall some concepts of convergence of a sequence of sets. • The convergence in the sense of Kuratowski-Painlev´e: The Kuratowski-Painlev´e lower and upper limits of (An ) are defined as Li An := {z ∈ Z , | z = lim zn , zn ∈ An for all large n}, n→∞ Ls An := {z ∈ Z , | z = lim zk , zk ∈ Ank for some (Ank ) ⊂ (An )}. n→∞ Clearly, Li An ⊂ Ls An . If Ls An ⊂ A ⊂ Li An , then we say that (An ) converges K to A in the sense of Kuratowski-Painlev´e and we denote An −→ A. From K closedness of Ls An and Li An imply that if An −→ A, then A is a closed subset. When we consider the limits in the weak topology on Z, we denote the lower and the upper limits above by w − Li An and w − Ls An . When w − Ls An ⊂ A ⊂ Li An , we say that (An ) converges to A in the sense of M Mosco and we denote An −→ A. • The convergence in the sense of Wijsman: We say that (An ) converges to A in the sense of Wijsman if lim d(An , x) = d(A, x)∀x ∈ Z, n→∞ where d(A, x) = inf d(a, x). a∈A • The convergence in the sense of Attouch-Wets: Stability of relative Pareto efficient sets 7 Let x ∈ Z and let A, B be nonempty subsets in Z. Define d(x, A) = inf d(x, a) (d(x, ∅) = ∞), a∈A e(A, B) = sup d(a, B) (e(∅, B) = 0, e(∅, ∅) = 0, e(A, ∅) = ∞), a∈A eρ (A, B) = e(A ∩ Bρ , B) Bρ = B(0, ρ), hρ (A, B) = max{eρ (A, B), eρ (B, A)}. We say that the sequence (An ) ⊂ Z converges to A in the sense Attouch-Wets if lim hρ (An , A) = 0 n→∞ for all ρ > 0. We can split this notion of convergence into an upper part and a lower part as follows lim eρ (An , A) = 0, n→∞ and lim eρ (A, An ) = 0. n→∞ Sonntag and Z˘ alinescu [26] shown that the upper part convergence in the sense of Attouch-Wets is equivalent to lim inf d(An , B) ≥ d(A, B), n→∞ for each nonempty bounded set B, where d(A, B) := inf inf d(a, b). a∈A b∈B For the relationships between the various notions of set convergence introduced here, see, e.g., [1, 26]. It is well known that, if Z is a finite dimensional space, the above quoted notions of set-convergence coincide whenever we consider a sequence (An ) of closed sets. Lemma 1 Let C be a convex subset in Z with nonempty relative interior. If z∈ / ri C, then there exists y ∈ C such that (1 − µ)y + µz ∈ /C for all µ > 1. Proof On the contrary, suppose that there exist z ∈ / ri C and µ > 1 such that (1 − µ)y + µz ∈ C (11) for all y ∈ C. Since ri C = ∅, it follows that there is x ∈ ri C. From (11) implies that (1 − µ)x + µz ∈ C. Thus there is y ∈ C satisfying (1 − µ)x + µz = y. Thus z = (1 − λ)x + λy, where 0 < λ = µ1 < 1. By [11, Lemma 3.1], we have z ∈ ri C, which contradicts the fact that z ∈ / ri C. ✷ 8 N. V. Tuyen Let (Cn ) be a sequence of convex cones in Z, and C ⊂ Z be a convex cone. For brevity, in the sequel we write ReMin A, ReMin An , Min A, Min An , WMin A, and WMin An instead of ReMin (A | C), ReMin (An | Cn ), Min (A | C), Min (An | Cn ), WMin (A | C), and WMin (An | Cn ), respectively. Theorem 1 Let (Cn ) and C be convex cones in Z with nonempty relative interior and Cn \(−Cn ) = ∅, C\(−C) = ∅. If K (i) An −→ A, (ii) Ls Cnc ⊂ C c ∪ {0}, where C c := Z\C, then Ls ReMin An ⊂ ReMin A. Proof Arguing by contradiction, assume that there is x ∈ Ls ReMin An \ReMin A. Since x ∈ Ls ReMin An and (i), we see that for each k ∈ N there exist xk ∈ ReMin Ank such that lim xk = x and x ∈ A. From x ∈ / ReMin A k→∞ implies that there exists a ∈ A satisfying x − a ∈ ri C, or x−a∈ / (ri C)c . (12) We claim that x−a∈ / Ls (ri Cn )c . Indeed, if otherwise, then there exist zk ∈ (ri Ck )c (13) where (ri Ck ) ⊂ (ri Cn ) satisfying lim zk = x − a. From (13) we see that for k→∞ each k ∈ N there exists yk ∈ Ck such that (1 − µ)yk + µzk ∈ / Ck for all µ > 1. Substituting µ = 1 + − 1 m (14) into the left side of (14) we obtain 1 1 yk + (1 + )zk ∈ / Ck , m m (15) for all m ∈ N. Taking m → ∞ in (15), we have zk ∈ cl (Ckc ) ∀k ∈ N. Taking k → ∞ in (16), we obtain x − a ∈ Ls [cl (Cnc )] . Furthermore, Ls [cl (Cnc )] = Ls Cnc ⊂ C c ∪ {0} ⊂ (ri C)c . Thus x − a ∈ (ri C)c , (16) Stability of relative Pareto efficient sets 9 contrary to (12). From Ls (ri Cnk )c ⊂ Ls (ri Cn )c implies that x−a ∈ / Ls (ri Cnk )c . Since a ∈ A and (i) imply that there exist an ∈ An satisfying lim an = a. n→∞ Thus lim ank = a, where ank ∈ Ank ∀k ∈ N. From k→∞ lim (xk − ank ) = x − a ∈ / Ls (ri Cnk )c , k→∞ implies that there exists k0 ∈ N such that xk0 − ank0 ∈ / (ri Cnk0 )c , or xk0 − ank0 ∈ ri Cnk0 , contradicting the fact that xk0 is efficient of Ank0 with respect to Cnk0 . The proof is complete. ✷ In Theorem 1, the condition (ii) cannot be replaced the weaker condition “Ls Cnc ⊂ cl (C c )”. To see this, we consider the following example. Example 1 Let Z = R2 and C = R+ × {0}. Let An = 1 z = (z1 , z2 ) ∈ R2 | z2 = − z1 , −1 ≤ z1 ≤ 1 , Cn = C ∀n ∈ N, n K and A = [−1, 1]×{0}. It is easy to see that An −→ A and Ls Cnc = cl (C c ) = R2 . However, we have ReMin An = An for all n ∈ N and ReMin A = {(−1, 0)}. Clearly, Ls ReMin An = A ReMin A. Corollary 3 Let (Cn ) and C be convex cones in Z with nonempty interior. If K (i) An −→ A, (ii) Ls Cnc ⊂ cl (C c ), then Ls WMin An ⊂ WMin A. Proof Note that when int C = ∅ then cl (C c ) ⊂ (int C)c . Analysis similar to that in the proof of Theorem 1 shows that Ls ReMin An ⊂ ReMin A. From ReMin An = WMin An for all n and ReMin A = WMin A imply that Ls WMin An ⊂ WMin A. The proof is complete. ✷ Remark 2 If Cn = C for all n ∈ N and int C = ∅, then the condition (ii) in Corollary 3 is satisfied and Corollary 3 coincides with [19, Proposition 3.1] 10 N. V. Tuyen Theorem 2 Let Cn and C be convex cones in Z with nonempty relative interior and Cn \(−Cn ) = ∅, C\(−C) = ∅. If K (i) An −→ A, (ii) Ls Cnc ⊂ C c ∪ {0}, ∞ (iii) for every bounded subset of ReMin An is relatively compact, n=1 then lim inf d(ReMin An , B) ≥ d(ReMin A, B), n→∞ (17) for each bounded subset B. Proof The conclusion of the Theorem is trivial lim inf d(ReMin An , B) = +∞. n→∞ Thus it suffices to consider the case lim inf d(ReMin An , B) is finite. Suppose n→∞ on the contrary that there is some bounded subset B ⊂ Z and some positive number γ > 0 such that lim inf d(ReMin An , B) < γ < α, (18) n→∞ where α := d(ReMin A, B). By taking a subsequence of (An ) if necessary we may assume that d(ReMin An , B) < γ ∀n ∈ N. This implies that there exists yn ∈ ReMin An satisfying d(yn , B) < γ ∀n ∈ N. ∞ Hence the sequence (yn ) is bounded. From (yn ) ⊂ ReMin An and (iii), n=1 without loss of generality, we can assume that the sequence (yn ) converges to y0 ∈ Z. Since (i), we have y0 ∈ A. From d(y0 , B) ≤ γ < α = d(ReMin A, B) implies that y0 ∈ / ReMin A. Thus there exists a ∈ A satisfying y0 − a ∈ ri C, or y0 − a ∈ / (ri C)c . (19) As in the proof of Theorem 1, inclusion (19) gives y0 − a ∈ / Ls (ri Cn )c . From a ∈ A and (i) imply that there is a sequence (an ), an ∈ An such that lim an = a. Consequently n→∞ lim (yn − an ) = y0 − a ∈ / Ls (ri Cn )c . n→∞ Stability of relative Pareto efficient sets 11 Thus there exists n0 ∈ N satisfying yn0 − an0 ∈ / (ri Cn0 )c , or yn0 − an0 ∈ ri Cn0 , contradicting the fact that yn0 is a relative efficient point of An0 with respect to Cn0 . The proof is complete. ✷ Corollary 4 (see [18, Theorem 2.1]) Let Cn and C be convex cones in Z with nonempty interior and Cn \(−Cn ) = ∅, C\(−C) = ∅. If the conditions (i)–(iii) of Theorem 2 are satisfied, then lim inf d(WMin An , B) ≥ d(WMin A, B), n→∞ (20) for each bounded subset B. Theorem 3 Let Z be a reflexive Banach space, let (Cn ) and C be convex pointed cones with nonempty relative interior. If (i) w − Ls An ⊂ A ⊂ w − Li An , (ii) w − Ls (ri Cn )c ⊂ (ri C)c , then lim inf d(ReMin An , x) ≥ d(ReMin A, x) ∀x ∈ Z. (21) n→∞ Proof Let x be an arbitrary element in Z. As in the proof of Theorem 2, we need only consider the case that lim inf d(ReMin An , x) is finite. Arguing by n→∞ contradiction, assume that lim inf d(ReMin An , x) < d(ReMin A, x) n→∞ for some x ∈ Z. By taking a subsequence of (An ) if necessary we can find yn ∈ ReMin An and a positive number γ satisfying d(yn , x) < γ < d(ReMin A, x) ∀n ∈ N. By the reflexivity of Z and the boundedness of (yn ), there exists a subsequence of (yn ) weakly converges to y0 ∈ Z. From (i) we obtain y0 ∈ A. Hence d(y0 , x) ≤ γ < d(ReMin A, x). This implies that y0 ∈ / ReMin A. Thus there exists a ∈ A such that y0 − a ∈ ri C. From (i) implies that there is a sequence (an ), an ∈ An for n large enough satisfying w − lim an = a. n→∞ We claim that there exists n0 such that yn − an ∈ ri Cn for all n ≥ n0 . Indeed, if otherwise, then there exists (ynk − ank ) ⊂ (yn − an ) satisfying ynk − ank ∈ (ri Cnk )c ∀k ∈ N. 12 N. V. Tuyen From w − lim (ynk − ank ) = y0 − a we have k→∞ y0 − a ∈ w − Ls (ri Cn )c ⊂ (ri C)c . Thus y0 − a ∈ (ri C)c , which contradicts the fact that y0 − a ∈ ri C. The proof is complete. ✷ 4 Lower convergence of relative Pareto efficient sets Let Z be a Banach space, ∅ = A ⊂ Z and let C ⊂ Z be a closed convex cone with C\(−C) = ∅. Put Θ := ri C ∪ {0}. Definition 4 We say that the relative domination property, denoted by (RDP ), holds for a set A ⊂ Z if for each x ∈ A, there is some a ∈ ReMin A such that x ∈ a + Θ. Remark 3 1) The relative domination property is an extended version of the weak domination property ((W DP ) for brevity). If int C = ∅, then Θ = int C ∪ {0} and the relative domination property coincides with the weak domination property. 2) From Difinition 4 implies that (RDP ) holds for A iff A ⊂ ReMin A + Θ. 3) Θ = ri C ∪ {0} is a correct cone. Indeed, we have cl Θ + Θ\l(Θ) = C + Θ\l(Θ) ⊂ C + Θ\{0} = C + ri C = ri C ⊂ Θ. Thus Θ is a correct cone. 4) From ReMin A = Min (A | Θ) and the correctness of Θ imply that (RDP ) holds for every compact set A. In this section we assume that Cn = C for all n ∈ N, where C is a closed convex cone with nonempty relative interior and C\(−C) = ∅. Theorem 4 Suppose that the following conditions hold K (i) An −→ A, (ii) (RDP ) holds for (An ) for all n large enough, ∞ (iii) ReMin An is relative compact. n=1 Then ReMin A is nonempty and ReMin A ⊂ Li ReMin An . Proof We first show that ReMin A is nonempty. Define A0 := Ls ReMin An . Then A0 is a closed subset in Z. From the nonemptyness of A and A ⊂ Li An imply that An is nonempty for all n large enough. Since (ii) and (iii), it follow that A0 is a nonempty compact set. From the correctness of Θ and Stability of relative Pareto efficient sets 13 the compactness of A0 imply that ReMin A0 = Min (A0 | Θ) is nonempty. We claim that ReMin A0 ⊂ ReMin A. On the contrary, suppose that there exists e ∈ ReMin A0 \ReMin A. Then there is a ∈ A such that e ∈ a + ri C. (22) By (i), there is a sequence (an ) such that an ∈ An for all n and lim an = a. n→∞ From (ii) implies that there exists en ∈ ReMin An satisfying an ∈ en + Θ for all n large enough. Consequently, an ∈ en + C for all n large enough. In view of (iii) we may assume that the sequence (en ) converges to some e0 ∈ Z. It is easy to see that e0 ∈ A0 and a ∈ e0 + C. From this and (22) imply that e ∈ e0 + C + ri C ⊂ e0 + ri C. Consequently, e ∈ e0 + ri C, which contradicts the minimality of e. Thus ReMin A0 ⊂ ReMin A and ReMin A is nonempty. We now claim that ReMin A ⊂ Li ReMin An . Taking any a ∈ ReMin A. From (i) implies that there is a sequence (an ) such that an ∈ An for all n ∈ N and lim an = a. Since (RDP ) holds for An , it follows that there exists n→∞ en ∈ ReMin An such that an ∈ en + Θ (23) for all n large enough. From (iii) we may assume that (en ) converges to some e ∈ A. We shall show that e = a. Indeed, assume that e = a. Taking n → ∞ in (23) we obtain a − e ∈ C. From lim (an − en ) = a − e = 0 we have n→∞ 0 = an − en ∈ Θ, for all large enough n. Consequently, 0 = an − en ∈ ri C, 14 N. V. Tuyen for all large enough n. Let x be an arbitrary element in C. Then there exists µ > 1 satisfying (1 − µ)x + µ(an − en ) ∈ C ∀n ∈ N. (24) Taking n → ∞ in (24), by the closedness of C, we obtain (1 − µ)x + µ(a − e) ∈ C. Thus a − e ∈ ri C, which contradicts the fact that a ∈ ReMin A. Thus a = e. From this and e ∈ Li ReMin An imply that a ∈ Li ReMin An . Thus ReMin A ⊂ Li ReMin An . ✷ The proof is complete. The next example shows that Theorem 4 cannot be deduced from [18, Theorem 3.1]. Example 2 Let Z = R2 and C = R+ × {0}. Let 1 n z1 , 0 ≤ z1 ≤ 1 + n+1 n 1 ∪ (z1 , z2 ) ∈ R2 | 0 ≤ z2 ≤ 1, z1 > 1 + , Cn = C ∀n ∈ N, n An = (z1 , z2 ) ∈ R2 | 0 ≤ z2 ≤ and A = (z1 , z2 ) ∈ R2 | 0 ≤ z2 ≤ z1 , 0 ≤ z1 ≤ 1 ∪ (z1 , z2 ) ∈ R2 | 0 ≤ z2 ≤ 1, z1 > 1 . We have ReMin A = {(z1 , z2 ) | z2 = z1 , 0 ≤ z1 ≤ 1} and ReMin An = (z1 , z2 ) ∈ R2 | z2 = 1 n z1 , 0 ≤ z1 ≤ 1 + n+1 n ∀n ∈ N. It is easy to check that all conditions in Theorem 4 are satisfied. Thus ReMin A ⊂ Li ReMin An . However, Ls Θ be employed. Θ, where Θ = {0} ∪ ri C. Thus Theorem 3.1 in [18] cannot Remark 4 In Theorem 4, the condition (iii) can be replaced by the following weaker condition: (iii ) If an ∈ An is such that lim an exists and n→∞ en ∈ ReMin An ∩ (an − Θ), then (en ) admits a convergent subsequence. Stability of relative Pareto efficient sets 15 Theorem 5 Suppose that the following conditions hold K (i) An −→ A, (ii)(RDP ) holds for (An ) for all n large enough, (iii) if an ∈ An is such that lim an exists and n→∞ en ∈ ReMin An ∩ (an − Θ), then (en ) admits a convergent subsequence. (iv) for any ρ > 0, ReMin A ∩ B(0, ρ) is relatively compact. Then for each ρ > 0 we have lim eρ (ReMin A, ReMin An ) = 0. n→∞ Proof We will follow the scheme of the proof of [18, Theorem 3.3]. Suppose that the conclusion of Theorem does not hold, then there exist ρ > 0, > 0 and a subsequence (Ank ) of (An ) such that eρ (ReMin A, ReMin Ank ) > ∀k ∈ N. (25) Thus for each k ∈ N there is ek ∈ ReMin A ∩ Bρ satisfying d(ek , ReMin Ank ) > . (26) By (iv), (ek ) admits a subsequence converging to some e ∈ Z and then from (26) there exists k0 such that d(e, ReMin Ank ) > 2 ∀k > k0 . (27) From Theorem 4 implies that ReMin A ⊂ Li ReMin An . Thus, for each k, there exists a sequence (eki ) such that eki ∈ ReMin Ani for all i ∈ N and lim eki = ek . Consequently, for each k ∈ N, there is a sequence (eki(k) ) such i→∞ that eki(k) ∈ ReMin Ani(k) and d(ek , eki(k) ) < 1 . k Clearly, (eki(k) ) converges to e, contrary to (27). ✷ 5 The relative containment property and the lower semicontinuity of the relative Pareto efficient multifunction Let A be a nonempty subset in a Banach space Z and B ⊂ A. Suppose that C ⊂ Z is a convex cone. Definition 5 We say that the domination property (DP ) holds for (A, B) ⊂ Z × Z, if A ⊂ B + C. (28) 16 N. V. Tuyen Definition 6 We say that the containment property (CP ) holds for (A, B) ⊂ Z × Z, if for each W ∈ N (0Z ) there exists V ∈ N (0Z ) such that [A\(B + W )] + V ⊂ B + C. (29) The following definition gives a weaker form of the notion containment property of a pair subset (A, B). Definition 7 We say that the relative containment property ((RCP ) for brevity) holds for (A, B) ⊂ Z × Z if for each W ∈ N (0Z ) there exists V ∈ N (0Z ) such that [A\(B + W )] + [V ∩ aff (C)] ⊂ B + C. (30) Note that if int C = ∅, then the relative containment property coincides with the containment property. Example 3 Let A = {(z1 , z2 ) ∈ R2 | 0 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1}, C = R+ × {0} and B = Min A. We have Min A = {0} × [0, 1] and Min A + C = {(z1 , z2 ) ∈ R2 | 0 ≤ z1 , 0 ≤ z2 ≤ 1}. It is easy to see that A ⊂ Min A + C. Thus (DP ) holds for (A, Min A). But (CP ) does not hold for this pair. Indeed, take W = B(0R2 , 21 ). We have A\(Min A + W ) = (z1 , z2 ) ∈ R2 | 1 ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1 . 2 Hence for any V ∈ N (0R2 ) then [A\(Min A + W )] + V Min A + C. We now claim that (RCP ) holds for (A, Min A). Let W = B(0R2 , ) be an arbitrary neighborhood of zero. An easy computation shows that A\(Min A+W ) = {(z1 , z2 ) ∈ R2 | ≤ z1 ≤ 1, 0 ≤ z2 ≤ 1} and aff C = R×{0}. Choose V = B(0R2 , 2 ) ∈ N (0R2 ). Then [A\(Min A + W )] + [V ∩ aff C] ⊂ Min A + C. Thus (RCP ) holds for (A, Min A). Proposition 4 Let A be a nonempty subset in Z and C ⊂ Z be a convex cone with ri C = ∅. If (RCP ) holds for (A, Min A), then Min A ⊂ ReMin A ⊂ cl Min A. Proof Due to Remark 1(d), to finish the proof, it suffices to show that ReMin A ⊂ cl Min A. Arguing by contradiction, assume that there is z¯ ∈ ReMin A\cl Min A. Since z¯ ∈ ReMin A, we have (¯ z − ri C) ∩ A = ∅. (31) Stability of relative Pareto efficient sets 17 We claim that (¯ z − ri C) ∩ (Min A + C) = ∅. (32) Otherwise, there exist θ ∈ ri C, a ∈ Min A and c ∈ C such that z¯ − θ = a + c. This imply that a = z¯ − (θ + c) ⊂ z¯ − (ri C + C) ⊂ z¯ − ri C. Thus a ∈ (¯ z − ri C) ∩ A, contrary to (31). From z¯ ∈ / cl Min A implies that there exists W ∈ N (0Z ) such that z¯ ∈ / (Min A + W ). Since(RCP ) holds for (A, Min A) there exists V ∈ N (0Z ) such that [A\(Min A + W )] + [V ∩ aff (C)] ⊂ Min A + C. Thus z¯ + [V ∩ aff (C)] ⊂ Min A + C. Clearly, (¯ z − ri C) ∩ z¯ + [V ∩ aff (C)] = ∅. Hence (¯ z − ri C) ∩ (Min A + C) = ∅, ✷ contrary to (32). Note that the set ReMin A may not be closed even in case that A is closed. Indeed, let A = {z = (z1 , z2 ) ∈ R2 | z12 + z22 ≤ 1} ∪ {(−2, 0)} and C = R+ × {0}. Then ReMin A = {z = (z1 , z2 ) ∈ R2 | z12 + z22 = 1, z1 ≤ 0, z2 = 0} ∪ {(−2, 0)} and it is not closed. However, if A ⊂ Rm is polyhedral, then ReMin A is closed. To prove this we need the following lemma. Lemma 2 Suppose that A is a polyhedral subset of Rm defined by A = {z ∈ Rm | ai , z ≤ bi , i = 1, 2, ..., N }, (33) where ai ∈ Rm and bi ∈ R for all i ∈ I =: {1, 2, ..., N }. Let C ⊂ Z be a pointed closed convex cone. Then ReMin A is nonempty if and only if Rec (A) ∩ (−ri C) = ∅, where Rec (A) is the recession cone of A and defined by Rec (A) = {z ∈ Rm | ai , z ≤ 0, i ∈ I}. 18 N. V. Tuyen Proof Denote Θ := {0} ∪ ri C. We have ReMin A = Min (A | Θ). By Theorem 3.18 of Chapter 2 [17], ReMin A is nonempty if and only if Rec (A) ∩ (−Θ) = {0}. (34) From the pointedness of C we see that 0 ∈ / ri C. Thus the condition (34) is equivalent to Rec (A) ∩ (−ri C) = ∅, completing the proof. ✷ Theorem 6 If A is a polyhedral subset in Rm given by (33) and C ⊂ Rm is a pointed closed convex cone, then ReMin A is closed. Proof We will follow the scheme of the proof of [8, Proposition 4.1]. Suppose that the conclusion of Theorem does not hold, then there exists a sequence (zn ) ⊂ ReMin A which converges to z¯ and z¯ ∈ / ReMin A. Thus there is z ∈ A satisfying z¯ − z ∈ ri C. For each n ∈ N put In := {i ∈ I | ai , zn = bi }. From I is finite and In ⊂ I for all n ∈ N imply that there exist infinitely many integers n such that In = I1 . Without loss of generality, we can assume that In = I1 for all n ∈ N. This means that ai , zn = bi , i ∈ I1 and ai , zn < bi , i ∈ I\I1 for all n ∈ N. Thus ai , z¯ = bi and ai , z¯ ≥ ai , z for i ∈ I1 . Furthermore, there exists i ∈ I\I1 such that ai , z¯ < ai , z . Otherwise, ai , z¯ ≥ ai , z for all i ∈ I, or ai , z − z¯ ≤ 0 for all i ∈ I. This implies that z−¯ z ∈ Rec (A). Hence z−¯ z ∈ [Rec (A)∩(−ri C)], which is impossible. Thus there are two index subsets J1 , J2 ⊂ I with J2 = ∅ satisfying ai , z − z¯ ≤ 0, i ∈ J1 ⊃ I1 , and ai , z − z¯ > 0, i ∈ J2 . For each n ∈ N put λn = min i∈J2 bi − ai , zn > 0, ai , z − z¯ and yn = zn + λn (z − z¯). We have ai , yn = ai , zn + λn ai , z − z¯ ≤ ai , zn + (bi − ai , zn ) ≤ bi for all i ∈ J2 . Clearly, ai , yn ≤ bi for all i ∈ I\J2 . Thus yn ∈ A for all n ∈ N. Moreover, we have yn − zn ∈ −ri C, contradicting the fact that zn ∈ ReMin A. The proof is complete. ✷ Stability of relative Pareto efficient sets 19 Corollary 5 Let A be a polyhedral subset in Rm and C be a pointed closed convex cone. If (RCP ) holds for (A, Min A), then ReMin A = cl Min A. Example 4 Let A and C be as in Example 3. It is easy to check that all conditions in Corollary 5 are satisfied. Thus ReMin A = cl Min A = Min A. Note that A is not a rotund set. Hence the sufficient condition for equation (7) given by Proposition 1 is not a necessary one. The following proposition gives a characterization of (RCP ) whenever ri C = ∅. Proposition 5 If ri C = ∅, then the following two properties are equivalent: (i) (RCP ) holds for (A, B); (ii) For each W ∈ N (0Z ) there is W0 ∈ N (0Z ) such that for all y ∈ A\(B + W ) there exist ηy ∈ B and cy ∈ C satisfying y = ηy + cy , (cy + W0 ) ∩ aff (C) ⊂ C. Proof (i) ⇒ (ii): For each W ∈ N (0Z ), put CW = {c ∈ C | (c + W ) ∩ aff (C) ⊂ C}. Clearly, ri C = W ∈N (0Z ) CW . We claim that for any V ∈ N (0Z ) there exists WV ∈ N (0Z ) such that z ∈ Z | z + [V ∩ aff (C)] ∈ B + C ⊂ B + CWV . (35) Indeed, since 0Z ∈ (−C) = cl (−ri C), it follows that there exists WV ∈ N (0Z ) satisfying V ∩ (−CWV ) = ∅. Obviously, −CWV ⊂ aff (C). Thus V ∩ (−CWV ) = V ∩ [(−CWV ) ∩ aff (C)] = [V ∩ aff (C)] ∩ (−CWV ) = ∅. Choose zV ∈ [V ∩ aff (C)] ∩ (−CWV ). Take any z ∈ {c ∈ Z | c + [V ∩ aff (C)] ⊂ B + C}, i.e., z + [V ∩ aff (C)] ⊂ B + C. We have z + zV ∈ B + C. In the other hand, we have C + CWV ⊂ CWV . Indeed, take any c1 ∈ C and c2 ∈ CWV . We claim that (c1 + c2 ) ∈ CWV , or equivalent to [(c1 + c2 ) + WV ] ∩ aff (C) ⊂ C. Let u be an arbitrary point in [(c1 + c2 ) + WV ] ∩ aff (C). Then there is w ∈ WV such that u = c1 +c2 +w and c1 +c2 +w ∈ aff (C). Since C is a convex cone, we have aff (C) + C = aff (C). Thus c2 + w = u − c1 ∈ aff (C) − C = aff (C). From this and c2 ∈ CWV imply that c2 +w ∈ C. Thus u = c1 +(c2 +w) ∈ C +C = C. This means that [(c1 + c2 ) + WV ] ∩ aff (C) ⊂ C for any c1 ∈ C and c2 ∈ CWV . Hence C + CWV ⊂ CWV . This implies that z ∈ B + C − zV ⊂ B + C + CWV ⊂ B + CWV . Next, take any W ∈ N (0Z ). Since (RCP ) holds for A there exists V ∈ N (0Z ) such that [A\(B + W )] + [V ∩ aff (C)] ⊂ B + C. (36) By virtue of (35) we can find WV ∈ N (0Z ) satisfying {z ∈ Z | z + [V ∩ aff (C)] ∈ B + C} ⊂ B + CWV . (37) 20 N. V. Tuyen For each y ∈ A\(B + W ), it follows from (36) and (37) that y + [V ∩ aff (C)] ⊂ B + C ⊂ B + CWV . Thus there exist ηy ∈ B and cy ∈ C satisfying y = ηy + cy , (cy + W0 ) ∩ aff (C) ⊂ C, where W0 := WV . (ii)⇒(i): Obviously. ✷ Corollary 6 [7, Proposition 2.2] If int C = ∅, then the following two properties are equivalent: (i) (CP ) holds for (A, Min A); (ii) For each W ∈ N (0Z ) there is W0 ∈ N (0Z ) such that for all y ∈ A\(Min A + W ) there exist ηy ∈ Min A and cy ∈ C satisfying y = ηy + cy , cy + W0 ⊂ C. Definition 8 Give a multifunction F : P ⇒ Z, where P is a topological space. We put F(p) = Min (F (p)), R(p) = ReMin (F (p)) and call F : P ⇒ Z and R : P ⇒ Z are the Pareto efficient multifunction and the relative Pareto efficient multifunction corresponding to the quadruplet {F, P, Z, C}, respectively. We recal some concepts of upper and lower continuities of a multifunction. Definition 9 Let F : P ⇒ Z be a multifunction and p0 ∈ P . (i) F is upper semicontinuous (usc for brevity) at p0 if for every open set V containing F (p0 ) there exists U0 ∈ N (p0 ) such that F (p) ⊂ V for all p ∈ U0 . (ii) F is lower semicontinuous (lsc) at p0 ∈ dom F if for any open set V ⊂ Z satisfying V ∩ F (p0 ) = ∅ there exists U0 ∈ N (p0 ) such that V ∩ F (p) = ∅ for all p ∈ U0 . (iii) F is Hausdorff upper semicontinuous (H-usc) at p0 if for every W ∈ N (0Z ) there exists U0 ∈ N (p0 ) such that F (p) ⊂ F (p0 ) + W for all p ∈ U0 . (iv) F is Hausdorff lower semicontinuous (H-lsc) at p0 if for every W ∈ N (0Z ) there exists U0 ∈ N (p0 ) such that F (p0 ) ⊂ F (p) + W for all p ∈ U0 . We now introduce some stronger forms of the properties lsc and H-usc of a multifunction, which will be used in this section. Definition 10 (i) F is said to be relatively lower semicontinuous (r-lsc for brevity) at p0 ∈ dom F if for any z¯ ∈ F (p0 ) and W ∈ N (0Z ) there exists U0 ∈ N (p0 ) such that [¯ z + (W ∩ aff (C))] ∩ F (p) = ∅ for all p ∈ U0 . (ii) F is said to be relatively Hausdorff upper semicontinuous (r-H-usc) at p0 if for every W ∈ N (0Z ) there exists U0 ∈ N (p0 ) such that F (p) ⊂ F (p0 ) + [W ∩ aff (C)] for all p ∈ U0 . Stability of relative Pareto efficient sets 21 The following observations and remarks are simple: 1. If F is upper semicontinuous (Hausdorff lower semicontinuous), then F is Hausdorff upper semicontinuous (lower semicontinuous). 2. If int C = ∅ then aff (C) = Z. Thus the properties r-usc (r-H-usc) and usc (H-usc) are coincide. 3. If F is relatively lower semicontinuous, then F is lower semicontinuous. The converse does not hold in general. For instance, let F : R ⇒ R2 be defined by F (p) = {(z1 , z2 ) ∈ R2 | 0 < z2 ≤ 1} ∀p ∈ R\{0}, F (0) = {0R2 } and C = R+ × {0}. Then F is lsc at every points but not r-lsc at p = 0. 4. If F is relatively Hausdorff upper semicontinuous, then F is Hausdorff upper semicontinuous. But not vice versa. For example, let P = R, Z = R2 , C = R+ × {0}. Let F : R ⇒ R2 be defined by F (0) = {(z1 , z2 ) | z1 = 0}\{0R2 } and F (p) = {(z1 , z2 ) | − |p| ≤ z1 ≤ |p|} ∀p ∈ R\{0}. Then for any δ > 0 we have F (p) ⊂ F (0) + B(0R2 , δ) ∀p ∈ R, |p| < δ . 2 Thus F is H-usc at p0 = 0. However, for each δ > 0, we have F (0) + [B(0R2 , δ) ∩ aff (C)] = {(z1 , z2 ) | |z1 | < δ and z1 = 0}. Hence F (p) F (0) + [B(0R2 , δ) ∩ aff (C)] ∀p ∈ R\{0}. This means that F is not r-H-usc at 0. Let F and G be two multifunctions from P to Z. We say that (RCP ) holds for pair (F, G) uniformly around a certain p0 if for any neighborhood W ∈ N (0Z ) there exist V ∈ N (0Z ) and U0 ∈ N (p0 ) such that [F (p)\(G(p) + W )] + [V ∩ aff (C)] ⊂ G(p) + C ∀p ∈ U0 . (38) Theorem 7 Suppose that ri C = ∅ and (RCP ) holds for pair (F, R) uniformly around p0 . If F is r-H-usc and r-lsc at p0 , then R is lcs at p0 . Proof Let z¯ ∈ R(p0 ) and W ∈ N (0Z ). The proof will be completed if we can show that there exists UW ∈ N (p0 ) such that (¯ z + W ) ∩ R(p) = ∅ ∀p ∈ UW . (39) Take any W1 ∈ NB (0Z ) satisfying W1 + W1 ⊂ W . Since (RCP ) holds for pair (F, R) uniformly around p0 , there exist W0 ∈ N (0Z ) and U0 ∈ N (p0 ) such that for all p ∈ U0 and y ∈ [F (p)\(R(p) + W1 )] we can find ηy ∈ R(p) and cy ∈ C satisfying y = ηy + cy , (cy + W0 ) ∩ aff (C) ⊂ C. (40) 22 N. V. Tuyen Choose W2 ∈ NB (0Z ) satisfying W2 + W2 ⊂ W0 . By the relatively lower semicontinuity of F at p0 , there exists a neighborhood U1 of p0 , U1 ⊂ U0 such that [¯ z + (W1 ∩ W2 ) ∩ aff (C)] ∩ F (p) = ∅ ∀p ∈ U1 . For each p ∈ U1 let yp ∈ [¯ z + (W1 ∩ W2 ) ∩ aff (C)] ∩ F (p). (41) By the relatively Hausdorff upper semicontinuity of F at p0 , there exists U2 ∈ N (p0 ), U2 ⊂ U1 , such that F (p) ⊂ F (p0 ) + [(W1 ∩ W2 ) ∩ aff (C)] ∀p ∈ U2 . (42) Suppose first that there exists U ∈ N (p0 ), U ⊂ U0 , such that yp ∈ R(p) + W1 ∀p ∈ U . (43) For each p ∈ U1 ∩ U , from (41) and (43) imply that there exist wp ∈ W1 ∩ W2 , ηp ∈ R(p) and w ¯p ∈ W1 satisfying yp = z¯ + wp = ηp + w ¯p . Thus ηp = z¯ + wp − w ¯p ∈ z¯ + W1 + W1 ⊂ z¯ + W. This means that (¯ z + W ) ∩ R(p) = ∅ ∀p ∈ U1 ∩ U , which proves assertion (39) with UW = U1 ∩ U . Next, suppose that for all U ∈ N (p0 ), U ⊂ U2 , there exists p ∈ U such that yp ∈ / R(p) + W1 . (44) Combining (44) with (40) we have yp ∈ [F (p)\(R(p) + W1 )]. By (40), there exist ηp ∈ R(p) and cp ∈ C satisfying yp = ηp + cp , (cp + W0 ) ∩ aff (C) ⊂ C. (45) From (42) and the relation ηp ∈ R(p) ⊂ F (p) imply that there exist z0 ∈ F (p0 ) and w0 ∈ (W1 ∩ W2 ) ∩ aff (C) such that ηp = z0 + w0 . (46) By (41), there is wp ∈ (W1 ∩ W2 ) ∩ aff (C) such that yp = z¯ + wp . Using (47), (45) and (46) we get z¯ + wp = ηp + cp = z0 + w0 + cp . This implies that z¯ = z0 + cp + w0 − wp . (47) Stability of relative Pareto efficient sets 23 Furthermore, cp + w0 − wp ∈ cp + [(W1 ∩ W2 ) ∩ aff (C)] − [(W1 ∩ W2 ) ∩ aff (C)] ⊂ cp + [W2 ∩ aff (C)] + [W2 ∩ aff (C)] ⊂ (cp + W0 ) ∩ aff (C) ⊂ C. Put k0 := cp +w0 −wp . Then k0 ∈ ri C and z0 = z¯−k0 . Thus F (p0 )∩(¯ z −ri C) = ∅, which contradicts the fact that z¯ ∈ R(p0 ). The proof of the theorem is now complete. ✷ Note that if C is a convex cone with C\(−C) = ∅ and k0 ∈ ri C, then k0 = 0. From z¯ − k0 ∈ F (p0 ) and k0 = 0 imply that F (p0 ) ∩ (¯ z − ri C) = {¯ z }. Consequently, F (p0 ) ∩ (¯ z − C) = {¯ z }. Thus, by replacing R by F, we have the following assertion. Corollary 7 Suppose that C is a convex cone with C\(−C) = ∅ and ri C = ∅, and (RCP ) holds for pair (F, F) uniformly around p0 . If F is r-H-usc and r-lsc at p0 , then F is lcs at p0 . Corollary 8 (see [4, Theorem 4]) Suppose that C is a convex pointed cone with int C = ∅, and (CP ) holds for pair (F, F) uniformly around p0 . If F is H-usc and lsc at p0 , then F is lcs at p0 . Example 5 Let P = [0, 1], Z = R2 , C = R+ × {0}. Let F : : P ⇒ R2 be defined by setting F (p) = {(z1 , z2 ) | f (z1 ) ≤ z2 ≤ −z1 + 1} for all p ∈ [0, 1], where   −t + p f (t) = 0   −t + 1 if t ≤ p if p < t ≤ 1 if t > 1 for all t ∈ R. For each p ∈ [0, 1] we have F(p) = {(z1 , z2 ) | z2 = −z1 + p, z1 ≤ p} ∪ {(z1 , z2 ) | z2 = −z1 + 1, z1 > 1}. It is easy to check that (locCP ) (see [12, Definition 3.1]) does not hold for F uniformly around p0 = 0. Note that F is r-H-usc and r-lsc at p0 . An easy computation shows that (RCP ) holds for F uniformly around p0 . Thus F is lsc at p0 . 24 N. V. Tuyen References 1. Aubin, J. P., Frankowska, H.: Set-Valued Analysis, Birkh¨ auser, Boston, Massachusetts, 1990. 2. Bao, T. Q., Mordukhovich, B. S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math. Program. 122 (2010), 101–138. 3. Bao, T. Q., Mordukhovich, B. S.: Extended Pareto optimality in multiobjective problems, Chapter 13 of the book Recent Advances in Vector Optimization (Q. H. Ansari and J.-C. Yao, eds.), pp. 467–516, Springer, Berlin, 2011. 4. Bednarczuk, E. M.: Berge-type theorems for vector optimization problems, Optimization 32 (1995), 373384. 5. Bednarczuk, E. M.: Some stability results for vector optimization problems in partially ordered topological vector, in: Proceedings of the first world congress on World congress of nonlinear analysts, volume III table of contents, Tampa, Florida, United States (1996), 2371–2382. 6. Bednarczuk, E. M.: A note on lower semicontinuity of minimal points, Nonelinear Anal. 50 (2002), 285–297. 7. Bednarczuk, E. M.: Upper H¨ older continuity of minimal points, J. Convex Anal. 9 (2002), 327–338. 8. Bednarczuk, E. M.: Continuity of minimal points with applications to parametric multiple objective optimization, European J. Oper. Res. 157 (2004), 59–67. 9. Bednarczuk, E. M.: Stability analysis for parametric vector optimization problems. Diss. Math. 442 (2007). 10. Borwein, J. M., Lewis, A. S.: Partially finite convex programming, Part I: Quasi relative interior and duality theory, Math. Program. 57 (1992), 15–48. 11. Borwein, J. M., Goebel, R.: Notions of relative interior in Banach spaces, J. Math. Sci. 115 (2003), 2542–2553. 12. Chuong, T. D., Yao, J. C., Yen, N. D.: Further results on the lower semicontinuity of efficient point multifunctions, Pacific J. Optim. 6, 405–422 (2010) 13. Dolecki, S., Malivert, C.: Stability of efficient sets: Continuity of mobile polarities, Nonlinear Anal., 12 (1988), pp. 1461–1486. 14. Dolecki, S., El Ghali, B.: Some old and new results on lower semicontinuity of minimal points, Nonlinear Anal. 39 (2000), 599–609. 15. Holmes, R. B.: Geometric Functional Analysis and Its Applications, Grad. Texts in Math. 24, Springer-Verlag, New York, 1975. 16. Huy, N. Q., Kim, D. S., Tuyen, N. V.: Existence theorems in vector optimization with generalized order, submited. 17. Luc, D. T.: Theory of Vector Optimization, Lecture Notes in Econom. and Math. Systems 319, Springer-Verlag, Berlin, Heidelberg, 1989. 18. Luc, D. T., Lucchetti, R., Malivert, C.: Convergence of the efficient sets, Set-Valued Anal., 2 (1994), pp. 207–218. 19. Lucchetti, R., Miglierina, E.: Stability for convex vector optimization problems, Optimization, 2 (2004), pp. 517–528 . 20. Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization, Math. Methods Oper. Res., 58 (2003), pp. 375–385. 21. Miglierina, E., Molho, E.: Convergence of the minimal sets in convex vector optimization. SIAM J. Optim. 15, 513–526 (2005) 22. Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory, Springer, Berlin, 2006. 23. Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications, Springer, Berlin, 2006. 24. Penot, J. P., Sterna-Karwat, A.: Parametrized multicriteria optimization: Continuity and closedness of optimal multifunction, J. Math. Anal. Appl., 120 (1986), pp. 150–168. 25. Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization. Mathematics in Science and Engineering, 176. Academic Press, Inc., Orlando, FL, 1985. 26. Sonntag, Y., Z˘ alinescu, C.: Set convergences: A survey and a classification, Set-Valued Anal. 2 (1994), 339–356. Stability of relative Pareto efficient sets 25 27. Stadler, W.: Initiators of multiobjective optimization. In: Stadler, W. (ed.) Multicriteria Optimization in Engineering and in Sciences, pp. 325. Series in Mathematical Concepts and Mathematics in Science and Engineering 37. Plenum Press, New York (1988). 28. Tolstonogov, A. A.: Differential Inclusions in a Banach Space, Mathematics and Its Applications, vol. 524, Kluwer Academic, Dordrecht, 2000. 29. Tuyen, N. V., Yen, N. D.: On the concept of generalized order optimality, Nonlinear Analysis 75 (2012), 1592–1601. [...]...Stability of relative Pareto efficient sets 11 Thus there exists n0 ∈ N satisfying yn0 − an0 ∈ / (ri Cn0 )c , or yn0 − an0 ∈ ri Cn0 , contradicting the fact that yn0 is a relative efficient point of An0 with respect to Cn0 The proof is complete ✷ Corollary 4 (see [18, Theorem 2.1]) Let Cn and C be convex cones in Z with nonempty interior and Cn \(−Cn ) = ∅, C\(−C) = ∅ If the conditions (i)–(iii) of Theorem... condition (34) is equivalent to Rec (A) ∩ (−ri C) = ∅, completing the proof ✷ Theorem 6 If A is a polyhedral subset in Rm given by (33) and C ⊂ Rm is a pointed closed convex cone, then ReMin A is closed Proof We will follow the scheme of the proof of [8, Proposition 4.1] Suppose that the conclusion of Theorem does not hold, then there exists a sequence (zn ) ⊂ ReMin A which converges to z¯ and z¯ ∈... ⊂ C Definition 8 Give a multifunction F : P ⇒ Z, where P is a topological space We put F(p) = Min (F (p)), R(p) = ReMin (F (p)) and call F : P ⇒ Z and R : P ⇒ Z are the Pareto efficient multifunction and the relative Pareto efficient multifunction corresponding to the quadruplet {F, P, Z, C}, respectively We recal some concepts of upper and lower continuities of a multifunction Definition 9 Let F :... ReMin A is nonempty and ReMin A ⊂ Li ReMin An Proof We first show that ReMin A is nonempty Define A0 := Ls ReMin An Then A0 is a closed subset in Z From the nonemptyness of A and A ⊂ Li An imply that An is nonempty for all n large enough Since (ii) and (iii), it follow that A0 is a nonempty compact set From the correctness of Θ and Stability of relative Pareto efficient sets 13 the compactness of A0 imply... closed convex cone Then ReMin A is nonempty if and only if Rec (A) ∩ (−ri C) = ∅, where Rec (A) is the recession cone of A and defined by Rec (A) = {z ∈ Rm | ai , z ≤ 0, i ∈ I} 18 N V Tuyen Proof Denote Θ := {0} ∪ ri C We have ReMin A = Min (A | Θ) By Theorem 3.18 of Chapter 2 [17], ReMin A is nonempty if and only if Rec (A) ∩ (−Θ) = {0} (34) From the pointedness of C we see that 0 ∈ / ri C Thus the condition... Pareto efficient sets 21 The following observations and remarks are simple: 1 If F is upper semicontinuous (Hausdorff lower semicontinuous), then F is Hausdorff upper semicontinuous (lower semicontinuous) 2 If int C = ∅ then aff (C) = Z Thus the properties r-usc (r-H-usc) and usc (H-usc) are coincide 3 If F is relatively lower semicontinuous, then F is lower semicontinuous The converse does not hold in... Notions of relative interior in Banach spaces, J Math Sci 115 (2003), 2542–2553 12 Chuong, T D., Yao, J C., Yen, N D.: Further results on the lower semicontinuity of efficient point multifunctions, Pacific J Optim 6, 405–422 (2010) 13 Dolecki, S., Malivert, C.: Stability of efficient sets: Continuity of mobile polarities, Nonlinear Anal., 12 (1988), pp 1461–1486 14 Dolecki, S., El Ghali, B.: Some old and. .. Thus y0 − a ∈ (ri C)c , which contradicts the fact that y0 − a ∈ ri C The proof is complete ✷ 4 Lower convergence of relative Pareto efficient sets Let Z be a Banach space, ∅ = A ⊂ Z and let C ⊂ Z be a closed convex cone with C\(−C) = ∅ Put Θ := ri C ∪ {0} Definition 4 We say that the relative domination property, denoted by (RDP ), holds for a set A ⊂ Z if for each x ∈ A, there is some a ∈ ReMin A such... optimization problems in partially ordered topological vector, in: Proceedings of the first world congress on World congress of nonlinear analysts, volume III table of contents, Tampa, Florida, United States (1996), 2371–2382 6 Bednarczuk, E M.: A note on lower semicontinuity of minimal points, Nonelinear Anal 50 (2002), 285–297 7 Bednarczuk, E M.: Upper H¨ older continuity of minimal points, J Convex... follow the scheme of the proof of [18, Theorem 3.3] Suppose that the conclusion of Theorem does not hold, then there exist ρ > 0, > 0 and a subsequence (Ank ) of (An ) such that eρ (ReMin A, ReMin Ank ) > ∀k ∈ N (25) Thus for each k ∈ N there is ek ∈ ReMin A ∩ Bρ satisfying d(ek , ReMin Ank ) > (26) By (iv), (ek ) admits a subsequence converging to some e ∈ Z and then from (26) there exists k0 such ... studied the Hausdorff upper semicontinuity, the C-Hausdorff upper semicontinuity and the lower (upper) semicontinuity of the efficient solution map and the efficient point multifunctions Recently,... Pareto efficient sets In Section 4, Stability of relative Pareto efficient sets we study the lower part of convergence in the sense of Kuratowski-Painlev´e of relative Pareto efficient sets In... efficient and the relationships between this notions and the traditional solution concepts In Section 3, we establish the upper part of convergence in the sense of Kuratowski-Painlev´e of relative Pareto