New Results on Second-Order Optimality Conditions in Vector Optimization Problems

DOI 10.1007/s10957-007-9242-9

New Results on Second-Order Optimality Conditions in Vector Optimization Problems

M. Hachimi · B. Aghezzaf

Published online: 25 July 2007

© Springer Science+Business Media, LLC 2007

Abstract In this paper, we study second-order optimality conditions for multiobjec- tive optimization problems. By means of different second-order tangent sets, various new second-order necessary optimality conditions are obtained in both scalar and vector optimization. As special cases, we obtain several results found in the literature (see reference list). We present also second-order sufficient optimality conditions so that there is only a very small gap with the necessary optimality conditions.

Keywords Multiobjective optimization · Efficient solutions · Constraint qualifications · Second-order tangent sets · Asymptotic second-order cones · Second-order necessary and sufficient conditions

1 Introduction

The investigation of the optimality conditions is one of the most attractive topics of optimization theory. For vector optimization, examination of most theoretical re- sults on the first-order necessary conditions reveals that Lin’s fundamental theorem (Theorem 5.1 in [1]) is usually a source for establishing the results (see, e.g. [2]). In recent years, there has been an increasing interest in the generalization of the second- order optimality conditions in multiobjective optimization problems and different ap-

Communicated by P.L. Yu.

The authors thank Professor P.L. Yu and the referees for valuable comments and helpful suggestions.

M. Hachimi

Faculté des Sciences Juridiques Economiques et Sociales, Universié Ibn Zohr, Agadir, Morocco B. Aghezzaf (

Département de Mathématiques et d’Informatique, Faculté des Sciences, Université Hassan II Aïn chock, Casablanca, Morocco

e-mail: [email protected]

proaches have been suggested (see, e.g. [3–16]). The tangent cone plays a fundamen- tal role in establishing first-order necessary optimality conditions. In [4], Aghezzaf and Hachimi obtained second-order necessary optimality conditions by means of a second-order tangent set which can be considered an extension of the tangent cone.

By extending the definition of tangent cone, it is possible to build other second-order tangent sets. Cambini et al. [11] and Penot [16] introduce a new second-order tan- gent set called the asymptotic second-order cone, which allows them to reduce the gap between necessary and sufficient conditions to an acceptable extent. Taking into account the importance of the asymptotic second-order cone, the aim of the present paper is to refine the necessary conditions obtained recently (see, e.g. [3–8]) and present new second-order optimality conditions for both scalar and vector optimiza- tion problems.

The outline of this paper is as follows. In Sect. 2, the notations are introduced and some preliminary results are given. In Sect. 3, we derive first the second-order necessary optimality conditions for a point to be a local (weak) efficient point of an arbitrary subset of the criteria space. We derive also second-order sufficient optimality conditions so that the gap with the necessary ones is reduced to an acceptable extent.

Second, we establish second-order necessary optimality conditions for a point to be a local (weak) efficient solution of a vector optimization problem with an arbitrary set (of the decision space) and a twice continuously differentiable objective function. We explain also the gap between a vector and scalar optimization; we deduce second- order necessary optimality conditions for scalar optimization problems. Using the relationship between a vector optimization problem and its corresponding individual scalar problems, we derive other second-order necessary optimality conditions for a point to be a local (weak) efficient solution of the vector optimization problem.

Finally, in Sect. 4, the feasible set is defined by inequality and equality constraints.

With the constraint qualification, we obtain a nonzero Lagrange multiplier associated with the objective function. In order to get positive Lagrange multipliers, we consider a regularity condition.

2 Preliminaries

In this section, we introduce notations and definitions which are used throughout the paper. Let R

be the n-dimensional Euclidean space. For x, y ∈ R

, by x y , we mean x

y

for all i; by x ≤ y , we mean x y and x = y ; and by x < y, we mean x

< y

for all i. We denote the inner product of x and y by xy = x

y = x

y

+ · · · + x

y

; here the superscript denotes the transpose of x. We denote by R

the set of vectors x ∈ R

which satisfy x

0 for all i.

For each S ⊂ R

, the sets int S, S, Fr S , cone S and conv S denote the interior, the closure, the boundary, the generated cone and the convex hull of S, respectively. We denote by B( x, ε) ¯ the open ball centered at x ¯ and of radius ε.

For any two vectors x = (x

, x

) and y = (y

, y

) in R

, x

y means that

x

< y

holds or x

= y

and x

y

. Similarly, x <

y means that x

< y

holds

or x

= y

and x

< y

; here the subscript lex is an abbreviation of lexicographic

order.

Now, for any twice continuously differentiable vector function g : R

→ R

and for any vectors x, y ∈ R

, we denote by ∇ g(x) and ∇

g(x)(y, y) the m × n Jacobian matrix and the m-dimensional vector whose ith component is y

∇

g

(x)y .

In this paper, we study a multiobjective optimization problem of the form (MOP) min f (x),

s.t. x ∈ S,

where f : R

→ R

is a function and S is a subset of the decision space R

. We do not yet fix the form of the constraint functions forming S, but refer to S in general.

We will refer to R

as the decision space and to R

as the criteria space.

Due to the conflicting nature of the objectives, a minimizer solution that simul- taneously minimizes all the objectives is usually not obtainable. We shall give the following concepts of solutions to problem (MOP). For other notions and their con- nections see [17, 18].

Definition 2.1 A point x ¯ ∈ S is called to be a local efficient [resp. local weak effi- cient] solution to problem (MOP) if there exists a neighborhood N of x ¯ such that no x ∈ S ∩ N satisfies f (x) ≤ f ( x) ¯ [resp. f (x) < f ( x)]. ¯

If the above definition holds with N = R

, the point x ¯ is called efficient [resp.

weak efficient] solution to problem (MOP). We also define efficiency in the criteria space.

Definition 2.2 A point y ¯ ∈ Z ⊂ R

is called to be a local efficient [resp. local weak efficient] point of Z if there exists a neighborhood V of y ¯ such that no y ∈ Z ∩ V satisfies y ≤ ¯ y [resp. y < y ¯ ].

Also, if the above definition holds with N = R

, the point y ¯ is called efficient [resp. weak efficient] point of Z.

Now, we focus on certain approximation sets to the feasible region. The following definition of tangent cone is used by Penot in [15]

Definition 2.3 Let S be a nonempty subset of R

. The tangent cone to S at x ¯ ∈ S is the set defined by

T (S, x) ¯ =

d ∈ R

| ∃ t

→ 0

, ∃ d

→ d such that x

= ¯ x + t

d

∈ S . Next, we define two second-order approximations of the feasible region S at x ¯ ∈ S which can be considered as two different extensions of the tangent cone T (S, x). ¯ These approximations are very useful for the formulation of the second-order opti- mality conditions.

Definition 2.4 Let S be a nonempty subset of R

. The second-order tangent set to S at x ¯ ∈ S is the set defined by

T

(S, x) ¯ =

(d, z) ∈ R

× R

| ∃ t

→ 0

, ∃ z

→ z such that x

= ¯ x + t

d + (1/2)t

z

∈ S

.

Definition 2.5 Let S be a nonempty subset of R

. The asymptotic second-order tan- gent cone to S at x ¯ ∈ S is the set defined by

T

(S, x) ¯ =

(d, z) ∈ R

× R

| ∃ t

→ 0

, ∃ r

→ 0

, ∃ z

→ z such that t

/r

→ 0, x

= ¯ x + t

d + (1/2)r

t

z

∈ S

. By definition, it follows immediately that the asymptotic second-order tangent cone is in fact a cone.

Other second-order approximations of the feasible region S at x ¯ ∈ S can be in- spired by a notion given in the work of Cambini et al. in [11]. Let k be any real number; we define the second-order tangent set of index k to S at x ¯ ∈ S by

T

(S, x) ¯ =

(d, z) ∈ R

× R

| ∃ t

→ 0

, ∃ r

→ 0

, ∃ z

→ z such that t

/r

→ k, x

= ¯ x + t

d + (1/2)r

t

z

∈ S

. The d -sections of T

(S, x) ¯ are denoted by

T

(S, x)(d) ¯ = { z ∈ R

| (d, z) ∈ T

(S, x) ¯ } .

Remark 2.1 It is easy to show that T

(S, x) ¯ = T

(S, x) ¯ and T

(S, x) ¯ = T

(S, x). ¯ Moreover, for every k 0, we observe that T

(S, x)(0) ¯ = T (S, x) ¯ and if d / ∈ T (S, x) ¯ then T

(S, x)(d) ¯ = ∅.

The following proposition shows the relationship between the second-order tan- gent set of index k > 0 and the classical second-order tangent set.

Proposition 2.1 Let S be a subset of X and let x ¯ ∈ S. Then, for all k > 0, T

(S, x) ¯ = (1/ k) T

(S, x). In particular, ¯ T

(S, x) ¯ = T

(S, x). ¯

Proof This follows immediately from the definition of T

(S, x). ¯

3 Geometric Optimality Conditions

In this section, we establish various second-order optimality conditions with the aid of the second-order tangent set and the asymptotic second-order tangent cone, both in the general case and in the differentiable case. First, we derive a second-order necessary optimality condition in the criteria space which allows us to deduce the rest.

Theorem 3.1 If y ¯ is a local (weak) efficient point of Z ⊂ R

, then for every k ∈ {0, 1}, the following holds:

T

(Z, y) ¯ ∩ = ∅ , (1)

where

= { (d, z) ∈ R

× R

| (d

, z

)

<

(0, 0)

, ∀ i } .

Proof Suppose to the contrary that (1) does not hold for some k ∈ {0, 1}, that is, there exist (t

, r

) → (0

, 0

) and z

→ z such that t

/r

→ k and

y

= ¯ y + t

d + (1/2)r

t

z

∈ Z. (2) If d

< 0, one has r

t

(z

)

= o(t

d

) so that (y

)

− ( y) ¯

< 0 when n is sufficiently large. If d

= 0, one has (z

)

< 0. Hence, for n large,

(y

)

− ( y) ¯

= (1/2)r

t

(z

)

< 0.

Consequently, y

< y ¯ for n sufficiently large, which contradicts the hypothesis of the

theorem.

Remark 3.1 As shown in the proof of Theorem 3.1, the condition (1) holds for every real number k. However, nothing essential is lost when k is restricted by the condition k ∈ { 0, 1 } . In fact, if k < 0, then T

(Z, y) ¯ is empty and (1) holds trivially; if k > 0, we have the following equivalence:

T

(Z, y) ¯ ∩ = ∅ ⇐⇒ (1/ k) T

(Z, y) ¯ ∩ (1/ k) = ∅

⇐⇒ T

(Z, y) ¯ ∩ = ∅ ,

because is a cone and (1/ k) T

(Z, y) ¯ = T

(Z, y) ¯ by Proposition 2.1.

The Theorem 3.1 is in the same spirit as the Aghezzaf and Hachimi Theorem 3.2 in [4], but with an additional information. This is illustrated by the following example.

Example 3.1 We consider the following set Z used by Cambini et al. in [11]:

Z = {(x, y) ∈ R

| y = x √

x}, where y ¯ = (0, 0).

It can be seen that

T

(Z, y) ¯ = T

(Z, y) ¯ = { (0, 0) } × { (z

, z

) ∈ R

| z

= 0}

and

T

(Z, y) ¯ = { (d, z) ∈ R

| d

= 0, d

= 0, and d

× z

0} ∪ T

(Z, y). ¯ Hence, condition (1) is satisfied for k = 1 but is not satisfied for k = 0, which implies that y ¯ is not efficient point of Z .

Corollary 3.1 If y ¯ is a local (weak) efficient point of Z ⊂ R

, then for every direction d ∈ T (Z, y) ¯ ∩ Fr ( R

) the following holds:

T

(Z, y)(d) ¯ ∩ (d) = ∅ , k ∈ {0, 1} , (3) where

(d) = { z ∈ R

| (d, z) ∈ } .

Proof It follows immediately from Theorem 3.1.

Lin [1] has established sufficient conditions for local efficiency. We extend Lin’s Theorem 5.2 by means of second-order tangent sets.

Theorem 3.2 A sufficient condition for a point y ¯ ∈ Z ⊂ R

to be a local efficient point of Z is that

T (Z, y) ¯ ∩ ( R

) ⊆ Fr ( R

) (4) and for any d ∈ H = T (Z, y) ¯ ∩ Fr ( R

) − {0}, if H is nonempty, we have

T

(Z, y)(d) ¯ ∩ d

∩ (d) = {0} , (5a) T

(Z, y)(d) ¯ ∩ d

∩ (d) = ∅ , (5b) where d

denotes the orthogonal subspace to d.

Proof First let us assume that T (Z, y) ¯ ∩ ( R

) = { 0 } . Then, y ¯ is a local efficient point of Z by Theorem 5.2 in [1]. Now, let us assume that T (Z, y) ¯ ∩ ( R

) is strictly larger than { 0 } and that condition (5) holds, but y ¯ is not a local efficient point of Z. By Definition 2.2, there would exist a sequence { y

} ⊂ Z such that

y

≤ ¯ y, y

→ ¯ y. (6)

We can assume that y

= ¯ y + t

d

, where d

= 1, d

→ d and t

= y

− ¯ y . Thus, d ∈ T (Z, y) ¯ with d = 1. It is obvious that d ∈ R

. Taking into account condition (4), we conclude that d ∈ T (Z, y) ¯ ∩ Fr ( R

) and d = 0.

Setting

z

= 2(d

− d)/t

, we have the following:

d

= d + (1/2)t

z

and y

= ¯ y + t

d + (1/2)t

z

. (7) We consider the case when { z

} is bounded and alternatively when it is not bounded.

First, we assume that { z

} is bounded. Then, taking a subsequence if necessary, there exist z such that z

→ z. So, from (7), we get

z ∈ T

(Z, y)(d) ¯ ∩ T (C, d), where

C = { v ∈ R

| v = 1 } .

Since T (C, d) = d

, we have z ∈ T

(Z, y)(d) ¯ ∩ d

. On the other hand, it is easy to

show that z ∈ (d), which contradicts condition (5b).

We assume now that { z

} is not bounded. Then, taking a subsequence if nec- essary, we can suppose that z

→ +∞ and that there exists z

= 0 such that z

= z

z

→ z

. From (7), we get

d

= d + (1/2)r

z

and y

= ¯ y + t

d + (1/2)r

t

z

, (8) where

r

= z

t

= 2 d

− d → 0

and t

/r

→ 0.

This implies that z

∈ T

(Z, y)(d) ¯ ∩ d

. It is also easy to show that z

∈ (d), which

again contradicts condition (5a).

The following example shows a case in which the Lin Theorem 5.2 is not applica- ble. However Theorem 3.2 can be applied.

Example 3.2 We consider the set Z defined by Z = { (x, y) ∈ R

| y = x

| x |} , with y ¯ = (0, 0).

It can be seen that

T (Z, y) ¯ = { (d

, d

) ∈ R

| d

= 0}

and that, for each d ∈ T (Z, y) ¯ and d = 0, we have

T

(Z, y)(d) ¯ = ∅, T

(Z, y)(d) ¯ = {z ∈ R

| z

· d

0}.

It is easy to show that condition (4) holds. Calculations show also that conditions (5) hold, which implies that y ¯ is an efficient point of Z .

In what follows, we assume that the objective function f is twice continuously differentiable. Now, we are in position to give the second-order necessary optimality conditions in the decision space.

Theorem 3.3 If x ¯ is a local (weak) efficient solution for problem (MOP), then the system

( ∇ f

( x)d, ¯ ∇ f

( x)z ¯ + k ∇

f

( x)(d, d)) < ¯

(0, 0), ∀ l, has no solution (d, z) in T

(S, x) ¯ for every k ∈ {0, 1}.

Proof According to Definitions 2.1 and 2.2, it is easy to see that, if x ¯ is a local (weak) efficient solution to problem (MOP), then there exists a neighborhood N of

¯

x such that f ( x) ¯ is (weak) efficient point of f (S ∩ N ). On the other hand, for any fixed k, let (d, z) be any element of T

(S, x). Then, taking into account ¯ T

(S, x) ¯ = T

(S ∩ N, x), there exist ¯ (t

, r

) → (0

, 0

) and z

→ z such that t

/r

→ k and

x

= ¯ x + t

d + (1/2)r

t

z

∈ S ∩ N. (9)

By the Taylor expansion, there exist ε

→ 0 such that f (x

) = f ( x) ¯ + t

∇ f ( x)(d ¯ + (1/2)r

z

+ (1/2)t

∇

f ( x)(d ¯ + (1/2)r

z

, d + (1/2)r

z

) + t

ε

. Thus, we have

f (x

) = f ( x) ¯ + t

∇ f ( x)d ¯ + (1/2)t

r

[∇ f ( x)(z ¯

)

+ k

∇

f ( x)(d ¯ + (1/2)r

z

, d + (1/2)r

z

) + 2k

ε

] , with k

→ k. Since f is twice continuously differentiable, then

∇ f ( x)(z ¯

) + k

∇

f ( x)(d ¯ + (1/2)r

z

, d + (1/2)r

z

)

n→+∞

−→ ∇ f ( x)z ¯ + k ∇

f ( x)(d, d), ¯

which implies that

( ∇ f ( x)d, ¯ ∇ f ( x)z ¯ + k ∇

f ( x)(d, d)) ¯ ∈ T

(f (S ∩ N ), f ( x)). ¯ Since f ( x) ¯ is a (weak) efficient point of f (S ∩ N ), Theorem 3.1 leads to

T

(f (S ∩ N ), f ( x)) ¯ ∩ = ∅ . Hence,

( ∇ f ( x)d, ¯ ∇ f ( x)z ¯ + k ∇

f ( x)(d, d)) ¯ ∈ .

This complete the proof.

Remark 3.2 It is easy to show that the set defined by

= {(d, z) ∈ R

| (∇f

(¯ x)d, ∇f

(¯ x)z + k∇

f

( x)(d, d)) < ¯

(0, 0), ∀l}

is a cone. Hence, for each k > 0, the following holds:

∩ T

(S, x) ¯ = ∅ ⇐⇒ ∩ (1/ k) T

(S, x) ¯ = ∅ ⇐⇒ ∩ T

(S, x) ¯ = ∅ . Definition 3.1 The set of the critical directions for problem (MOP) at x ¯ ∈ S is the set defined by

K = { d ∈ R

| d ∈ T (S, x), ¯ ∇ f ( x)d ¯ 0, ∇ f

( x)d ¯ = 0, at least one i } . (10) For any feasible point x ¯ ∈ S and d ∈ R

, we set

I

= { 1, . . . , } , I

( x, d) ¯ = { l ∈ I

| ∇ f

( x)d ¯ = 0 } .

Corollary 3.2 If x ¯ is a local (weak) efficient solution to problem (MOP), then for each critical direction d ∈ K, the system

∇ f

( x)z ¯ + k ∇

f

( x)(d, d) < ¯ 0, l ∈ I

( x, d), ¯ (11) has no solution z in T

(S, x)(d) ¯ for every k ∈ {0, 1}.

Proof This follows immediately from Theorem 3.3.

Now, we consider the scalar optimization problems and we extend Corollary 3.2 to such problems. The following lemma is useful (see [19]) and makes things easy.

Its proof is obvious and is omitted here.

Lemma 3.1 Let and T be sets in R

. Suppose that is a open half space. Then, T ∩ = ∅ ⇒ conv T ∩ = ∅,

where conv T is the closure of the convex hull of T .

Now, we can state the second-order optimality conditions for scalar optimization problems.

Theorem 3.4 Let = 1. If x ¯ is a (local) minimizer solution to problem (MOP), then for each critical direction d ∈ K, the system

∇ f ( x)z ¯ + k ∇

f ( x)(d, d) < ¯ 0 has no solution z in conv [ T

(S, x)(d) ¯ ] for every k ∈ {0, 1}.

Proof For k ∈ {0, 1}, let

= { z ∈ R

| ∇ f ( x)z ¯ + k ∇

f ( x)(d, d) < ¯ 0} .

The set may be either empty or R

or an open half space. In each case, Lemma 3.1 holds. Consequently, the result follows from Corollary 3.2 and Lemma 3.1.

Now, let us return to vector optimization problems. Before deriving a variant of Corollary 3.2, we needs the following notations and lemma.

For each i = 1, 2, . . . , , we define the nonempty sets Q

and Q by

Q

= { x ∈ R

| x ∈ S, f

(x) f

( x), l ¯ = 1, 2, . . . , and l = i } , (12) Q = { x ∈ R

| x ∈ S, f

(x) f

( x), l ¯ = 1, 2, . . . , } , (13) and the ith objective constraint problem (P

) by

(P

) min f

(x),

s.t. x ∈ Q

.

The following lemma gives the relationship between the multiobjective problem (MOP) and the corresponding individual scalar problems.

Lemma 3.2 A point x ¯ is a (local) efficient solution to problem (MOP) if and only if it is a (local) minimizer to problem (P

) for every i = 1, . . . , .

Proof This follows immediately from proof of Theorem 3.4.11 in [20].

Theorem 3.5 If x ¯ is (local) efficient solution to problem (MOP), then for each critical direction d ∈ K, the system

∇ f

( x)z ¯ + k ∇

f

( x)(d, d) < ¯ 0, for at least one l ∈ I

( x, d), ¯ (14) has no solution z in

i=1

conv [ T

(Q

, x)(d) ¯ ] for every k ∈ {0, 1}.

Proof It follows immediately from Theorem 3.4 and Lemma 3.2.

The following theorem shows that we can obtain sufficient optimality conditions by replacing the strict inequalities in (11) by nonstrict inequalities.

Theorem 3.6 A sufficient condition for a point x ¯ ∈ S to be a local efficient solution to problem (MOP) is that

T (S, x) ¯ ∩ { d ∈ R

| ∇ f ( x)d ¯ 0} ⊆ K (15) and for any critical direction d ∈ K − {0}, if K − {0} is nonempty, the following two systems have no solution z ∈ R

:

∇ f

( x)z ¯ 0, l ∈ I

( x, d), ¯ z ∈ T

(S, x)(d) ¯ ∩ d

, z = 0, and

∇ f

( x)z ¯ + ∇

f

( x)(d, d) ¯ 0, l ∈ I

( x, d), ¯ z ∈ T

(S, x)(d) ¯ ∩ d

.

Proof The proof of this theorem makes use of the arguments of Theorem 3.2.

4 John and Kuhn–Tucker Optimality Conditions

In what follows, we consider the feasible region S defined by inequality and equality constraints. Let g : R

→ R

and h : R

→ R

be twice continuously differentiable functions and let I

= {1, . . . , p } and I

= {1, . . . , q } be the corresponding index sets.

We will denote the feasible region by

S = { x ∈ R

| g(x) 0, h(x) = 0} . For any feasible point x ¯ ∈ S and d ∈ R

, we set

I

( x) ¯ = { i ∈ I

| g

( x) ¯ = 0} , I

( x, d) ¯ = { i ∈ I

( x) ¯ | ∇ g

( x)d ¯ = 0} , I

= {1, . . . , } , I

( x, d) ¯ = { l ∈ I

| ∇ f

( x)d ¯ = 0} .

Definition 4.1 The kth second-order linearizing set to S at x ¯ is L

(S, x) ¯ =

(d, z) ∈ R

|

( ∇ g

( x)d, ¯ ∇ g

( x)z ¯ + k ∇

g

( x)(d, d)) ¯

(0, 0), i ∈ I

( x), ¯ ( ∇ h

( x)d, ¯ ∇ h

( x)z ¯ + k ∇

h

( x)(d, d)) ¯ = (0, 0), j ∈ I

. Definition 4.2 The kth weak second-order linearizing set to S at x ¯ is

W L

(S, x) ¯ =

(d, z) ∈ R

|

( ∇ g

( x)d, ¯ ∇ g

( x)z ¯ + k ∇

g

( x)(d, d)) < ¯

(0, 0), i ∈ I

( x), ¯ ( ∇ h

( x)d, ¯ ∇ h

( x)z ¯ + k ∇

h

( x)(d, d)) ¯ = (0, 0), j ∈ I

.

In order to derive the second-order necessary optimality conditions for problem (MOP), we prove the following lemmas, which shows the relationship between the second-order tangent sets and the second-order linearizing sets.

Lemma 4.1 Let x ¯ ∈ S be any feasible solution to problem (MOP). Then,

T

(S, x) ¯ ⊆ L

(S, x), ¯ k ∈ { 0, 1 } . (16) Moreover, if the vectors {∇ h

( x), ¯ j ∈ I

} are linearly independent, then

W L

(S, x) ¯ ⊆ T

(S, x), ¯ k ∈ { 0, 1 } . (17) Proof It is similar to the proof of Lemma 4.1 in [5]. See also [9].

It follows immediately from (16) that, for every direction d ∈ R

,

T

(S, x)(d) ¯ ⊆ L

(S, x)(d), ¯ k ∈ {0, 1} , (18) where

L

(S, x)(d) ¯ = { z ∈ R

| (d, z) ∈ L

(S, x) ¯ } . Since L

(S, x)(d) ¯ is a closed convex set for each d ∈ R

, (18) leads to

conv [ T

(S, x)(d) ¯ ] ⊆ L

(S, x)(d), ¯ k ∈ {0, 1} . (19)

Lemma 4.2 Let x ¯ ∈ S be any feasible solution to problem (MOP). Then,

i=1

conv [ T

(Q

, x)(d) ¯ ] ⊆ L

(Q, x)(d), ¯ k ∈ { 0, 1 } . (20)

Proof Taking into account the fact that

i=1

L

(Q

, x)(d) ¯ = L

(Q, x)(d), the proof ¯

follows from condition (19).

4.1 John Type Necessary Optimality Conditions

Now, we reduce the geometric necessary optimality conditions of Corollary 3.2 in terms of the gradients and Hessians of the objective and the constraint functions.

Theorem 4.1 Let k ∈ { 0, 1 } . Suppose that the vectors {∇ h

( x), ¯ j ∈ I

} are linearly independent. If x ¯ is a local (weak) efficient solution to problem (MOP), then for each critical direction d ∈ K, the system

∇ f

( x)z ¯ + k ∇

f

( x)(d, d)) < ¯ 0, l ∈ I

( x, d), ¯ (21a)

∇ g

( x)z ¯ + k ∇

g

( x)(d, d)) < ¯ 0, i ∈ I

( x, d), ¯ (21b)

∇ h

( x)z ¯ + k ∇

h

( x)(d, d)) ¯ = 0, j ∈ I

, (21c) has no solution z in R

.

Proof It follows from Corollary 3.2 and condition (17) of Lemma 4.1.

In order to obtain the multiplier rules, the following lemma will be useful.

Lemma 4.3 Let A, B, C be given real matrices, with A = 0. Let a , b, c be real vectors. If the system

(ϒ ) Ax + a < 0, Bx + b 0, Cx + c = 0

has no solution x in R

, then exactly one of the two following systems (I) and (II) has a solution:

(I) λ

A + μ

B + ν

C = 0, λ

a + μ

b + ν

c 0, λ ≥ 0, μ 0,

(II) Az < 0, Bz 0, Cz = 0.

Proof It is easy to shows that (I) and (II) cannot hold simultaneously. Now, let us

assume that (I) has no solution. It is equivalent to the inconsistency of the following

system

λ

A + μ

B + ν

C + ξ 0 = 0, λ

a + μ

b + ν

c − ξ = 0, λ ≥ 0, μ 0, ξ 0.

By the Motzkin theorem of the alternative (see [21]), there exist z and t 0 satisfying Az + at < 0, Bz + bt 0, Cz + ct = 0. (22) Since the system (ϒ) has no solution, we have t = 0. So, (II) has a solution z.

Theorem 4.2 Let k ∈ {0, 1}. If x ¯ is a local (weak) efficient solution to problem (MOP), then for each critical direction d ∈ K, there exist λ ∈ R

, μ ∈ R

, ν ∈ R

not all zero such that

l=1

λ

∇ f

( x) ¯ +

i=1

μ

∇ g

( x) ¯ +

j=1

ν

∇ h

( x) ¯ = 0, (23a)

k

l=1

λ

∇

f

( x) ¯ +

i=1

μ

∇

g

( x) ¯ +

j=1

ν

∇

h

( x) ¯ (d, d) 0, (23b)

μ

g

( x) ¯ = 0, for each j ∈ I

, (23c)

λ

∇ f

( x)d ¯ = 0, for each l ∈ I

, (23d)

μ

∇ g

( x)d ¯ = 0, for each i ∈ I

. (23e)

Proof If the vectors {∇ h

( x), ¯ j ∈ I

} are linearly dependent, then one can find vec- tors λ 0, μ 0, ν not all zero such that the conditions (23) hold. Now suppose that the vectors {∇ h

( x), ¯ j ∈ I

} are linearly independent. Then, for d ∈ K, the sys- tem (21) has no solution z. Let us assume that the system (23) does not hold. By Lemma 4.3, there exits z such that

∇ f

( x)z < ¯ 0, l ∈ I

( x, d), ¯ ∇ g

( x)z < ¯ 0, i ∈ I

( x, d), ¯ ∇ h( x)z ¯ = 0.

On the other hand,

∇ f

( x)d ¯ = 0, l ∈ I

( x, d), ¯ ∇ f

( x)d < ¯ 0, l / ∈ I

( x, d), ¯

∇ g

( x)d ¯ = 0, i ∈ I

( x, d), ¯ ∇ g

( x)d < ¯ 0, i / ∈ I

( x, d), ¯ ∇ h( x)d ¯ = 0, because d is a critical direction. Thus, it follows that

∇ f ( x)(d ¯ + t z) < 0, ∇ g

( x)(d ¯ + t z) < 0, i ∈ I ( x), ¯ ∇ h( x)(d ¯ + t z) = 0, for any sufficiently small t > 0, which contradicts the first-order necessary conditions

for efficiency.

Remark 4.1 When k = 0, even if Theorem 4.2 states the characterization of local (weak) efficient solutions in terms of only first-order derivatives, it requires second- order differentiability, which is the price to be paid to obtain the additional optimality condition (23d, 23e).

In scalar objective optimization, the difference between the Fritz John and the Kuhn-Tucker conditions is that the multiplier λ of the objective function is assumed to satisfy one of the following conditions in the latter case:

(i) λ 0, λ = 0 (see for example [12]), (ii) λ > 0 (see for example [22]).

It is clear that, in scalar objective optimization, conditions (i) and (ii) are equivalent.

However, in multiobjective optimization, the equivalence between (i) and (ii) is not conserved. So, we are in position to state two types of Kuhn–Tucker conditions in multiobjective optimization.

4.2 Kuhn–Tucker Type 1 Necessary Optimality Conditions

It should be remarked in Theorem 4.2 that there is no guarantee that λ

= 0 for at least one l ∈ I

. In cases where λ = 0, the objective function does not play any role in the necessary conditions of efficiency. In order to avoid this undesirable situation, some assumptions on the feasible region have to be made.

Definition 4.3 The kth Abadie second-order constraint qualification (ASOCQ)

holds at x ¯ ∈ S in the direction d ∈ R

iff

T

(S, x)(d) ¯ = L

(S, x)(d). ¯ (24) If (ASOCQ)

holds at x ¯ ∈ S in every direction d ∈ R

, we say that (ASOCQ)

holds at x ¯ ∈ S.

The kth Guignard second-order constraint qualification (GSOCQ)

holds at x ¯ ∈ S in the direction d ∈ R

iff

conv [ T

(S, x)(d) ¯ ] = L

(S, x)(d). ¯ (25) It may be noted that, in the special case d = 0, conditions (24) and (25) are reduced to the first-order Abadie [24] and Guignard [25] constraint qualifications.

Theorem 4.3 Let k ∈ {0, 1}. Suppose that (ASOCQ)

holds at x ¯ ∈ S in the critical direction d ∈ K . If x ¯ is a local (weak) efficient solution to problem (MOP), then the system

∇ f

( x)z ¯ + k ∇

f

( x)(d, d)) < ¯ 0, l ∈ I

( x, d), ¯ (26a)

∇ g

( x)z ¯ + k ∇

g

( x)(d, d)) ¯ 0, i ∈ I

( x, d), ¯ (26b)

∇ h

( x)z ¯ + k ∇

h

( x)(d, d)) ¯ = 0, j ∈ I

, (26c)

has no solution z in R

.

Proof It follows from Corollary 3.2 and condition (24).

Theorem 4.4 Let k ∈ {0, 1}. Suppose that (ASOCQ)

holds at x ¯ ∈ S in the critical direction d ∈ K. If x ¯ is a local (weak) efficient solution to problem (MOP), then there exist multipliers λ ∈ R

, μ ∈ R

, ν ∈ R

, with λ = 0 such that the conditions (23) of Theorem 4.2 hold.

Proof The proof follows from Lemma 4.3 applied to the system (26).

Now, we consider the scalar optimization problems and we refine Theorems 4.3 and 4.4 for such problems.

Theorem 4.5 When = 1, Theorems 4.3 and 4.4 still hold if the second-order con- straint qualification (ASOCQ)

is replaced by the weaker second-order constraint qualification (GSOCQ)

.

Proof It follows on the lines of Theorem 4.3 and Theorem 4.4; it suffices to take into

account Theorem 3.4 instead Corollary 3.2.

4.3 Kuhn–Tucker Type 2 Necessary Optimality Conditions

It should be remarked in Theorem 4.4 that there is no guarantee that λ > 0 even if the constraint qualifications hold. In cases where λ

= 0 for some l ∈ I

, the corre- sponding objective function has no role in the necessary conditions of efficiency. In order to avoid this undesirable situation, some assumptions on the problem have to be made.

Definition 4.4 The kth Guignard second-order regularity condition (GSORQ)

holds at x ¯ ∈ S in the direction d ∈ R

iff

i=1

conv [ T

(Q

, x)(d) ¯ ] = L

(Q, x)(d). ¯ (27)

The term “regularity condition” is used instead of “constraint qualifications” be- cause both the objective and the constraint functions are involved.

Theorem 4.6 Let k ∈ {0, 1}. Suppose that (GSORQ)

holds at x ¯ ∈ S in the critical direction d ∈ K . If x ¯ is a (local) efficient solution to problem (MOP), then the system

∇ f

( x)z ¯ + k ∇

f

( x)(d, d) ¯ 0, l ∈ I

( x, d), ¯ (28a)

∇ f

( x)z ¯ + k ∇

f

( x)(d, d) < ¯ 0, for at least one s, (28b)

∇ g

( x)z ¯ + k∇

g

( x)(d, d) ¯ 0, i ∈ I

(¯ x, d), (28c)

∇ h

( x)z ¯ + k ∇

h

( x)(d, d) ¯ = 0, j ∈ I

, (28d)

has no solution z in R

.

Proof It follows from Theorem 3.5 and condition (27).

In order to obtain multiplier rules, the following lemma will be useful.

Lemma 4.4 Let A, B, C be given real matrices, with A = 0. Let a , b, c be real vectors. If the system

() Ax + a ≤ 0, Bx + b 0, Cx + c = 0,

has no solution x in R

, then exactly one of the two following systems (I) and (II) has a solution:

(I) λ

A + μ

B + ν

C = 0, λ

a + μ

b + ν

c 0, λ > 0, μ 0,

(II) Az ≤ 0, Bz 0, Cz = 0.

Proof The proof is similar to the proof of Lemma 4.3.

Theorem 4.7 Let k ∈ {0, 1}. Suppose that (GSORQ)

holds at x ¯ ∈ S in the critical direction d ∈ K. If x ¯ is a (local) efficient solution to problem (MOP), then there exist multipliers λ ∈ R

, μ ∈ R

, ν ∈ R

, with λ

> 0 for each l ∈ I

( x, d), such that the ¯ conditions (23) of Theorem 4.2 hold.

Proof The proof follows from Lemma 4.4 applied to the system (28).

A similar result was given in [3], in which only critical directions satisfying

∇ f ( x)d ¯ = 0 were considered, and improved in [5] where the authors relaxed the regularity condition considered in Theorem 3.2 of [3] and gave optimality conditions for each critical direction. Our results coincide with Theorem 5.5 in [5] in the case k = 1, but the case k = 0 is new.

It is worth noting that Theorems 4.6 and 4.7 embrace also the first-order optimal- ity conditions given by Maeda [23], which can be obtained by just considering the particular case d = 0.

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