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

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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 newconcepts of extended Pareto efficient by using some kinds of relative interiorreplace for interior of ordering cones The aim of this paper is to presentnew results on the convergence of relative Pareto efficient sets and the lowersemicontinuity of relative Pareto efficient multifunctions under perturbations.Our results extend the results of Bednarczuk [4], Chuong and Yen [12], andLuc [18] Examples are given to illustrate the results obtained

Keywords Stability · Relative Pareto efficient · Kuratowski-Painlev´econvergence · 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 mization and of vector optimization, Kruger and Mordukhovich (see [23, Sub-section 5.5.18] and the related references) have introduced the notion of thelocally (f ; Θ)-optimal solution, where f is a single-valued mapping betweenBanach spaces and Θ is a set (may not be convex and/or conic) containing thezero vector In [29], Tuyen and Yen gave a detailed analysis of the notion ofgeneralized order optimality and compared it with the traditional notions of

opti-This work is funded by Vietnam National Foundation for Science and Technology ment (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 (VI- ASM) He would like to thank the VIASM for providing a fruitful research environment and working condition.

Develop-N V Tuyen

Department of Mathematics, Hanoi Pedagogical Institute No 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam.

E-mail: tuyensp2@yahoo.com.

Pareto efficiency, weakly Pareto efficiency and Slater efficiency Recently, Baoand Mordukhovich have introduced the concept of relative Pareto efficient toextended 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 weakerthan the classical minimality with respect to a cone and is actually a general-ization of weak minimality, which can be taken into consideration only if theinterior 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 ative to the closed affine hull of C It is well known that ri C is nonemptyfor every nonempty convex set C in finite dimensions However, it is not thecase in many infinite-dimensional settings To improve this situation, Borweinand 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 provedthat qri C is nonempty for any nonempty closed and convex set C in a separa-ble Banach space Further properties of quasi relative interiors of convex sets

rel-in Banach spaces can be found rel-in [10, 11] and the references thererel-in

Stability analysis is one of the most important and interesting subjectsand its role has been widely recognized in the theory of optimization In theliterature, two classical approaches can be found to study stability in vectoroptimization One is to investigate the set-convergence of efficient sets of per-turbed 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, chetti and Malivert [18] investigated the stability of vector optimization interms of the convergence of the efficient sets Miglierina and Molho [20, 21]obtained some results on stability of convex vector optimization problems byconsidering the convergence of efficient sets For more results concerning use ofconvexity in stability analysis, we refer readers to [17, 19] Various stability re-sults 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] studiedthe Hausdorff upper semicontinuity, the C-Hausdorff upper semicontinuity andthe lower (upper) semicontinuity of the efficient solution map and the efficientpoint multifunctions Recently, by using the approach of Bednarczuk [4,6] andintroducing the new concepts of local containment property, K-local domina-tion property and uniformly local closedness of a multifunction around a givenpoint, Chuong and Yen [12] obtain further results on the lower semicontinuity

Luc-of efficient point multifunctions taking values in Hausdorff topological vectorspaces

In this paper, following the ideas of [4, 12, 18] we study the stability of avector optimization problem using the notion of relative Pareto efficiency Therest of paper is organized as follows Section 2 presents the notion of relativePareto efficient and the relationships between this notions and the traditionalsolution concepts In Section 3, we establish the upper part of convergence inthe sense of Kuratowski-Painlev´e of relative Pareto efficient sets In Section 4,

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 calledrelative containment property, relative lower semicontinuous and relative upperHausdorff semicontinuous of a multifunctions around a given point Then, wederive some sufficient conditions for the lower semicontinuity of efficient pointmultifunctions under perturbations for cones with possibly empty interior Thenew 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 beused throughout: int A, cl A (or A), bd A, ri A, qri A and aff (A) stay for theinterior, closure, boundary, relative interior, quasi relative interior and affinehull 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 dering Θ containing the origin, A be a nonempty subset in Z and ¯z ∈ A Wesay that

or-(i) ¯z is a local extended Pareto efficient of A with respect to Θ, or it is a localΘ-efficient point of A if there exist a neighborhood V of ¯z such that

Z ordered by a closed and convex cone C ⊂ Z satisfying C\(−C) 6= ∅, 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 6= ∅ (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 6= ∅, (5)

where riC is the collection of interior points of C with respect to the closedaffine hull of C.

• ¯z is a quasi relative efficient point of A if

A ∩ (¯z − qriC) = ∅ provided that qriC 6= ∅, (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 relativeefficient point in (3)–(6) can be unified by a Θ-efficient point, where Θ is, foreach 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 thecondition int C 6= ∅ yeilds ri C = int C Furthermore, each relative efficientpoint is a quasi relative efficient point, since the ri C 6= ∅ yields qri C = ri C.(c) The condition C\(−C) 6= ∅ 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 allthe Pareto efficient poitnts of A with respect to C is denoted by Min (A | C)

It is easy to see 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]}

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 + e ∈ (¯z − ri C)

Thus

(¯x + e) ∈ A ∩ (¯z − ri C),contrary to ¯z ∈ ReMin (A | C)\Min (A | C) The proof is complete 2Corollary 1 (see [21, Proposition 4.3]) Let Z be a Banach space and C be aconvex pointed cone with int C 6= ∅ 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 generalizedorder optimality in [23, Definition 5.53] and gave a detailed analysis of thenotion of generalized order optimality and compared it with the traditionalnotions 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 kzkk → 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 isdenoted by GE(A | C)

Proposition 2 Suppose that C is a convex cone with qri C 6= ∅ If ¯z ∈ A is aquasi relative efficient point of A, then ¯z is a generalized efficient point of Awith respect to C

Proof Suppose that ¯z is a quasi relative efficient point of A with respect to

C Since qri C 6= ∅ we can select an element z0∈ qri C For each k ∈ N, put

zk= (k + 1)−1z0 From the convexity of C and z0∈ qri C imply that

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

Obviously, kzkk → 0 as k → ∞ This and (9) imply that ¯z is a generalized

Corollary 2 (see [29, Proposition 2.9]) Suppose that C is a convex cone with

ri C 6= ∅ If ¯z ∈ A is a relative efficient point of A, then ¯z is a generalizedefficient point of A with respect to C

Proposition 3 (see [29, Proposition 2.11]) If C is a convex cone with int C 6=

∅, then

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 ofsets

• 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

n→∞zn, zn∈ An for all large n},

Ls An:= {z ∈ Z , | z = lim

n→∞zk, zk ∈ Ank for some (Ank) ⊂ (An)}.Clearly, Li An⊂ Ls An If Ls An⊂ A ⊂ Li An, then we say that (An) converges

to A in the sense of Kuratowski-Painlev´e and we denote An −→ A FromKclosedness of Ls An and Li An imply that if An

K

−→ A, then A is a closedsubset When we consider the limits in the weak topology on Z, we denotethe 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 ofMosco and we denote An −→ A.M

• The convergence in the sense of Wijsman:

We say that (An) converges to A in the sense of Wijsman if

lim

n→∞d(An, x) = d(A, x)∀x ∈ Z,where d(A, x) = inf

a∈Ad(a, x)

• The convergence in the sense of Attouch-Wets:

Let x ∈ Z and let A, B be nonempty subsets in Z Define

d(x, A) = inf

a∈Ad(x, a) (d(x, ∅) = ∞),e(A, B) = sup

a∈A

d(a, B) (e(∅, B) = 0, e(∅, ∅) = 0, e(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-Wetsif

lim

n→∞hρ(An, A) = 0for all ρ > 0 We can split this notion of convergence into an upper part and

a lower part as follows

lim

n→∞eρ(An, A) = 0,and

a∈Ainf

b∈Bd(a, b)

For the relationships between the various notions of set convergence duced here, see, e.g., [1, 26] It is well known that, if Z is a finite dimensionalspace, the above quoted notions of set-convergence coincide whenever we con-sider a sequence (An) of closed sets

intro-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 /∈ Cfor all µ > 1

Proof On the contrary, suppose that there exist z /∈ ri C and µ > 1 such that

Let (Cn) be a sequence of convex cones in Z, and C ⊂ Z be a convex cone Forbrevity, in the sequel we write ReMin A, ReMin An, Min A, Min An, WMin A,and WMin Aninstead 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 relativeinterior and Cn\(−Cn) 6= ∅, C\(−C) 6= ∅ If

xk ∈ ReMin Ank such that lim

k→∞xk = x and x ∈ A From x /∈ ReMin Aimplies that there exists a ∈ A satisfying

x − a ∈ ri C,or

We claim that

x − a /∈ Ls (ri Cn)c.Indeed, if otherwise, then there exist

where (ri Ck) ⊂ (ri Cn) satisfying lim

k→∞zk = x − a From (13) we see that foreach k ∈ N there exists yk∈ Ck such that

Ls [cl (Cnc)] = Ls Cnc ⊂ Cc∪ {0} ⊂ (ri C)c.Thus

x − a ∈ (ri C)c,

contrary to (12) From Ls (ri Cnk)c⊂ Ls (ri Cn)cimplies that x−a /∈ Ls (ri Cnk)c.Since a ∈ A and (i) imply that there exist an ∈ An satisfying lim

n→∞an = a.Thus lim

k→∞ank= a, where ank∈ Ank∀k ∈ N From

lim

k→∞(xk− ank) = x − a /∈ Ls (ri Cnk)c,implies that there exists k0∈ N such that

xk0− ank0 ∈ (ri C/ nk0)c,or

xk0− ank0 ∈ ri Cnk0,contradicting the fact that xk0 is efficient of Ank0 with respect to Cnk0 The

In Theorem 1, the condition (ii) cannot be replaced the weaker condition

“Ls Cc

n⊂ cl (Cc)” To see this, we consider the following example

Example 1 Let Z = R2 and C = R+× {0} Let

and A = [−1, 1]×{0} It is easy to see that An

K

−→ A and Ls Cc

n= cl (Cc) = R2.However, we have ReMin An = An for all n ∈ N and ReMin A = {(−1, 0)}.Clearly,

Remark 2 If Cn = C for all n ∈ N and int C 6= ∅, then the condition (ii) inCorollary 3 is satisfied and Corollary 3 coincides with [19, Proposition 3.1]

Theorem 2 Let Cn and C be convex cones in Z with nonempty relative terior and Cn\(−Cn) 6= ∅, C\(−C) 6= ∅ If

Proof The conclusion of the Theorem is trivial lim inf

n→∞ d(ReMin An, B) = +∞.Thus it suffices to consider the case lim inf

n→∞ d(ReMin An, B) is finite Suppose

on the contrary that there is some bounded subset B ⊂ Z and some positivenumber γ > 0 such that

lim inf

n→∞ d(ReMin An, B) < γ < α, (18)where α := d(ReMin A, B) By taking a subsequence of (An) if necessary wemay assume that

y0− a ∈ ri C,or

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 thatlim

n→∞an= a Consequently

lim(yn− an) = y0− a /∈ Ls (ri Cn)c

Thus there exists n0∈ N satisfying

yn0− an0 ∈ (ri C/ n0)c,or

yn0− an0 ∈ ri Cn0,contradicting the fact that yn0 is a relative efficient point of An0 with respect

Corollary 4 (see [18, Theorem 2.1]) Let Cn and C be convex cones in Z withnonempty interior and Cn\(−Cn) 6= ∅, C\(−C) 6= ∅ If the conditions (i)–(iii)

of Theorem 2 are satisfied, then

lim inf

n→∞ d(WMin An, B) ≥ d(WMin A, B), (20)for each bounded subset B

Theorem 3 Let Z be a reflexive Banach space, let (Cn) and C be convexpointed cones with nonempty relative interior If

n→∞ d(ReMin An, x) is finite Arguing bycontradiction, assume that

lim inf

n→∞ d(ReMin An, x) < d(ReMin A, x)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 y/ 0− a ∈

ri C From (i) implies that there is a sequence (an), an∈ Anfor n large enoughsatisfying

w − lim

n→∞an= a

We claim that there exists n0such that yn− an ∈ ri Cn for all n ≥ n0 Indeed,

if otherwise, then there exists (ynk− ank) ⊂ (yn− an) satisfying

yn − an ∈ (ri Cn )c ∀k ∈ N

From w − lim

k→∞(ynk− ank) = y0− a we have

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

4 Lower convergence of relative Pareto efficient sets

Let Z be a Banach space, ∅ 6= A ⊂ Z and let C ⊂ Z be a closed convex conewith C\(−C) 6= ∅ 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 theweak domination property ((W DP ) for brevity) If int C 6= ∅, then Θ = int C ∪{0} and the relative domination property coincides with the weak dominationproperty

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 closedconvex cone with nonempty relative interior and C\(−C) 6= ∅

Theorem 4 Suppose that the following conditions hold

ReMin An is relative compact

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 A is a nonempty compact set From the correctness of Θ and