Sectionwise connected sets in vector optimization

Operations Research Letters 37 (2009) 295–298

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Sectionwise connected sets in vector optimization E. Miglierina a,∗ , E. Molho b a

Dipartimento di Economia, Università degli Studi dell’Insubria, via Monte Generoso, 71, Varese, Italy

b

Dipartimento di Economia Politica e Metodi Quantitativi, Università degli Studi di Pavia, via S. Felice, 5, Pavia, Italy

article

info

Article history: Received 8 November 2008 Accepted 31 March 2009 Available online 15 April 2009 Keywords: Vector optimization Generalized convexity Stability

abstract We introduce the notion of sectionwise connected set as a new tool to investigate nonconvex vector optimization. Indeed, the image of a K -convex set through a K -quasiconnected vector function is proved to be sectionwise connected. Some properties of the minimal frontiers of sectionwise connected sets are studied. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In this work we introduce a generalization of the notion of convex set that seems to have interesting properties in the field of vector optimization. This notion is based on the arcwise connectedness of the conical sections of a set with respect to the ordering cone. The new class of sectionwise connected sets includes many sets that play an important role in vector optimization theory. It contains the so-called free disposal sets, introduced by G. Debreu in the field of economic theory and recently used to study the structure of the solution set of a vector optimization problem (see, e.g., [1,2]). Moreover, we prove that the image of a convex set through a K -quasiconnected vector function is sectionwise connected. Thus we develop a new tool to investigate nonconvex vector optimization through an image space approach. A natural comparison arises with the study of convex vector optimization in the image space. A key role in this field is played by the notion of K -convex set. We show that not every K -convex set is sectionwise connected. Hence, a K -convex set which is not sectionwise connected cannot be considered as the image of a convex set through a K -convex vector function. Therefore we conclude that the image of a convex vector optimization problem enjoys stronger properties than the simple K -convexity, since it is not only K -convex but also sectionwise connected. In this work we study some properties of the minimal frontiers of sectionwise connected sets in a finite dimensional framework. We prove that in a sectionwise connected set local and global

minimal points coincide. Moreover, every minimal point is also a strict minimal point in the sense of the definition introduced by Bednarczuk in [3]. This can be considered as a sort of stability property of the minimal frontier of a sectionwise connected set with respect to perturbations of the order structure. Indeed, strict minimality can be considered as one of the weakest forms of proper minimality since, in a finite dimensional setting, it avoids some pathological asymptotic behavior (see, [4]). Moreover, we develop a stability analysis of minimal frontiers of sectionwise connected sets. Indeed, we consider a sequence of sectionwise connected sets {Qn } converging in the sense of Kuratowski–Painlevé to a given set Q and we prove the lower convergence of the minimal frontiers of the perturbed sets Qn to the minimal frontier of Q . This result can be considered as an extension to the nonconvex case of some results proved in [5]. In what follows we use the following notations. We denote by Bδ (q) ⊂ Rl the closed ball of radius δ centered at q. Let K ⊂ Rl be a pointed (i.e. K ∩−K = {0}) closed and convex cone with int K 6= ∅. The cone K induces the order relation ≤K in Rl defined as follows: for every y1 , y2 ∈ Rl y1 ≤K y2 ⇔ y2 ∈ y1 + K .

Let Q ⊂ R be a nonempty subset, we define the set of minimal points as min Q = {y ∈ Q : Q ∩ (y − K ) = {y}} .

Corresponding author. E-mail address: [email protected] (E. Miglierina).

0167-6377/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2009.03.012

(2)

We will consider the vector optimization problem K − min f (x) x∈A

∗

(1)

l

(P)

where f : A ⊂ Rm → Rl is the objective function and A ⊂ Rm is the feasible region. The solution of this problem is the counterimage of min f (A).

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2. Sectionwise connected sets The aim of this section is to define and study a new generalization of the notion of convex set. First, we recall the classical definition of K -convexity, frequently used in the framework of vector optimization. Definition 1. Let K ⊂ Rl be a closed and convex cone. A set Q ⊂ Rl is said to be K -convex if the set Q + K is convex.

Definition 7 (See, e.g., [7]). Let K ⊂ Rl be a closed and convex cone and S ⊂ Rm be a convex set. The function f : S → Rl is said to be K -convex if, for y ∈ Rl and x1 , x2 ∈ S, we have f (α x1 + (1 − α) x2 ) ∈ f (α x1 ) + (1 − α) f (x2 ) − K

∀ α ∈ [0, 1] .

(6) l

The function f : S → R is said to be K -quasiconvex if, for y ∈ Rl and x1 , x2 ∈ S, we have

We propose a generalization of the notion of convex set following a different approach which is also suitable to study vector optimization problems.

f (x1 ), f (x2 ) ∈ y − K ⇒ f (α x1 + (1 − α) x2 ) ∈ y − K

Definition 2. Let K ⊂ Rl be a closed and convex cone. A set Q ⊂ Rl is said to be sectionwise connected if, for every y ∈ Rl , the set (y − K ) ∩ Q is arcwise connected.

We also introduce the notion of K -quasiconnected function, that can be considered as a vector version of the known notion of quasiconnected scalar function (see [8]).

It is easy to see that convexity implies both K -convexity and sectionwise connectedness of a given set. The following examples show that the two notions defined above are independent.

Definition 8. Let K ⊂ Rl be a closed and convex cone and let S ⊂ Rm be an arcwise connected set. The function f : S → Rl is said to be a K -quasiconnected function if for every x1 , x2 ∈ S there exists a continuous path γ (t ; x1 , x2 ) : [0, 1] → S with γ (0; x1 , x2 ) = x1 and γ (1; x1 , x2 ) = x2 , such that the following implication holds:

Example 3. Let K = R2+ = (x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 0 and





A1 = (x1 , x2 ) ∈ R2 : x1 = 0, 0 ≤ x2 ≤ 1 ,





f (x1 ), f (x2 ) ≤K y H⇒ f (γ (t ; x1 , x2 )) ∈ y − K

A2 = (x1 , x2 ) ∈ R2 : 0 ≤ x1 ≤ −1, x2 = 1 ,







The set A = A1 ∪ A2 ∪ A3 is K -convex but not sectionwise connected. Example 4. Let K = R2+ and A1 = (x1 , x2 ) ∈ R2 : x1 ≥ 0, x2 ≥ 1 ,





(4)

A2 = (x1 , x2 ) ∈ R : x1 ≥ 1, x2 ≥ 0 .



2

(7)

∀ t ∈ [0, 1] .

(8)

(3)

A3 = (x1 , x2 ) ∈ R2 : x1 = −1, 0 ≤ x2 ≤ 1 .



∀ α ∈ [0, 1] .



The set A = A1 ∪ A2 is sectionwise connected but not K -convex. Now let us show that the class of sectionwise connected sets is reasonably wide. Indeed, it contains, not only the convex sets, but also the so-called free disposal sets. This notion was introduced in the setting of Mathematical Economics by Debreu [6]. It was recently used to investigate the topological structure of the solution set of a vector optimization problem (see, e.g., [1]) and the approximate solutions in vector optimization [2].

It is well known that every K -convex function is also a K quasiconvex function. Moreover every K -quasiconvex function is also K -quasiconnected (it is sufficient to consider the continuous path γ (t ; x1 , x2 ) = tx1 + (1 − t )x2 ). Nevertheless the class of K quasiconnected functions is larger than the class of K -quasiconvex functions, as shown by the following example. Example 9. Let K = R2+ and let f (x, y) = (f1 (x, y), f2 (x, y)) : S → R2 where S = R2 and f1 (x, y) = f2 (x, y) = x2 y2 .

(9)

The function f (x, y) is K -quasiconnected but not K -quasiconvex. An important feature of the class of sectionwise connected sets is that it contains the images of arcwise connected sets through K -quasiconnected functions.

Definition 5. Let K ⊂ Rl be a closed and convex cone. A set Q ⊂ Rl is said to be free disposal if Q + K = Q .

Proposition 10. Let S ⊂ Rm be an arcwise connected set and let f : S → Rl be a continuous K -quasiconnected function, then f (S ) is a sectionwise connected set.

Proposition 6. Let K ⊂ Rl be a closed and convex cone. If the set Q ⊂ Rl is a free disposal set then Q is a sectionwise connected set.

Proof. Let y ∈ Rl and y1 , y2 ∈ (y − K ) ∩ f (S ). Therefore there exist x1 , x2 ∈ S such that f (x1 ) = y1 and f (x2 ) = y2 and a continuous path γ (t ; x1 , x2 ) defined for every t ∈ [0, 1] such that γ (0; x1 , x2 ) = x1 and γ (1; x1 , x2 ) = x2 . Now we consider the continuous path α(t ) = f (γ (t ; x1 , x2 )). It holds that α(0) = y2 , α(1) = y1 and α (t ) ∈ (y − K ) ∩ f (S ) for every t ∈ [0, 1], since f is a K -quasiconnected function. 

Proof. Let y ∈ Rl and y1 , y2 ∈ (y − K )∩ Q . Since Q is free disposal, y ∈ Q . Now let us consider the path

α(t ) =

  (1 − 2t )y1 + 2ty

for 0 ≤ t ≤

 (2 − 2t ) y + (2t − 1)y2

for

1 2

1 2

(5)

≤t≤1

joining y1 and y2 . It holds that α(t ) ∈ (y1 + K ) ∪ (y2 + K ) for every t ∈ [0, 1]. Hence α(t ) ∈ Q for every t ∈ [0, 1] because Q is a free disposal set. Finally, α(t ) ∈ (y − K ), which completes the proof.  Let us remark that the converse implication does not hold. Indeed, it is easy to see that any convex and bounded subset of Rl is sectionwise connected (with respect to any closed convex cone K with K 6= {0}) but it is not a free disposal set. Now let us consider the notion of convex vector valued function and some of its generalizations.

We recall that the image of a convex set by a K -convex function is a K -convex set and the image of a convex set by a continuous K - quasiconnected function is sectionwise connected. A K -convex set is not necessarily sectionwise connected as shown in Example 3 and thereby a K -convex set is not necessarily the image of a convex set by a K -convex function. Indeed, if a set is the image of a convex set by a K -convex function, it should be sectionwise connected since a K -convex function is always K -quasiconnected. Now we use the notion of sectionwise connected set to study some properties of minimal frontiers in vector optimization. The first result is a local–global property of the minimal points that recalls those proved in [8] in the scalar case: all locally minimal points are also minimal.

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Proposition 11. Let Q ⊂ Rl be a sectionwise connected set and q ∈ Q . If there exists a positive real number δ such that

(Q ∩ Bδ (q)) ∩ (q − K ) = {q}

(10)

297

The set Ls An is called the upper limit of the sequence of sets {An }, while the set Li An is called the lower limit of {An }. We say that a sequence of sets {An } converges in the sense of Kuratowski to the set A if Ls An ⊂ A ⊂ Li An , and we denote this convergence by K

then q ∈ min Q . Proof. By contradiction we suppose that there exists an element y ∈ Q ∩ (q − K \ {0}). Then, sinceQ is a sectionwise connected set, there exists a continuous path γ (t ) such that γ (0) = q, γ (1) = y and γ (t ) ∈ Q ∩ (q − K ) for every t ∈ [0, 1]. Hence for every δ > 0 there exists tδ ∈ (0, 1] such that γ (tδ ) ∈ (Q ∩ Bδ (q)) ∩ (q − K \ {0}), a contradiction.  Finally, the notion of sectionwise connected set is also related to the notion of strict minimality. This concept is a refinement of the notion of minimality introduced in [3] in order to avoid some undesirable features of minimal points in the image space. Let Q ⊂ Rl , a point y ∈ Q is said to be a strictly minimal point if for every real number ε > 0, there exists a real number δ > 0 such that

(Q − y) ∩ (Bδ (0) − K ) ⊂ Bε (0).

(11)

We denote the set of strictly minimal points of Q by St min Q . We can prove that in a sectionwise connected set all the minimal points are strictly minimal. Hence, the special geometrical structure of these sets allows us to avoid the existence of minimal points with pathological asymptotic behavior. Proposition 12. Let Q ⊂ Rl be a closed sectionwise connected set, then min Q = St min Q . Proof. By contradiction, we suppose that there exists an element q ∈ min Q \St min Q .

(12)

Without loss of generality, let q = 0. Hence there exist a positive real number η and a sequence {qn } ⊂ Q such that qn ∈ B 1 (0) − K

An → A. The inclusion Ls An ⊂ A is known as the upper part of convergence, while the inclusion A ⊂ Li An is the lower part of the convergence. Theorem 13. Let Qn , n = 1, 2, . . . be sectionwise connected subsets of Rl such that Qn ∩ (q − K ) is a closed set for every q ∈ Qn and for K

every n. If Qn → Q then min Q ⊂ Li min Qn . Proof. Let q ∈ min Q . Without loss of generality, let q = 0. Then, K

since Qn → Q , there exists a sequence {qn } such that qn ∈ Qn and qn → 0. Now let ε > 0. We claim that Sn = (qn − K ) ∩ Qn ⊂ Bε (0). Indeed, otherwise, there exists a sequence {yk } such that yk ∈ Snk , where {nk } is a subsequence of the integers, and kyk k > ε . Since Snk is an arcwise connected set, a continuous path α k (t ) exists such that α k (0) = yk and α k (1) = qnk . Moreover, an element t˜ exists such that α k (t˜) = ε . Now let y˜ k = α k (t˜). By compactness of Bε (0), we can find a convergent subsequence y˜ kj of y˜ k and we denote by y˜ its limit. We can easily observe that y˜ ∈ (q−K \{0})∩Q , which contradicts the minimality of q and proves the claim. Hence there exists n¯ such that for every n > n¯ it holds that Sn ⊂ Bε (0). Since Sn is a closed set, it is compact (for every n > n¯ ). Then min Sn 6= ∅ (see, e.g., Cor. 3.6, p. 48 in [7]). Let sn ∈ min Sn , then sn → 0. Since min Sn ⊂ min Qn , the thesis follows. 







Remark 14. The assumption on sectionwise connectedness of the set Qn can be weakened. Indeed, it is sufficient to suppose that (q − K ) ∩ Qn is arcwise connected for every q ∈ Qn . Theorem 13 allows us to state the following result concerning the stability properties of the image of the solution set of Problem (P).

n

and kqn k > η. Moreover there exists a sequence {kn } ⊂ int K such that kn → 0 and B 1 (0) ⊂ kn − K . Therefore qn ∈ (kn − K ) ∩ Q . n

Since 0 ∈ (kn − K ) ∩ Q , we find a continuous arc αn : [0, 1] → (kn − K ) ∩ Q such that αn (0) = 0 and αn (1) = qn . Now we can find a sequence {tn } ⊂ (0, 1) such that kαn (tn )k = η. For any n, set an = αn (tn ). By compactness of the sphere Sη (0) =   x ∈ Rl : kxk = η , there exists a subsequence ans such that ans → a where kak = η. Since Q is a closed set a ∈ Q ∩(−K \ {0}), which is in contradiction with the minimality at 0.  3. Convergence of minimal frontiers of sets In this section we will show that the notion of sectionwise connected set can also be used to prove some results concerning the stability of the minimal frontier. The main result is the lower convergence in the sense of Kuratowski–Painlevé of the minimal frontiers of a converging sequence of sectionwise connected sets. This theorem extends a similar result proved in [5] for convex sets. In order to study the stability properties of minimal frontiers we will use the well-known notion of Kuratowski–Painlevé set convergence. Let {An }n∈N be a sequence of subsets of a Euclidean space Rl . Set:

 Ls An =

y ∈ R : y = lim yk , yk ∈ Ank , l

Corollary 15. Let An , n = 1, 2, . . . be convex subsets of Rm and fn : An → Rl , be K -quasiconnected functions such that fn (An ) ∩ (fn (a) − K ) is a closed set for every a ∈ An and for every n. If there exist a set A ⊂ Rm and a function f : A → Rl such that K

fn (An ) → f (A) then min f (A) ⊂ Li min fn (An ). Remark 16. To ensure that fn (An ) ∩ (fn (a) − K ) is a closed set it is sufficient to suppose that for every n and for every a ∈ An the set {x ∈ A : f (x) ≤K f (a)} does not contain an unbounded sequence {xn } such that the sequence {f (xn )} is bounded. The proof follows immediately from 4.16 in [9]. A similar result in the convex case was proved in [5]. Recently, some results on the convergence of minimal frontiers of perturbed quasiconvex vector optimization problems were proved in [10] under some assumptions involving the boundedness of the sublevel sets of the limit objective function. Our approach allows us to consider a more general class of problems, as shown in the following example. Example 17. Let fn : R → R2 be fn = (f1n , f2n ) defined as follows: f1n (x) =

k→+∞



0 x

x≤0 x>0

 {nk } a subsequence of N  Li An =

(13)



y ∈ R : y = lim yn , yn ∈ An for all large n . l

n→+∞

(14)

 0   1 f2n (x) = x3 − 1 + n

x≤0 x > 0.

(15)

Let f : R → R2 be f = (f1 , f2 ) defined as follows: f1 (x) =



0 x

x≤0 , x>0

f2 (x) =



0

−x 3

x≤0 x > 0.

(16)

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E. Miglierina, E. Molho / Operations Research Letters 37 (2009) 295–298

Let An = A = R and let K = R2+ . The results in [10] cannot be used since f has unbounded level sets, but Theorem 13 can be applied with Qn = fn (An ) and Q = f (A). Indeed, it is easy to see that K

Qn → Q . Hence it holds that min f (A) ⊂ Li min fn (An ). References [1] J. Benoist, N. Popovici, The structure of the efficient frontier of finitedimensional completely shaded sets, Journal of Mathematical Analysis and Applications 250 (2000) 98–117. [2] E. Miglierina, E. Molho, F. Patrone, S.H. Tijs, Axiomatic approach to approximate solutions in multiobjective optimization, Decisions in Economics and Finance 31 (2008) 95–115.

[3] E. Bednarczuk, A note on lower semicontinuity of minimal points, Nonlinear Analysis Series A 50 (2002) 285–297. [4] A. Zaffaroni, Degrees of efficiency and degrees of minimality, SIAM Journal on Control and Optimization 42 (2003) 1071–1086. [5] R.E. Lucchetti, E. Miglierina, Stability for convex vector optimization problems, Optimization 53 (2004) 517–528. [6] G. Debreu, Theory of Value, John Wiley, New York, 1959. [7] D.T. Luc, Theory of Vector Optimization, in: Lecture Notes in Economics and Mathematical Systems, vol. 319, Springer-Verlag, Berlin Heidelberg, 1989. [8] M. Avriel, W.E. Diewert, S. Schaible, I. Zang, Generalized Concavity, Plenum Press, New York, London, 1988. [9] R.T. Rockafellar, R.J.-B. Wets, Variational Analysis, in: Grundlehren der matematischen Wissenschaften, vol. 317, Springer-Verlag, Berlin, 1998. [10] G.P. Crespi, M. Rocca, M. Papalia, Extended well-posedness of quasiconvex vector optimization problems, Journal of Optimization Theory and Applications 141 (2009) 285–297.