Journal of Convex Analysis

Volume 9 (2002),No.1,97Û116

Alternative Theorems and Necessary Optimality

Conditions for Directionally DiÞerentiable

Multiobjective Programs

B.Jimæenez

Departamento de Matemæatica Aplicada,Universidad Nacional de

Educaciæon a Distancia,Apartado 60149,28080 Madrid,Spain

bjimen1@encina.pntic.mec.es

V.Novo

Departamento de Matemæatica Aplicada,Universidad Nacional de

Educaciæon a Distancia,Apartado 60149,28080 Madrid,Spain

vnovo@ind.uned.es

Received May 4,2001

Revised manuscript received September 20,2001

In this paper we study,in a unißed way,some alternative theorems that involve linear and sublinear

functions between ßnite dimensional spaces and a convex set,and we propose several generalizations of

them.These theorems are applied to obtain,under diÞerent constraint qualißcations,several necessary

conditions for a point to be Pareto optimum,both Fritz John and Kuhn-Tucker type,in multiobjective

programming problems which are deßned by directionally diÞerentiable functions and which include three

types of constraints:inequality,equality and set constraints.In particular,these necessary conditions

are applicable to convex programs and to diÞerentiable programs.

Keywords:Multiobjective programming,alternative theorems,necessary conditions for Pareto mini-

mum,Lagrange multipliers

2000 Mathematics Subject Classißcation:90C29,90C46

1.Introduction

Alternative theorems are indispensable tools in mathematical programming since they

allow the transformation of inequality systems,of a diácult direct approach,into equality

systems.

The ßrst alternative theorems come from the beginning of the century and from then

numerous generalizations have been proposed until today.For example,see Mangasarian

[10] for a study of the classic theorems.These deal with a ßnite number of linear functions

fromR

n

to R,and fromthere,the generalizations have included a ßnite or inßnite number

of convex or with some type of generalized convexity functions and even multifunctions

and increasingly abstract spaces.See Jeyakumar [7] for a review.

These theorems are used in the optimization theory to obtain necessary conditions,in

terms of Lagrange multipliers (dual form),so that a point will be an optimum for a

mathematical programming problem.Usually a necessary condition expressed through

the incompatibility of a system of equations and inequations,formed with the directional

derivatives of the functions involved in the problem (primal form),is transformed by an

ISSN 0944-6532/$ 2.50

c

Heldermann Verlag

98 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

alternative theorem in the checking of the existence of some multipliers,or what is the

same,in the checking of the compatibility of a system of equations,whose verißcation is

usually much more simple.Hence the great relevance of these theorems.

In this work the results obtained are not directly applied to the functions of the problem,

but to their directional derivatives,that is why,in the systems considered,the inequalities

are given by sublinear (positively homogeneous and convex) functions and the equalities,

by linear functions.

After introducing the notations in Section 2,we study,in Section 3,various generalizations

of some of the classic alternative theorems,working with linear or sublinear functions in

spaces of ßnite dimension and substituting the gradient by the subdiÞerential of the

Convex Analysis.The emphasis has been put in a unißed treatment of the diÞerent

situations,including a convex set constraint and giving a diÞerentiated treatment to the

equality constraints,which could be treated as two inequalities in some occasions,but in

others we obtain advantages from specißc treatment,otherwise we would obtain trivial

or inapplicable results.The equality constraints considered are given by linear functions,

suácient in many cases,since they are referred to derivatives in the applications.

Finally,in Section 4,the obtained alternative theorems are applied to the demonstration of

diÞerent necessary optimality conditions for multiobjective programs (in ßnite dimension)

with directionally diÞerentiable functions (not necessarily diÞerentiable neither convex).

This permits us to generalize,for example,the results of Singh [16] and of Giorgi and

Guerraggio [3],that deal with diÞerentiable functions,and also several results on convex

programs,in particular those of Kanniappan [8] and Islam [6].

2.Notations

Let x and y be two points of R

n

.Throughout this paper,we shall use the following

notations.

x ´ y if x

i

´ y

i

;i = 1;:::;n;x < y if x

i

< y

i

;i = 1;:::;n:

Let S be a subset of R

n

,as usual,cl S,int S,ri S,co S,aÞS,cone S,linS,will denote

the closure,interior,relative interior,convex hull,aáne hull,generated cone and linear

span by S,respectively.B(x

0

;®) is the open ball centered at x

0

and radius ® > 0.

Given a point x

0

2 S and a function f:R

n

!R

p

,the following multiobjective optimiza-

tion problem is considered

Minff(x):x 2 Sg.

It is said that x

0

is to be a weak Pareto minimumif there exists no x 2 S such that f(x) <

f(x

0

).The point x

0

is to be a local weak Pareto minimum,written x

0

2 LWMin(f;S),

if the previous condition is verißed on S\B(x

0

;®),for some ® > 0.The usual notion of

Pareto minimum is also used,will be denoted Min(f;S).

The following cones (Deßnition 2.1) and directional derivatives (Deßnition 2.2) are con-

sidered.

Deßnition 2.1.Let S º R

n

,x

0

2 cl S,

(a) The tangent cone (or contingent cone) to S at x

0

is

T(S;x

0

) = fv 2 R

n

:9t

k

> 0;9x

k

2 S;x

k

!x

0

such that t

k

(x

k

x

0

)!vg.

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 99

(b) The cone of linear directions (or radial tangent cone) is

Z(S;x

0

) = fv 2 R

n

:9® > 0 such that x

0

+tv 2 S 8t 2 (0;®]g.

(c) The cone of sequential linear directions (or sequential radial tangent cone,Penot

[12,Deßnition 2.3]) is

Z

s

(S;x

0

) = fv 2 R

n

:9t

k

!0

+

such that x

0

+t

k

v 2 S 8k 2 Ng.

We have that Z(S;x

0

) º Z

s

(S;x

0

) º T(S;x

0

).

Let D º R

n

,the polar cone to D is D

£

= fv 2 R

n

:hv;di ´ 0 8d 2 Dg and the strict

polar cone is D

s

= fv 2 R

n

:hv;di < 0 8d 2 D;d 6= 0g.If D is a subspace,then

D

£

= D

?

,orthogonal subspace to D.The normal cone to S at x

0

is the polar of tangent

cone:N(S;x

0

) = T(S;x

0

)

£

.If S is a convex set,one has that N(S;x

0

) = (S x

0

)

£

.

Deßnition 2.2.Let f:R

n

!R

p

;x

0

;v 2 R

n

.

(a) The Dini derivative (or directional derivative) of f at x

0

in the direction v is

Df(x

0

;v) = lim

t!0

+

[f(x

0

+tv) f(x

0

)]=t.

(b) The Hadamard derivative of f at x

0

in the direction v is

df(x

0

;v) = lim

(t;u)!(0

+

;v)

[f(x

0

+tu) f(x

0

)]=t.

(c) f is Dini diÞerentiable or directionally diÞerentiable (resp.Hadamard diÞerentiable)

at x

0

if its Dini derivative (resp.Hadamard derivative) exists in all directions.

The following properties hold:

Û If f is Fræechet diÞerentiable at x

0

,with Fræechet derivative rf(x

0

),then df(x

0

;v) =

rf(x

0

)v.

Û If df(x

0

;v) exists,then also Df(x

0

;v) exists and they are equal.

Û Df(x

0

;v) (resp.df(x

0

;v)) is the vector of components Df

i

(x

0

;v) (resp.df

i

(x

0

;v)),

i = 1;:::;p.

Let f:R

n

!R be a Dini diÞerentiable function at x

0

.The concept of subdiÞerential is

well known (see Penot [12]).

Deßnition 2.3.The Dini subdiÞerential of f at x

0

is

@

D

f(x

0

) = f¸ 2 R

n

:h¸;vi ´ Df(x

0

;v) 8v 2 R

n

g:

If Df(x

0

;v) is a convex function in v,then its subdiÞerential (in the Convex Analysis

sense) at v = 0 exists,and it is denoted @Df(x

0

;¡)(0).This is a nonempty,compact and

convex set of R

n

and the following asserts are true:

@

D

f(x

0

) = @Df(x

0

;¡)(0);

Df(x

0

;v) = Maxfh¸;vi:¸ 2 @

D

f(x

0

)g:

If Df(x

0

;v) is not convex,then @

D

f(x

0

) can be empty.

A function f whose Dini derivative Df(x

0

;¡) is convex was called by Pshenichnyi qua-

sidiÞerentiable at x

0

.This concept was later extended by Demyanov (see Demyanov and

Rubinov [2]).

100 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

3.Alternative Theorems

In this section,several generalizations of some alternative theorems are demonstrated.In

particular,we extend the classic theorems of Gordan and Motzkin [10] and the results of

Robinson [14,Theorem 3],and Ishizuka and Shimizu [5,Lemma 2].In the ßrst place,the

case of equalities is considered and,then,we work with both equalities and inequalities.

The proof of the following Lemma 3.1 is omitted because it is very simple.

Lemma 3.1.Let B be a nonempty set and C a cone of R

n

with 0 2 C.Then

0 =2 B +C,( B)\C =;,0 =2 B and ( cone B)\C = f0g.

Lemma 3.2.Let Q º R

n

be a convex set with 0 2 Q and h:R

n

!R

r

linear with

component functions h

k

;k 2 K = f1;:::;rg given by h

k

(u) = hc

k

;ui.Suppose the aáne

hull of Q is given by aÞQ = fx 2 R

n

:hd

j

;xi = 0;j = 1;:::;lg,being d

1

;:::;d

l

linearly

independent.Consider the following propositions:

(h1) 0 2

P

r

k=1

·

k

c

k

+N(Q;0);· 2 R

r

) · = 0.

(h2)

P

r

k=1

·

k

h

k

(u) µ 0 8u 2 Q;· 2 R

r

) · = 0.

(h3) (h3.1) c

1

;:::;c

r

;d

1

;:::;d

l

are linearly independent and (h3.2) Ker h\ri Q 6=;.

(h4) (h4.1) c

1

;:::;c

r

are linearly independent and (h4.2) linfc

k

:k 2 Kg\N(Q;0) =

f0g.

(h5) 0 2 int h(Q).

(h6) Ker h\ri Q 6=;.

(h7) 0 2 ri h(Q).

(h8) linfc

k

:k 2 Kg\(ri Q)

s

=;.

Then

(i) (h6) to (h8) are equivalent.

(ii) (h1) to (h5) are equivalent.

(iii) Each proposition (h1) to (h5) implies (h6),(h7) and (h8).

(iv) If int h(Q) 6=;then

1) (h4.1) holds.

2) (h1) to (h8) are equivalent.

Proof.(i) (h7) )(h8).Suppose that (h8) is false and take ¶ 2 linfc

k

:k 2 Kg\(ri Q)

s

.

Then ¶ =

P

r

k=1

·

k

c

k

and h

P

r

k=1

·

k

c

k

;qi < 0 8q 2 ri Qn f0g.

Let':R

r

!R be the linear application deßned by'(y) = h·;yi and À ='® h,which

is given by À(x) = h¶;xi.One has

À(q) < 0 8q 2 ri Qn f0g:(1)

By convexity of Q,Q º cl ri Q,and by the continuity of À,À(q) ´ 0 8q 2 Q.Thus

'(y) ´ 0 8y 2 h(Q).Therefore,y = 0 is a maximum of the convex function'on the

convex h(Q).By hypothesis,0 2 ri h(Q).But if a convex function reaches its maximum

at a relative interior point of its domain,then the function is constant (Rockafellar [15,

Theorem 32.1]).It follows that'(y) = 0 8y 2 h(Q),that is,'(h(q)) = 0 8q 2 Q,which

contradicts (1).

(h6),(h7).It is clear if we take into account that ri h(Q) = h(ri Q) [15,Theorem 6.6].

Not (h6) ) Not (h8).As Ker h\ri Q =;and Ker h is a convex cone,using Theorems

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 101

11.3 and 11.7 in [15],Ker h and Q are separated properly by a hyperplane M through the

origin.That is,there exists ¶ 2 R

n

;¶ 6= 0 such that M = Kerh¶;¡i and

h¶;qi ´ 0 ´ h¶;xi 8q 2 Q;8x 2 Ker h:

Hence ¶ 2 (Ker h)

£

= (Ker h)

?

= linfc

k

:k 2 Kg.It follows that Ker h º M,and as

M separates properly to Q and Ker h,there exists q

0

2 Q such that h¶;q

0

i < 0.As a

matter of fact

h¶;qi < 0 8q 2 ri Q;(2)

because if for some q

1

2 ri Q,h¶;q

1

i = 0,then the convex function h¶;¡i has a maximum

on Q at q

1

and,therefore,it is constant on Q,this is h¶;qi = 0 8q 2 Q,which contradicts

that M separates properly.

Accordingly,from (2),¶ 2 linfc

k

:k 2 Kg\(ri Q)

s

.

(ii) (h1),(h2).This is obvious.

(h1) )(h4).Let us prove the linear independence,since the second part is obvious.Let

P

r

k=1

·

k

c

k

= 0.Taking d = 0 2 N(Q;0) one has

P

r

k=1

·

k

c

k

+d = 0.By (h1),· = 0.

(h4) ) (h3).Let us observe in the ßrst place that (h4.2) ) (h8) because (ri Q)

s

º

Q

£

n f0g.Now,by (i),(h8),(h6)=(h3.2).We have to prove (h3.1).

Let

P

r

k=1

·

k

c

k

+

P

l

j=1

«

j

d

j

= 0.As Q º aÞQ = (linfd

1

;:::;d

l

g)

?

results that

N(aÞQ;0) = (aÞQ)

?

= linfd

1

;:::;d

l

g º N(Q;0).

Hence,d =

P

l

j=1

«

j

d

j

=

P

r

k=1

·

k

c

k

2 linfc

k

:k 2 Kg\N(Q;0);by (h4.2),d = 0.

Since the vectors d

j

are linearly independent,« = 0,and by (h4.1),· = 0.

(h3) )(h2).Suppose that the hypothesis of (h2) holds.Thus

h¶;ui = h·;h(u)i µ 0 8u 2 Q,being ¶ =

P

r

k=1

·

k

c

k

.Take q 2 Ker h\ri Q.It is verißed

that the concave function h¶;¡i has a minimum of value 0 on Q at q 2 ri Q.Hence it is

constant on Q,then h¶;ui = 0 8u 2 Q and,therefore,h¶;ui = 0 8u 2 aÞQ.Thereby

¶ 2 (aÞQ)

?

= linfd

1

;:::;d

l

g.Consequently,¶ =

P

l

j=1

«

j

d

j

=

P

r

k=1

·

k

c

k

,taking into

account (h3.1),« = 0 and · = 0.

The equivalence of (h2) with (h5) is proved in a similar way to the equivalence of (b) with

(e) in Theorem 3.9.

(iii) It is obvious since (h3.2)=(h6).

(iv) 1) int h(Q) 6=;)(h4.1).In fact,the hypothesis implies that h(R

n

) = R

r

,that is,h

has rank r.The conclusion follows from observing that the vectors c

1

;:::;c

r

are the rows

of the matrix of h in the canonical bases.

2) It is immediate since (h7) becomes (h5).

Lemma 3.3.If (h6),(h7) or (h8) is verißed then linfc

k

:k 2 Kg +N(Q;0) is closed.

Proof.Since ri Ker h = Ker h,one has by hypothesis (h6),ri Ker h\ri Q 6=;.By

Corollary 23.8.1 in [15],it follows

N[Ker h\Q;0] = N[Ker h;0] +N(Q;0) = linfc

k

:k 2 Kg +N(Q;0),

which is closed.

102 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

Let us observe that (h4.2) is a form more interesting than (h8),since (ri Q)

s

,which is,

in general,neither open nor closed,is substituted by N(Q;0),which is closed.

Let us prove with an example that in (h8) of Lemma 3.2 (ri Q)

s

cannot be substituted,

in general,by Q

£

(Example 3.4(a)),either by Q

s

,or by ri(Q

£

) (Example 3.4(b)) and that

the converse of (iii) is false.Also it is easy to prove that (h4.1) is neither a necessary nor

suácient condition so that (h6) to (h8) are verißed.

Example 3.4.

(a) Let h:R

3

!R

2

given by h(x

1

;x

2

;x

3

) = (x

1

;x

2

) and Q = fx = (x

1

;x

2

;x

3

) 2 R

3

:

x

1

= 0;x

3

µ 0g.It is easy to prove the following results.

1.L = linfc

k

:k 2 Kg = linf(1;0;0);(0;1;0)g = fx:x

3

= 0g,Ker h =

fx:x

1

= 0;x

2

= 0g.

2.h(Q) = fy = (y

1

;y

2

) 2 R

2

:y

1

= 0g,ri Q = fx:x

1

= 0;x

3

> 0g,

Q

£

= N(Q;0) = fx:x

2

= 0;x

3

´ 0g,ri(Q

£

) = fx:x

2

= 0;x

3

< 0g,Q

s

=;.

3.(h7) in Lemma 3.2 holds,and logically its equivalents (h6) and (h8).But,

(1;0;0) 2 L\Q

£

6= f0g,this means that (h4.2) does not hold.Then,the

converse of (iii) is false.Let us observe that int h(Q) =;.

(b) We change the set Q.It will be now Q = fx:x

1

= 0;x

2

µ 0;x

3

µ 0g.It is

proved without diáculty:

4.h(Q) = fy:y

1

= 0;y

2

µ 0g,ri Q = fx:x

1

= 0;x

2

> 0;x

3

> 0g,

Q

£

= fx:x

2

´ 0;x

3

´ 0g,ri(Q

£

) = Q

s

= fx:x

2

< 0;x

3

< 0g,

(ri Q)

s

= Q

£

n fx:x

2

= 0;x

3

= 0g.

5.(h7) in Lemma 3.2 does not hold,neither does (h8),obviously.However,L\

Q

s

=;and L\ri(Q

£

) =;.Note that c

1

;c

2

are linearly independent,therefore,

this is not a suácient condition so that (h6) to (h8) are fulßlled.

(c) In part (a) we change h.Now h(x

1

;x

2

;x

3

) = (x

1

;x

2

;x

1

+x

2

).3 of part (a) holds

since the sets of 1 and 2 are all maintained except h(Q) which now is h(Q) = fy 2

R

3

:y

1

= 0;y

2

= y

3

g and one has that 0 2 ri h(Q),that is,(h6),(h7) and (h8)

are fulßlled.The vectors c

1

= (1;0;0);c

2

= (0;1;0);c

3

= (1;1;0) are linearly

dependent,hence,this is not a necessary condition so that (h6) to (h8) are fulßlled.

Notice that Theorem 21.2 of Rockafellar [15] is not applicable to the situation described

by Lemma 3.2 since there are no strict inequality constraints.Example 3.4(a) above shows

that the two incompatible alternatives of the aforesaid theoremwould be verißed if it were

applicable.In fact,the vector u = (0;0;1) 2 ri Q verißes the system h

1

(u) ´ 0;h

2

(u) ´ 0

and considering µ

1

= 1;µ

2

= 0 we have the inequality µ

1

h

1

(u) +µ

2

h

2

(u) µ 0 8u 2 Q.

Theorem 3.5.Let f

1

;:::;f

p

be sublinear functions from R

n

to R with p µ 1,f =

(f

1

;:::;f

p

),h:R

n

!R

r

linear given by h = (h

1

;:::;h

r

) being h

k

(u) = hc

k

;ui,k 2 K =

f1;:::;rg,and Q º R

n

a convex set with 0 2 Q.Consider the following propositions:

(a) 0 2

P

p

i=1

µ

i

@f

i

(0) +

P

r

k=1

·

k

c

k

+N(Q;0);µ µ 0 implies µ = 0.

(b)

p

X

i=1

µ

i

f

i

(u) +

r

X

k=1

·

k

h

k

(u) µ 0 8u 2 Q;µ µ 0 (3)

implies µ = 0.

(c) There exists v 2 R

n

such that f(v) < 0;h(v) = 0;v 2 Q.

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 103

(d) 0 =2 co([

p

i=1

@f

i

(0)) +linfc

k

:k 2 Kg +N(Q;0).

Then

(i) (a),(b) and (d) are equivalent.

(ii) (c) ) (b).

(iii) If 0 2 ri h(Q),then (b) ) (c) and,consequently,the four are equivalent.

Proof.(b) )(a).Suppose that the hypothesis of (a) holds.Then there exist a

i

2 @f

i

(0)

and d 2 N(Q;0) such that

p

X

i=1

µ

i

a

i

+

r

X

k=1

·

k

c

k

+d = 0:(4)

As f

i

(u) µ ha

i

;ui 8u 2 R

n

and 0 µ hd;ui 8u 2 Q,it follows:

P

p

i=1

µ

i

f

i

(u) +

P

r

k=1

·

k

h

k

(u) µ

P

p

i=1

µ

i

ha

i

;ui +

P

r

k=1

·

k

hc

k

;ui +hd;ui = 0 8u 2 Q.

By (b),µ = 0.

(a) )(b).Suppose that (3) is verißed.Then,u = 0 is a minimum of the convex function

'=

P

p

i=1

µ

i

f

i

+

P

r

k=1

·

k

h

k

on Q.Hence,

0 2 @'(0) +N(Q;0) =

P

p

i=1

µ

i

@f

i

(0) +

P

r

k=1

·

k

c

k

+N(Q;0).

By (a),µ = 0.

Not (a),Not (d).Suppose that (4) holds for some µ µ 0,µ 6= 0,a

i

2 @f

i

(0),d 2

N(Q;0).We can assume that

P

p

i=1

µ

i

= 1,otherwise we just divide by

P

p

i=1

µ

i

> 0.

Hence,(4) means that 0 2 co([

p

i=1

@f

i

(0)) +linfc

k

:k 2 Kg +N(Q;0).The converse is

now obvious.

(c) ) (b).Suppose that (3) holds for some µ 6= 0 and let v be a vector satisfying (c).

Then

P

p

i=1

µ

i

f

i

(v)+

P

r

k=1

·

k

h

k

(v) < 0 in contradiction with what is obtained in (3) taking

u = v.Hence µ = 0.

(iii) Suppose that (c) is not fulßlled.Using Lemma 3.2(i) on has Ker h\ri Q 6=;

and then we can apply [15,Theorem 21.2 ] to f,h and h,resulting that there exist

(µ;«;¬) 2 R

p

¢R

r

¢R

r

such that (µ;«;¬) µ 0;µ 6= 0 and

P

p

i=1

µ

i

f

i

(u) +

P

r

k=1

«

k

h

k

(u)

P

r

k=1

¬

k

h

k

(u) µ 0 8u 2 Q,

therefore,(b) is not verißed taking ·

k

= «

k

¬

k

.

The following proposition shows the connection with subsequent theorems.Corollary 3.7

simplißes this theorem when there are no linear constraints.

Proposition 3.6.In the hypotheses of Theorem 3.5,if (d) holds and Ker h\ri Q 6=;,

then

D:= cone co([

p

i=1

@f

i

(0)) +linfc

k

:k 2 Kg +N(Q;0) is closed.

Proof.By Lemma 3.1,proposition (d) is equivalent to

(e)

(

(e1) 0 =2 co([

p

i=1

@f

i

(0))

(e2) [ cone co([

p

i=1

@f

i

(0))]\[linfc

k

:k 2 Kg +N(Q;0)] = f0g;

104 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

but (e1),by [4,Proposition 1.4.7,Chap.3],implies that cone co([

p

i=1

@f

i

(0)) is closed and

by Lemma 3.3,linfc

k

:k 2 Kg + N(Q;0) is closed.Taking into account (e2),by [15,

Corollary 9.1.3],we deduce that D is closed.

Corollary 3.7.Let f

1

;:::;f

p

be sublinear functions from R

n

to R,f = (f

1

;:::;f

p

) and

Q º R

n

a convex set which contains to 0.The following propositions are equivalent:

(a) 0 2

P

p

i=1

µ

i

@f

i

(0) +N(Q;0);µ µ 0 ) µ = 0.

(b)

P

p

i=1

µ

i

f

i

(u) µ 0 8u 2 Q;µ µ 0 ) µ = 0.

(c) There exists v 2 R

n

such that f(v) < 0;v 2 Q.

(d) 0 =2 co([

p

i=1

@f

i

(0)) +N(Q;0).

If some of the conditions (a)-(d) with Q = R

n

is satisßed,we will say that the sets

@f

1

(0);:::;@f

p

(0) are positively linearly independent.

This corollary is a generalization of the Gordan alternative theorem [10] since we obtain

it if Q = R

n

and if @f

i

(0) = fa

i

g,this is,if f

i

(u) = ha

i

;ui is linear.

Remark 3.8.It can be seen in Theorem 3.5 (with Q = R

n

,and thus,propositions (a)

to (d) are equivalent) a characterization of the compatibility of a system with inßnite

equations through the compatibility of inßnite systems with a ßnite number of equations.

In fact,proposition (c) is equivalent to

(c

0

) There exists a solution u 2 R

n

of the system

º

Max

a

i

2A

i

ha

i

;ui < 0;i = 1;:::;p

hc

k

;ui = 0;k = 1;:::;r

(5)

being A

i

= @f

i

(0),since f

i

(u) = Maxfha

i

;ui:a

i

2 A

i

g.

Proposition (a) can be formulated as:

(a

0

) For every a

i

2 A

i

;i = 1;:::;p,there exist no (µ;·) 2 R

p

¢R

r

;µ µ 0;µ 6= 0 such

that

P

p

i=1

µ

i

a

i

+

P

r

k=1

·

k

c

k

= 0,

and,by the classic Motzkin alternative theorem,this is equivalent to

(a

00

) For every a

i

2 A

i

;i = 1;:::;p,there exists a solution u 2 R

n

of the system

º

ha

i

;ui < 0;i = 1;:::;p

hc

k

;ui = 0;k = 1;:::;r:

(6)

(c

0

) expresses that a system of inßnite equations has a solution and it is equivalent to

(a

00

),that expresses that the inßnite ßnite systems of type (6) have a solution.

If in the equivalence (c

0

),(a

00

) the equality constraints are removed,we obtain Theorem

2.1 in Wang,Dong and Liu [18].

Note that every solution u of (5) it is also of (6),but not inversely.The solutions of (6)

depend on the election on a

1

;:::;a

p

.

The results obtained in Lemma 3.2 and Theorem 3.5 are combined in the following theo-

rem.

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 105

Theorem 3.9.In the hypotheses of Theorem 3.5,assume that the aáne hull to Q is

aÞQ = fx 2 R

n

:hd

j

;xi = 0;j = 1;:::;lg,being d

1

;:::;d

l

linearly independent.The

following propositions are equivalent:

(a) 0 2

P

p

i=1

µ

i

@f

i

(0) +

P

r

k=1

·

k

c

k

+N(Q;0);µ µ 0 ) µ = 0;· = 0.

(b)

P

p

i=1

µ

i

f

i

(u) +

P

r

k=1

·

k

h

k

(u) µ 0 8u 2 Q;µ µ 0 ) µ = 0;· = 0.

(c) (c1) c

1

;:::;c

r

;d

1

;:::;d

l

are linearly independent and (c2) there exists v 2 R

n

such

that

f(v) < 0;h(v) = 0;v 2 ri Q.

(d) (d1) c

1

;:::;c

r

are linearly independent,(d2) linfc

k

:k 2 Kg\N(Q;0) = f0g and

(d3) 0 =2 co([

p

i=1

@f

i

(0)) +linfc

k

:k 2 Kg +N(Q;0).

(e) 0 2 int[(f ¢h)(Q) +R

p

+

¢f0

r

g].

(f ) (f ¢h)(cone Q) +R

p

+

¢f0

r

g = R

p

¢R

r

.

For the demonstration of this theorem we need the following lemma.

Lemma 3.10.

(c2),

º

(c

0

) there exists w 2 R

n

such that h(w) = 0;w 2 ri Q

(c

00

) there exists v 2 R

n

such that f(v) < 0;h(v) = 0;v 2 Q:

Proof.It is clear that (c2) ) (c

0

) and (c

00

).For the other implication let w and v be

vectors satisfying (c

0

) and (c

00

),respectively,and v

µ

= µv +(1 µ)w.By linearity of h,

h(v

µ

) = 0,and by [4,Lemma 2.1.6,Chap.2],we deduce that v

µ

2 ri Q 8µ 2 [0;1).As

lim

µ!1

v

µ

= v,by the continuity of f,f(v

µ

) < 0 for µ near enough 1 and,consequently,all

these vectors v

µ

verify (c2).

Proof of Theorem 3.9.One has,obviously,the following equivalences:(a),(h1) and

(3.5a),(b),(h2) and (3.5b) and (d),(h4) and (3.5d),where (h1) denotes the same

proposition of Lemma 3.2,(3.5a) denotes proposition (a) of Theorem 3.5,etc.By Lemma

3.2(ii) and Theorem 3.5(i),it follows the equivalence of (a),(b) and (d).

By Lemma 3.10 one has that (c2),(h3.2) and (3.5c),and as (c1)=(h3.1) it follows

(c) = (c1) and (c2),(h3.1) and (h3.2) and (3.5c).

By Theorem 3.5(ii),(3.5c) ) (3.5a);by Lemma 3.2,(h3.1) and (h3.2),(h1) and by

Theorem 3.5(iii),(h3.2) and (3.5a) )(3.5c).Hence,(h3.1) and (h3.2) and (3.5c),(h1)

and (3.5a),(a).Consequently,(c),(a).

Let us prove that (b) )(e).Let

A = (f ¢h)(Q) +R

p

+

¢f0

r

g

= f(x;y) 2 R

p

¢R

r

:9u 2 Q such that f(u) ´ x;h(u) = yg:

A is a non-empty convex set and verißes that int A 6=;.In fact,if int A =;,A is

contained in a hyperplane,that is,there exist (µ;·) 2 R

p

¢R

r

n f(0;0)g such that A º

Kerh(µ;·);(¡;¡)i.Therefore,hµ;f(u)i +h·;h(u)i = 0 8u 2 Q and,by (b),(µ;·) = (0;0),

which is a contradiction.

Let us prove that 0 2 int A.Otherwise,as 0 2 A,there exists a supporting hyperplane to

A at 0 and by a similar reasoning to the previous one it also results a contradiction.

106 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

(e) ) (b).Suppose that the hypothesis of (b) holds,this is that hµ;f(u)i +h·;h(u)i µ

0 8u 2 Q.Then hµ;xi +h·;yi µ 0 8(x;y) 2 A.Hence,the concave function h(µ;·);(¡;¡)i

has a minimum on the convex A at (0;0) 2 int A (from hypothesis).Therefore,this

function is constant on A,namely,hµ;xi + h·;yi = 0 8(x;y) 2 A.Since int A 6=;,it

follows that (µ;·) = (0;0).

Finally,(e),(f).By [15,Corollary 6.4.1],condition (e) is equivalent to say that for all

u 2 R

p

¢R

r

there exists t > 0 such that 0 +tu 2 A,this means,cone A = R

p

¢R

r

.Now,

it is immediate that

cone A = cone[(f ¢h)(Q) +R

p

+

¢f0

r

g] = (f ¢h)(cone Q) +R

p

+

¢f0

r

g,

with what one has the equivalence of (e) and (f).

If some of the six equivalent conditions of Theorem 3.9 with Q = R

n

is verißed,we

will say f@

D

f

i

(0):i = 1;:::;pg is posindependent of fc

1

;:::;c

r

g,or,that f

1

;:::;f

p

are

posindependent of h

1

;:::;h

r

.This notion generalizes that of positive-linearly independent

vectors in Qi and Wei [13,Deßnition 2.1].

Theorem 3.10 extends Theorem 3 in Robinson [14] who supposes f linear and only con-

siders the propositions (b),(c) and (e).

Remark 3.11.

(1) Taking into account Lemmas 3.2 and 3.10,condition (c) can be expressed:

(c) 0 2 int h(Q) and there exists v 2 R

n

such that f(v) < 0;h(v) = 0;v 2 Q.

(2) When f is linear,condition (e) becomes the so-called Robinson constraint qualiß-

cation [14],(f) becomes Zowe-Kurcyusz constraint qualißcation [19] and,if further-

more Q = R

n

,(c) becomes the classic Mangasarian-Fromovitz constraint qualißca-

tion [11].

Finally we approach the Motzkin alternative theorem in the most general situation that

includes inequality constraints,both strict and not strict,equality constraints and convex

set constraint.

Theorem 3.12.Let f

1

;:::;f

p

;g

1

;:::;g

m

be sublinear functions from R

n

to R with p µ 1

and m µ 0,f = (f

1

;:::;f

p

),g = (g

1

;:::;g

m

),h:R

n

!R

r

linear given by h =

(h

1

;:::;h

r

),h

k

(u) = hc

k

;ui,k 2 K = f1;:::;rg,and Q º R

n

convex subset with 0 2 Q.

Consider the following propositions:

(a) 0 2

P

p

i=1

µ

i

@f

i

(0) +

P

m

j=1

¶

j

@g

j

(0) +

P

r

k=1

·

k

c

k

+N(Q;0);(µ;¶) µ 0 ) µ = 0.

(b)

P

p

i=1

µ

i

f

i

(u) +

P

m

j=1

¶

j

g

j

(u) +

P

r

k=1

·

k

h

k

(u) µ 0 8u 2 Q;(µ;¶) µ 0 ) µ = 0.

(c) There exists v 2 R

n

such that f(v) < 0;g(v) ´ 0;h(v) = 0;v 2 Q.

(d) 0 =2 co([

p

i=1

@f

i

(0)) +cone co([

m

j=1

@g

j

(0)) +linfc

k

:k 2 Kg +N(Q;0).

Then

(i) (a),(b) and (d) are equivalent.

(ii) (c) ) (b).

(iii) If the condition

(c

0

) there exists w 2 R

n

such that g(w) < 0;h(w) = 0;w 2 ri Q

holds,then (b) ) (c) and,consequently,the four are equivalent.

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 107

Proof.(i) and (ii) are proved in a similar way to (i) and (ii) of Theorem 3.5.

(iii) Suppose that (c) is not true.Then there exists no v 2 R

n

such that

f(v) < 0;g(v) < 0;h(v) = 0;v 2 Q.

By Theorem 3.5(iii),(which is applicable since by (c

0

),Ker h\ri Q 6=;,and by Lemma

3.2(i),0 2 ri h(Q)),there exist (µ;¶;·) 2 R

p

¢R

m

¢R

r

such that (µ;¶) µ 0;(µ;¶) 6= 0

and

p

X

i=1

µ

i

f

i

(u) +

m

X

j=1

¶

j

g

j

(u) +

r

X

k=1

·

k

h

k

(u) µ 0 8u 2 Q:(7)

By (b),µ = 0.Let w be a vector verifying (c

0

),then

P

m

j=1

¶

j

g

j

(w) +

P

r

k=1

·

k

h

k

(w) < 0,

in contradiction with which is obtained applying (7) to u = w (with µ = 0).

If in this theorem we take m= 0,we obtain Theorem 3.5.

In the next theorem it is proved that the implication (b) ) (c) of Theorem 3.12 is also

verißed with other diÞerent conditions to (iii) in the aforesaid theorem.

Theorem 3.13.In the hypotheses of Theorem 3.12,assume that some of the equivalent

conditions (a),(b),(d) of that theorem holds.

If D = cone co([

m

j=1

@g

j

(0)) +linfc

k

:k 2 Kg +N(Q;0) is closed and cone Q is closed,

then (c) holds,so the four are equivalent.

Proof.Let C = co([

p

i=1

@f

i

(0)).C is a non-empty compact convex set and D is a closed

convex cone.By assumption (d),0 =2 C +D,and by Lemma 3.1,C\( D) =;.By the

strong separation theorem,there exist v 2 R

n

n f0g and « 2 R such that

hv;xi < « < hv;yi 8x 2 C;8y 2 D:(8)

Taking y = 0 2 D it follows that « < 0.If for some y 2 D we have hv;yi < 0,then

lim

t!+1

hv;tyi = 1 with ty 2 D and (8) does not hold.Hence,hv;yi µ 0 8y 2 D,or

rather,hv;yi ´ 0 8y 2 D.In particular:

1) 8b

j

2 @g

j

(0) one has hv;b

j

i ´ 0.As g

j

(v) = Maxfhb

j

;vi:b

j

2 @g

j

(0)g,it follows

g

j

(v) ´ 0;for j = 1;:::;m:(9)

2) hv;c

k

i ´ 0 and hv; c

k

i ´ 0,hence

hv;c

k

i = 0 8k 2 K:(10)

3) hv;di ´ 0 8d 2 Q

£

,therefore v 2 Q

££

= cl cone Q and since cone Q is closed,

v 2 cone Q:(11)

From (7),hv;xi < « < 0 8x 2 C.In particular,8a

i

2 @f

i

(0) hv;a

i

i < 0.Thus,

f

i

(v) < 0;for i = 1;:::;p:(12)

From (11) there exist t > 0,u 2 Q such that v = tu (since v 6= 0) and substituting v by

tu in (9),(10) and (12),these equations are verißed for u in the place of v,this means

that u is solution of system (c).

108 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

According to Theorem 3.5 and Proposition 3.6 if 0 2 ri h(Q) and there exists v 2 R

n

such that g(v) < 0;h(v) = 0;v 2 Q,then D is closed and,if furthermore cone Q is

closed,Theorem 3.13 can be applied,but in this case we get no advantage since Theorem

3.12(iii) is applicable (by Lemma 3.10) with slightly weaker hypotheses (it is not required

that cone Q be closed).Nevertheless,there are,obviously,cases in which Theorem 3.13

is applicable and Theorem 3.12 is not applicable as in Example 3.4(b) considering,for

example,f(x

1

;x

2

;x

3

) = x

3

.

This theorem generalizes Lemma 2 in Ishizuka and Shimizu [5] which does not include

equality constraints or constraint set.This theorem also generalizes the classic Motzkin

alternative theorem,because if f

i

(u) = ha

i

;ui and g

j

(u) = hb

j

;ui are linear and Q = R

n

,

then cone cofb

j

:j = 1;:::;mg +linfc

k

:k 2 Kg is closed.

4.Optimality conditions

In this section,as an application of the results of the previous section,we obtain both Fritz

John and Kuhn-Tucker type necessary optimality conditions for multiobjective optimiza-

tion problems.These problems are deßned by Dini or Hadamard diÞerentiable functions

with convex derivative in the direction,and they include three types of constraints:in-

equality,equality and set constraints.For this purpose,we need a constraint qualißcation

of extended Abadie type,if the objective function is Hadamard diÞerentiable,or one of

extended Zangwill type,if only it is Dini diÞerentiable.To obtain the Kuhn-Tucker type

conditions we need the addition of other regularity conditions.If there are no equality

constraints (Theorem 4.8) the extended qualißcations (Abadie or Zangwill) are not nec-

essary.These results can be applied to study the diÞerentiable programs and the convex

programs,so that the results of Singh [16] and of Giorgi and Guerraggio [3],for diÞer-

entiable programs,and those of Kanniappan [8],Islam [6] and Kouada [9],for convex

programs,are just particular cases.

Consider the following multiobjective optimization problem

(P) Minff(x):x 2 S\Qg,

where f:R

n

!R

p

,S = fx 2 R

n

:g(x) ´ 0;h(x) = 0g,g:R

n

!R

m

,h:R

n

!R

r

and

Q is an arbitrary set of R

n

.

Let f

i

;i 2 I = f1;:::;pg,g

j

;j 2 J = f1;:::;mg,h

k

;k 2 K = f1;:::;rg be the

component functions of f,g and h,respectively.Given x

0

2 S,the set of active indexes

at x

0

is J

0

= fj 2 J:g

j

(x

0

) = 0g.The sets deßned by the constraints g and h are

denoted,respectively,G = fx 2 R

n

:g(x) ´ 0g,H = fx 2 R

n

:h(x) = 0g,accordingly,

S = G\H,and the set of points ÕbetterÔ than x

0

is F = fx 2 R

n

:f(x) ´ f(x

0

)g.

Hereafter we will suppose that the involved functions are Dini diÞerentiable at x

0

.We

will consider the following cones:

C

0

(G) = fv 2 R

n

:Dg

j

(x

0

;v) < 0 8j 2 J

0

g;

C(G) = fv 2 R

n

:Dg

j

(x

0

;v) ´ 0 8j 2 J

0

g;

C

0

(S) = C

0

(G)\Ker Dh(x

0

;¡),C(S) = C(G)\Ker Dh(x

0

;¡);

C

0

(F) = fv 2 R

n

:Df

i

(x

0

;v) < 0 8i 2 Ig;

C(F) = fv 2 R

n

:Df

i

(x

0

;v) ´ 0 8i 2 Ig.

(C(S) is called the linearized cone).Let us point out that the active inequality constraints

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 109

take part with Õ<Ô or Õ´Ô,according to be C or C

0

and those of equality,with Õ=Ô,and

that the point x

0

is omitted to cut it short.

It is assumed that the functions g

j

;j 2 J n J

0

are continuous at x

0

.

Theorem 4.1.Let x

0

2 S\Qbe a feasible point of problem(P) and suppose the following:

(a) T(Q;x

0

) is convex.

(b) g

j

;j 2 J

0

are Dini diÞerentiable at x

0

with convex derivative at x

0

and h is Dini

diÞerentiable at x

0

with linear derivative given by Dh

k

(x

0

;¡) = hc

k

;¡i;k 2 K.

(c) The extended Abadie constraint qualißcation (EACQ) is verißed:

C(S)\T(Q;x

0

) º T(S\Q;x

0

).

(d) x

0

2 LWMin(f;S\Q).

(e) f is Hadamard diÞerentiable at x

0

with convex derivative at x

0

.

Then

(i) There exist (µ;¶;·) 2 R

p

¢R

m

¢R

r

such that (µ;¶) µ 0;(µ;¶;·) 6= 0 and

0 2

p

X

i=1

µ

i

@

D

f

i

(x

0

) +

m

X

j=1

¶

j

@

D

g

j

(x

0

) +

r

X

k=1

·

k

c

k

+N(Q;x

0

);

¶

j

g

j

(x

0

) = 0;j = 1;:::;m:

(ii) If linfc

k

:k 2 Kg +N(Q;x

0

) is closed,then (i) is satisßed with (µ;¶) 6= 0.

(iii) If

cone co([

j2J

0

@

D

g

j

(x

0

)) +linfc

k

:k 2 Kg +N(Q;x

0

) is closed,(13)

then (i) is satisßed with µ 6= 0.

Proof.It is known that if the objective function is Hadamard diÞerentiable and (d)

holds,then T(S\Q;x

0

)\C

0

(F) =;.From here,by the extended Abadie constraint

qualißcation,it follows that there exists no v 2 R

n

such that

8

>

>

<

>

>

:

Df

i

(x

0

;v) < 0 8i 2 I

Dg

j

(x

0

;v) ´ 0 8j 2 J

0

Dh

k

(x

0

;v) = 0 8k 2 K

v 2 T(Q;x

0

):

(14)

Therefore,we also ßnd the incompatibility in v 2 R

n

of the systems

(a)

8

>

>

<

>

>

:

Df

i

(x

0

;v) < 0 8i 2 I

Dg

j

(x

0

;v) < 0 8j 2 J

0

Dh

k

(x

0

;v) = 0 8k 2 K

v 2 T(Q;x

0

)

(b)

8

>

>

<

>

>

:

Df

i

(x

0

;v) < 0 8i 2 I

Dg

j

(x

0

;v) < 0 8j 2 J

0

Dh

k

(x

0

;v) = 0 8k 2 K

v 2 ri T(Q;x

0

):

(15)

By Theorem 3.9 applied to system (15)(b),part (i) is obtained (taking ¶

j

= 0,for j 2

J n J

0

,as usual).

Parts (ii) and (iii) are deduced,respectively,by Theorem 3.13 (with m = 0) applied to

system (15)(a) and by Theorem 3.13 applied to system (14).

110 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

Part (iii) of this theorem generalizes Theorem 3.1 of Singh [16] (which is identical to

Theorem 2 of Wang [17]),valid for diÞerentiable functions and without constraint set

(Q = R

n

),and therefore @

D

f

i

(x

0

) = frf

i

(x

0

)g,@

D

g

j

(x

0

) = frg

j

(x

0

)g,c

k

= rh

k

(x

0

) and

condition (i) becomes the one used by Singh.Condition (13) is always satisßed,since

cone co([

j2J

0

@

D

g

j

(x

0

)) +linfc

k

:k 2 Kg

= cone cofrg

j

(x

0

):j 2 J

0

g +linfrh

k

(x

0

):k 2 Kg

is a polyhedral convex set and therefore,it is closed.Condition (c) in this case becomes

C(S) º T(S;x

0

),which is the constraint qualißcation used by Singh.

Remark 4.2.

(1) If

0 2 ri Dh(x

0

;¡)(T(Q;x

0

));(16)

then (i) is satisßed with (µ;¶) 6= 0.This follows from Lemma 3.3 (applied to the

convex T(Q;x

0

)) and part (ii).

(2) If the constraint qualißcation

C

0

(S)\ri T(Q;x

0

) 6=;;(17)

is satisßed,then (13) holds by Theorem 3.5 and Proposition 3.6.Therefore,(13) is

more general than (17),but this is simpler to check.

(3) Notice that hypotheses of part (ii) can be false and (13) can be true though.There-

fore these criteria are of independent application.On the other hand,(17) implies

(16) by Lemma 3.2.

The following example shows that,in fact,there are situations in which part (iii) is

applicable but (17) is not.

Example 4.3.In R

2

,let x

0

= (0;0);f(x;y) = 2y;g

1

(x;y) = 2x and g

2

the support

function of the set B = f(x;y):(x 2)

2

+y

2

´ 2;y µ 0g,that is to say

g

2

(x;y) =

º

2x +

p

2x

2

+2y

2

if y µ 0

2x +

p

2x

2

if y < 0:

Obviously Dg

2

(x

0

;v) = g

2

(v);@

D

g

2

(x

0

) = B.

The feasible set is G = f(0;y):y ´ 0g.The point x

0

is an (absolute) minimum of f on

G.We have:C

0

(G) =;(hence,(17) is false),C(G) = G,T(G;x

0

) = G = C(G) and,

therefore,the Abadie constraint qualißcation holds at x

0

.Furthermore,condition (13)

holds,because

cone co([

j2J

0

@

D

g

j

(x

0

)) = cone co(B [ f( 2;0)g) = f(x;y):y µ 0g is closed.

Hence,Theorem4.1(iii) is applicable.Concretely the conclusion is satisßed with (µ;¶

1

;¶

2

)

= (1;1;2) and the element b = (1;1) 2 B (there are inßnite solutions).

When the constraint set Q is not present (or if Q is a closed convex set) and the functions

are continuously Fræechet diÞerentiable,no regularity condition is required in obtaining the

usual Fritz John conditions.However,under the hypotheses of Theorem4.1,the following

example shows that (EACQ) cannot be eliminated to obtain part (i).

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 111

Example 4.4.Let h:R

2

!R given by

h(x;y) =

º

y

2

if x = 0;y < 0

x +x

2

otherwise,

x

0

= (0;0) and f(x;y) = y.It is clear that x

0

is a local minimum of f on H:= h

1

(0) =

f(0;y):y µ 0g [ f(x;y):x = 1g.However,the Fritz John conditions

µrf(x

0

) +·rh(x

0

) = (0;0) with (µ;·) 6= (0;0);µ µ 0;

are not satisßed.Notice that (EACQ) is not verißed:Ker rh(x

0

) 6º T(H;x

0

).

Similar conditions to (EACQ) are used by other authors.For example,Bender,who

considers Hadamard diÞerentiable functions with linear derivative,uses the following [1,

Condition (3)]:

Ker Dh(x

0

;¡)\T(Q;x

0

) º T(H\Q;x

0

):(18)

If the inequality constraints are not considered,(EACQ) becomes (18),but if they are

present,there is not implications between (EACQ) and (18).Thus,if we incorporate the

constraint g(x;y) = y in the example 4.4,then (EACQ) holds but (18) is not verißed.

On the other hand,if we consider h(x;y) = y,g(x;y) = y x

3

and Q = f(x;y):y µ 0g

then though (18) holds,(EACQ) is not true.

It is still possible to hold up the conclusions of Theorem4.1 if f is only Dini diÞerentiable.

In return,the constraint qualißcation must be more restrictive.

Theorem 4.5.Suppose that conditions (a),(b),(d) of Theorem 4.1 are verißed and the

following ones:

(c) The extended Zangwill constraint qualißcation is satisßed:

cl Z

s

(S\Q;x

0

) = C(S)\T(Q;x

0

).

(e) f is Dini diÞerentiable at x

0

with convex derivative.

Then the conclusions are the same (i)-(iii) that in Theorem 4.1.

Proof.It is known ([12,Proposition 4.1]) that if f is Dini diÞerentiable and (d) is verißed,

then

Z

s

(S\Q;x

0

)\C

0

(F) =;.

Therefore,cl Z

s

(S\Q;x

0

)\int C

0

(F) =;.As Df(x

0

;¡) is continuous,C

0

(F) is open,

hence,cl Z

s

(S\Q;x

0

)\C

0

(F) =;,and taking into account (c) we deduce that C(S)\

T(Q;x

0

)\C

0

(F) =;,this means that system (14) is incompatible.From here,it is

continued just as in the proof of Theorem 4.1.

Remark 4.6.

(1) Notice that if the extended Zangwill constraint qualißcation holds,also the extended

Abadie constraint qualißcation holds,since Z

s

(S\Q;x

0

) º T(S\Q;x

0

).

(2) In Theorems 4.1 and 4.4 can be used any convex subcone T

1

(Q) of T(Q;x

0

) instead of

the T(Q;x

0

) itself,in whose case the normal cone N(Q;x

0

) would be substituted by

the polar cone T

1

(Q)

£

,condition (c) of Theorem4.1 would be C(S)\T

1

(Q) º T(S\

Q;x

0

) and condition (c) of Theorem 4.5 would be C(S)\T

1

(Q) º cl Z

s

(S\Q;x

0

).

Of course,the best results are obtained choosing the largest convex subcone.

112 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

The following proposition provides us with a suácient condition for the extended Zangwill

constraint qualißcation to hold.

Proposition 4.7.If Q is a convex set,g

j

;j 2 J

0

are Dini diÞerentiable at x

0

with

convex derivative,Z

s

(H;x

0

) = Ker Dh(x

0

;¡) and C

0

(S)\ri cone(Q x

0

) 6=;,then the

extended Zangwill constraint qualißcation holds.

Proof.For the set G,we have C

0

(G) º Z(G;x

0

) º C(G).Therefore,

C

0

(G)\Ker Dh(x

0

;¡) º Z(G;x

0

)\Z

s

(H;x

0

) º Z

s

(G\H;x

0

) º C(G)\Ker Dh(x

0

;¡),

hence C

0

(S) º Z

s

(S;x

0

) º C(S),so

C

0

(S)\cone(Q x

0

) º Z

s

(S;x

0

)\Z(Q;x

0

) º Z

s

(S\Q;x

0

) º C(S)\T(Q;x

0

).

The conclusion follows by taking closure,since cl[C

0

(S)\cone(Q x

0

)] = C(S)\T(Q;x

0

)

by [4,Proposition 2.1.10,Chap.3].

In the following theorem,equality constraints are removed.

Theorem 4.8.Let x

0

2 G\Q and suppose the following:

(a) T

1

(Q) is a convex subcone of T(Q;x

0

).

(b) x

0

2 LWMin(f;G\Q).

Then

(i) If f and g

j

;j 2 J

0

are Hadamard diÞerentiable at x

0

with convex derivative,then

there exist (µ;¶) 2 R

p

¢R

m

,(µ;¶) 6= 0 such that

(µ;¶) µ 0;

0 2

P

p

i=1

µ

i

@

D

f

i

(x

0

) +

P

m

j=1

¶

j

@

D

g

j

(x

0

) +T

1

(Q)

£

;

¶

j

g

j

(x

0

) = 0;j = 1;:::;m:

9

=

;

(19)

(ii) If f and g

j

;j 2 J

0

are Dini diÞerentiable at x

0

with convex derivative and T

1

(Q) º

Z

s

(Q;x

0

),then (19) holds with (µ;¶) 6= 0.

(iii) If,moreover,

C

0

(G)\T

1

(Q) 6=;;(20)

then (19) holds with µ 6= 0.

Proof.(i) Since f is Hadamard diÞerentiable and (b) holds,as it has been pointed out

in the proof of Theorem 4.1,

T(G\Q;x

0

)\C

0

(F) =;:(21)

Let us show that

C

0

(G)\T(Q;x

0

) º T(G\Q;x

0

):(22)

In fact,if v 2 C

0

(G)\T(Q;x

0

),then there exist sequences x

n

2 Q;t

n

!0

+

such that

x

n

x

0

t

n

= v

n

!v.As g

j

;j 2 J

0

are Hadamard diÞerentiable,one has

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 113

dg

j

(x

0

;v) = lim

n!1

g

j

(x

0

+t

n

v

n

) g

j

(x

0

)

t

n

= lim

n!1

g

j

(x

n

)

t

n

< 0;8j 2 J

0

,

because v 2 C

0

(G).Hence,for n large enough,g

j

(x

n

) < 0.For j 2 J n J

0

,by the

continuity of g

j

at x

0

,also it is g

j

(x

n

) < 0,for n large.Therefore,x

n

2 G\Q and

consequently v 2 T(G\Q;x

0

).

From (21),taking into account (22) and hypothesis (a),it follows that C

0

(G)\T

1

(Q)\

C

0

(F) =;,that is,there exists no v 2 R

n

such that

Df(x

0

;v) < 0;Dg

j

(x

0

;v) < 0 8j 2 J

0

;v 2 T

1

(Q):

By Corollary 3.7 one has the conclusion.

(ii) Since f is Dini diÞerentiable and (c) holds,as has been pointed out in the proof of

Theorem 4.5,

Z

s

(G\Q;x

0

)\C

0

(F) =;:(23)

On the other hand,since C

0

(G) º Z(G;x

0

),we deduce that

C

0

(G)\Z

s

(Q;x

0

) º Z(G;x

0

)\Z

s

(Q;x

0

) º Z

s

(G\Q;x

0

):(24)

From (23),taking into account (24) and hypothesis of (ii),it follows that C

0

(G)\T

1

(Q)\

C

0

(F) =;.From here we would continue as in part (i) above.

(iii) By reduction to the absurd,suppose that µ = 0.Then we have

0 2

P

j2J

0

¶

j

@

D

g

j

(x

0

) +T

1

(Q)

£

;¶ µ 0;¶ 6= 0.

By Corollary 3.7,C

0

(G)\T

1

(Q) =;,which contradicts (20).

Remark 4.9.

(1) If Q is a convex set,the sharpest results in Theorem 4.8 are obtained for T

1

(Q) =

cone(Q x

0

),with what T

1

(Q)

£

= N(Q;x

0

) and (20) is equivalent to

C

0

(G)\(Q x

0

) 6=;.

(2) The condition obtained in (i) is Fritz John type.If we wish one of Kuhn-Tucker

type,a constraint qualißcation can be used,as for example C

0

(G)\T

1

(Q) 6=;,but

in this case,if T

1

(Q) º Z

s

(Q;x

0

),it is preferable to use (iii) which is less restrictive.

Part (i) generalizes Theorem 5 in Giorgi and Guerraggio [3] in which the functions f

and g are diÞerentiable.Parts (i) and (ii) generalize Theorem 3.2 in Kanniappan [8] and

Theorem 3.1 in Islam [6] in which it is supposed that the functions f and g are convex

on R

n

and Q is convex.In fact,if a function is convex,it is Dini diÞerentiable with

convex derivative,and in addition,it is locally Lipschitz,with what it is also Hadamard

diÞerentiable,and (i) and (ii) can be applied by taking T

1

(Q) = cone(Q x

0

) (the results

of these authors are obtained since,for a convex real function',@

D

'(x

0

) = @'(x

0

)).

Theorems 3.4 and 3.2 of the same authors can be deduced frompart (iii).These theorems

state:ÕIf f and g are convex,Q is convex,x

0

2 Min(f;G\Q) and the Slater constraint

qualißcation (SCQ) holds,that is,for each i = 1;:::;p there exists x

i

such that f

k

(x

i

) <

f

k

(x

0

) 8k 6= i,g

j

(x

i

) < 0 8j 2 J;x

i

2 Q,then (19) holds with µ > 0.Ô In fact,let

114 B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions

us observe in the ßrst place that if':R

n

!R is convex and'(x) <'(x

0

) for some

x;x

0

2 R

n

,then

D'(x

0

;x x

0

) < 0:(25)

By convexity,

'(x

0

+t(x x

0

)) ´'(x

0

) +t('(x) '(x

0

)) 8t 2 (0;1).

Hence,

'(x

0

+t(x x

0

)) '(x

0

)

t

´'(x) '(x

0

) < 0.Taking the limit when t!0

+

,

(25) is obtained.

Therefore,applying (25),by (SCQ),for each i = 1;:::;p there exists v = x

i

x

0

such

that

Df

k

(x

0

;v) < 0 8k 6= i;Dg

j

(x

0

;v) < 0 8j 2 J

0

;v 2 Q x

0

:(26)

Theorem 4.6(iii) (with T

1

(Q) = cone(Q x

0

)) can be applied,since any Pareto minimum

is a weak local Pareto minimum,resulting (19) with µ 6= 0.If p = 1,one has µ > 0.

If p µ 2,suppose that some µ

i

= 0 (with some µ

j

6= 0,since µ 6= 0).By Corollary 3.7

(applied to the p 1 functions Df

k

(x

0

;¡);k 6= i and to Dg

j

(x

0

;¡);j 2 J

0

) there exists no

vector v satisfying system(26),in contradiction to the existence of the solution v = x x

i

.

Therefore,µ > 0.

Notice that in (SCQ) it is suácient to require the condition g

j

(x

i

) < 0 for every j 2 J

0

(instead of for all j 2 J).Observe also that if the weak Slater constraint qualißcation

holds:there exists x 2 Q such that g

j

(x) < 0 8j 2 J

0

,then (19) holds with µ 6= 0.

Finally,notice that at the same time it has been proved that if (SCQ) holds,every Pareto

minimum is a solution for a scalarized problem,Minfhµ;f(x)i:x 2 G\Qg,with µ > 0

([9,Theorem 5]).

The next lemma provides us with a simple expression for the tangent cone to a set which

is intersection of two sets:the ßrst convex and the second is deßned by quasiconvex or

quasilinear functions.This expression will allow us to obtain an optimality criterion with

this type of functions.Let us recall these concepts previously.

Let È º R

n

be a convex set,':È!R,x

0

2 È.'is quasiconvex at x

0

on È if

8x 2 È;'(x) ´'(x

0

) )'(µx +(1 µ)x

0

) ´'(x

0

) 8µ 2 (0;1).

'is quasilinear at x

0

if'and 'are quasiconvex at x

0

.

Lemma 4.10.Suppose that Q º R

n

is a convex set and g

j

,j 2 J

0

are quasiconvex at x

0

on a neighborhood of x

0

and h is quasilinear at x

0

on a neighborhood of x

0

.Then

cl Z(S\Q;x

0

) = T(S\Q;x

0

).

Proof.Let B(x

0

;®) be a neighborhood of x

0

on which g

j

,j 2 J

0

are quasiconvex at x

0

and h is quasilinear.Hence,8x 2 B(x

0

;®) if g

j

(x) ´ g

j

(x

0

),then

g

j

(µx +(1 µ)x

0

) ´ g

j

(x

0

) 8µ 2 [0;1].

Therefore,if x 2 G\B(x

0

;®) we derive g

j

(µx + (1 µ)x

0

) ´ 0 8µ 2 [0;1];8j 2 J

0

.

If j 2 J n J

0

,by the continuity of g

j

,there exists a neighborhood B(x

0

;®

1

) such that

g

j

(x) < 0 8x 2 B(x

0

;®

1

);8j 2 J n J

0

.Taking ®

0

= Minf®;®

1

g,both conditions are

verißed and,therefore [x

0

;x] º G 8x 2 G\B

0

,being B

0

= B(x

0

;®

0

).Similarly,since

B.Jimæenez,V.Novo/Alternative Theorems and Optimality Conditions 115

h and h are quasiconvex at x

0

,[x

0

;x] º H.Hence [x

0

;x] º S 8x 2 S\B

0

.As Q is

convex,one has [x

0

;x] º S\Q 8x 2 S\Q\B

0

.Hence,v = x x

0

2 Z(S\Q;x

0

).It

follows

cone(S\Q\B

0

x

0

) º Z(S\Q;x

0

) º T(S\Q;x

0

).

Now,T(S\Q;x

0

) = T(S\Q\B

0

;x

0

) º cl cone(S\Q\B

0

x

0

) º cl Z(S\Q;x

0

).

In conclusion,by closedness of the tangent cone,

T(S\Q;x

0

) = cl Z(S\Q;x

0

) = cl cone(S\Q\B

0

x

0

).

Theorem 4.11.In the hypotheses of Lemma 4.10.Suppose that conditions (b),(c) and

(d) of Theorem 4.1 are verißed and that f is Dini diÞerentiable at x

0

with convex deriva-

tive.Then the conclusions are the same (i)-(iii) that in Theorem 4.1.

Proof.By Lemma 4.10,condition (c) becomes the extended Zangwill constraint qualiß-

cation,i.e.,condition (c) of Theorem 4.5,and it is enough to apply this theorem.

Acknowledgements.The authors are grateful to the anonymous referee for his helpful

comments and suggestions which have been included in the ßnal version of the paper.

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