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.

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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

1 Introduction

Kruger and Mordukhovich [16, Definition 5.53] have introduced the new con-cept 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 opti-mality solutions to vector optimization problems with generalized order (see, e.g., [8, 16, 17]) In [16, Theorem 5.59], Mordukhovich established some pre-liminary 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 Develop-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.

N V Tuyen

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

E-mail: tuyensp2@yahoo.com.

some necessary conditions to set-valued optimization problems with equilib-rium 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 con-vexity assumptions Recently, Bao and Mordukhovich [3, 4, 7] gave some new sufficient conditions for global weak Pareto (global Pareto) solutions to set-valued optimization problems However, we are not familiar with any sufficient optimality conditions for nonconvex vector optimization problems with gen-eralized 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 2 investigates the notion of generalized order optimality and preliminaries from variational anal-ysis In Section 3, we establish some necessary and sufficient conditions for global generalized solutions in vector optimization

2 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 6= ∅ and U ∩ AC6= ∅, with AC:= Z\A The set of all the boundary points of A denoted by bd (A) Definition 1 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 kzkk → 0 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 1 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 6= ∅, then the above optimality concept covers the conventional concept of optimality

Definition 2 Suppose that Θ is a convex cone with riΘ 6= ∅ 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

The set of all the relative efficient points of A is denoted by RE(A | Θ)

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 3 Let A be a nonempty subset in Z

(a) Suppose that intΘ 6= ∅ 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 1 (see [17, Proposition 2.11]) If Θ is a convex cone, then the following holds:

(i) If intΘ 6= ∅, then GE(A | Θ) ⊂ W E(A | Θ)

(ii) If intΘ 6= ∅, then W E(A | Θ) ⊂ RE(A | Θ)

(iii) If riΘ 6= ∅, 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 2 (see [11, Proposition 2]) Suppose that Θ is a convex cone with Θ\l(Θ) 6= ∅ 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

Proposition 3 (see [11, Theorem 2.3]) Let Z be a Banach space, A be a nonempty set in Z, and 0 ∈ Θ ⊂ Z Then

Proof Let ¯z ∈ GE(A | Θ) Then, there exists (zk) ⊂ Z with kzkk → 0 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 0 ∈ Θ

we have ¯z ∈ (A − Θ) Thus U ∩ (A − Θ) 6= ∅ Since lim

k→∞(¯z − zk) = ¯z, we 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 6= ∅ 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,1 k



∩ (A − Θ)C

6= ∅ ∀k ∈ N,

where B ¯z,1
:= z ∈ Z | kz − ¯zk ≤ 1 For each k ∈ N, let xk∈ B ¯z,1
∩

(A − Θ)C We have lim

k→∞xk = ¯z and {xk} ⊂ (A − Θ)C

For each k ∈ N, put

zk= ¯z − xk, then

lim

k→∞zk= 0 and z − z¯ k = xk ∈ (A − Θ)C

∀k ∈ N,

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

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

Lim sup

z→¯ z

F (z) := {z∗∈ Z∗| ∃ zk→ ¯z, z∗k w

∗

−→ z∗, z∗k∈ F (zk) ∀k ∈ N} Definition 4 (see [15, Difinition 1.1]) Let Ω ⊂ Z, ¯z ∈ Ω, and  > 0

(i) The set of -normals of Ω at ¯z is defined by

ˆ

N(¯z; Ω) :=

(

z∗∈ Z∗| lim sup

z →¯Ωz

hz∗, z − ¯zi

k z − ¯z k 6 

) ,

where the notation z −Ω→ ¯z means that z → ¯z and z ∈ Ω We call the closed convex cone ˆN (¯z; Ω) := ˆN0(¯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

z→¯ z

↓0

b

N(x, Ω),

that is,

N (¯z; Ω) =nz∗∈ Z∗| ∃ k→ 0+, zk→ ¯Ω z, zk∗w

∗

→ z∗, zk∗∈ ˆNk(zk; Ω) ∀ko

(6) Definition 5 (see [15, Difinition 1.20]) A set Ω ⊂ Z is said to be sequentially normally compact (SNC) at ¯z if for any sequences k ↓ 0, zk

Ω

−→ ¯z and z∗

k ∈ ˆ

Nk(zk, Ω) it holds



z∗k w

∗

−−→ 0 as k → 0=⇒kzk∗k → 0 as k → 0, where 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

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 6 (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 → 0 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 1 (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

xi∈ Ωi∩ (¯z + B), x∗i ∈ ˆN (xi, Ωi) f or i = 1, 2,

kx∗

1+ x∗2k ≤ , and 1 −  ≤ kx∗

1k + kx∗

2k ≤ 1 + 

3 Main results

Theorem 2 (Necessary condition) Let Z be an Asplund space, and let

∅ 6= A ⊂ Z, 0 ∈ Θ ⊂ Z be closed subsets If ¯z ∈ GE(A | Θ), then there exists

0 6= 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 0

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→ 0 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− ak ) 6= ∅

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 1 to the system {Ω1, Ω2, (¯z, ¯z)} shows that for each k ∈ N, there are elements (uik, zik) and (u∗ik, −z∗ik) for i = 1, 2 satisfying the relationships











(uik, zik) ∈ Ωi with (uik, zik) → (¯z, ¯z), and (u∗ik, −z∗ik) ∈ ˆN ((uik, zik), Ωi) , i = 1, 2, with k(u∗1k, z1k∗ )k + k(u∗2k, z2k∗ )k → 1, and k(u∗

1k, z∗ 1k) + (u∗

2k, z∗ 2k)k → 0

(11)

From

(u∗2k, −z∗2k) ∈ ˆN ((u2k, z2k), Ω2)

∈ ˆN ((u2k, z2k), A × {¯z})

= ˆN (u2k, A) × ˆN (¯z, {z})

it follows that u∗2k∈ ˆN (u2k, A) Since

(u∗1k, −z1k∗ ) ∈ ˆN ((u1k, z1k), Ω1)

= ˆN ((u1k, z1k), Z × (A − Θ)), (13)

we have

(u∗1k− z∗

1k, −z∗1k) ∈ ˆN ((u1k, 0), Z × (−Θ))

= ˆN (u1k, Z) × ˆN (0, −Θ)

= {0} × (− ˆN (0, Θ)) (14) Hence, we get u∗1k− z∗

1k = 0 and z∗1k ∈ ˆN (0, Θ) The Asplund property of

Z and the boundedness of the sequence (u∗1k, z1k∗ , u∗2k, z∗2k) in (11) allow us to find a quadruple (u∗1, z1∗, u∗2, z∗2) such that

(u∗ik, −z∗ik)w

∗

→ (u∗i, −zi∗) for i = 1, 2 along some subsequences Employing (11)–(14)gives us

(u∗1, −z1∗) = (−u∗2, z∗2) = (z∗, −z∗), −z∗∈ N (¯z, A) and z∗∈ N (0, Θ)

To complete the proof of the theorem, it remains to show that z∗ 6= 0 in (8) under assumeed SNC Arguing by contradiction, suppose that z∗ = 0 The imposed SNC assumptions give us two cases:

Case1 : A is SNC at ¯z From u∗2k ∈ ˆN (u2k, A), u2k→ ¯z and u∗2kw

∗

→ 0 it follows thatku∗2kk → 0 Since

ku∗1kk ≤ ku∗1k+ u∗2kk + ku∗2kk (15)

≤ k(u∗1k, z1k∗ ) + (u∗2k, z2k∗ )k + ku∗2kk for all k ∈ N, (16)

we have ku∗1kk → 0 as k → ∞ Thus kz∗

1kk → 0 and kz∗

2kk → 0 This contra-dicts the nontriviality in the third line of (11)

Case2 : Θ is SNC at 0 From u∗1k = z∗1k, z∗1k ∈ ˆN (0, Θ) for all k ∈ N, and z∗

1k

w∗

→ 0 imply that kz∗

1kk → 0 and ku∗

1kk → 0 Since (11), we have k(u∗

Corollary 1 Let Z be an Asplund space, and let ∅ 6= A ⊂ Z be a closed subset Suppose that Θ is a closed convex cone and ri Θ 6= ∅ If ¯z ∈ RE(A | Θ) then there exists 0 6= 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 2 Let Z be an Asplund space, and let ∅ 6= A ⊂ Z be a closed subset Suppose that Θ is a closed convex cone and intΘ 6= ∅ If ¯z ∈ W E(A | Θ) then there exists 0 6= 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 3 Let Z be an Asplund space, and let ∅ 6= A ⊂ Z be a closed subset Suppose that Θ is a closed convex cone and Θ\l(Θ) 6= ∅ If ¯z ∈ E(A | Θ) then there exists 0 6= 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 0

Proof We have E(A | Θ) ⊂ GE(A | Θ) by Proposition 2 Assertion (19) is

Corollary 4 Let Z be an Asplund space and ∅ 6= A ⊂ Z, 0 ∈ Θ ⊂ Z,

¯

z ∈ GE(A | Θ) Suppose that A − Θ and Θ are closed subsets and the following condition

holds true Then, there exists z∗∈ Z∗ such that

0 6= −z∗∈ N (¯z, A − Θ) ∩ (−N (0, Θ)) (21) provided that either A − Θ is SNC at ¯z or Θ is SNC at 0

Proof We first show that if the condition (20) is satisfied, then

Indeed, suppose that ¯z ∈ GE(A | Θ) Then, there exists a sequence (zk) ∈ Z such that zk → 0 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 0 6= z∗∈ Z∗ such that

−z∗∈ N (¯z, A − Θ) ∩ (−N (0, Θ))

Theorem 3 (Necessary and sufficient condition) Let Z be an Asplund space, ∅ 6= A ⊂ Z, 0 ∈ 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

0 6= −z∗∈ N (¯z, A − Θ) ∩ (−N (0, Θ)) (23) Proof (⇒): Suppose that ¯z ∈ GE(A | Θ) Then, there exists a sequence zk → 0

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 0 6= z∗∈ Z∗ such that

−z∗∈ N (¯z, A − Θ) ∩ N (¯z, {¯z}), (25)

or, equivalent to

As in the proof of [5, Theorem 3.1], equation (26) gives

−z∗∈ (−N (0, Θ))

Hence, there exists z∗∈ Z∗ such that

0 6= −z∗∈ N (¯z, A − Θ) ∩ (−N (0, Θ)) (⇐): Arguing by contradiction, assume that there is z∗∈ Z∗such that

0 6= −z∗∈ N (¯z, A − Θ) ∩ (−N (0, Θ)) , (27) but ¯z /∈ GE(A | Θ) Therefore ¯z /∈ bd (A − Θ) by Lemma 3 From this and

¯

z ∈ (A − Θ) imply that ¯z ∈ int(A − Θ) Thus N (¯z, A − Θ) = {0}, contrary to

Example 1 Let Z = R2, A = {z = (z1, z2) ∈ R2 | z2 = −z1, 0 ≤ 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− 2 ≤ z2≤ z1, z1≤ 1}

∪ {z ∈ R2| z1≤ −|z2| + 2, 1 ≤ 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 3

Now we compare Theorem 3 with [16, Theorem 5 89], which characterizes the linear suboptimality of set systems via the relations of the exact extremal principle Given two subsets Ω1 and Ω2of a Banach space Z Put

ϑ(Ω1, Ω2) := sup{υ ≥ 0 | υB ⊂ Ω1− Ω2} (28) The constant ϑ(Ω1, Ω2) describing the measure of overlapping for these sets

Ω1and 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(Ω ∩ B (¯z), Ω ∩ B (¯z)) = 0 for some r > 0 (29)

Definition 7 (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

x 1 Ω1

−→¯ z

x 2 Ω2

−→¯ z r↓0

ϑ ([Ω1− x1] ∩ rB, [Ω2− x2] ∩ rB)

where the measure of overlapping ϑ(·, ·) is defined in (28)

The next result characterizes the linear suboptimality of set systems

Theorem 4 (see [16, Theorem 5 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

0 6= z∗ ∈ N (¯z; Ω1) ∩ (−N (¯z; Ω2)) (31) Furthermore, condition (31) is necessary and sufficient for the linear subex-tremality of {Ω1, Ω2} around ¯z if dim Z < ∞

The following examples show that the sufficient condition in Theorem 4 is not sufficient for an extremal system even in the finite dimensional case

Example 2 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} 6= {0}

Thus the system {A, Θ} is linearly subextremal around the point ¯z by Theorem

4 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 3 now shows that ¯z

is not an extremal point of the system {A, Θ}

Example 3 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}