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Via Cosimo Ridolfi, 10 – 56124 PISA – Tel. Segr. Amm. 050 945231 Segr. Stud. 050 945317 Fax 050 945375

Università degli Studi di Pisa

Dipartimento di Statistica e Matematica

Applicata all’Economia

Report n. 212

Necessary Optimality Conditions

in Vector Optimization

Riccardo Cambini

Pisa, Settembre 2001

(2)

-Necessary Optimality Conditions

in Vector Optimization

Riccardo Cambini

Dept. of Statistics and Applied Mathematics, University of Pisa

Via Cosimo Ridolfi 10, 56124 Pisa, ITALY

E-mail: cambric@ec.unipi.it

September 2001

Abstract

This paper deals with vector optimization problems having a vec-tor valued objective function and three kinds of constraints: inequality constraints, equality constraints, and a set constraint (which covers the constraints which cannot be expressed by means of neither equalities nor inequalities). Necessary optimality conditions and sufficient ones are given in the image space for the nonsmooth case (when the continu-ity is not required), while necessary conditions in the image and in the decision spaces are given for the nondifferentiable case (when just the Hadamard directional differentiability is assumed). The new concept of U-regularity is introduced in order to study necessary optimality conditions in the decision space. Finally, the results are specialized under differentiability hypothesis, thus obtaining conditions generaliz-ing the so called “maximum principle conditions”.

Keywords Vector Optimization, Optimality Conditions, Image Space, Maximum Principle Conditions.

AMS - 2000 Math. Subj. Class. 90C29, 90C46, 90C30

JEL - 1999 Class. Syst. C61, C62

1

Introduction

The aim of this paper is to study optimality conditions for vector problems

having a vector valued objective function and three kinds of constraints:

in-equality constraints, in-equality constraints, and a set constraint (which covers

the constraints which cannot be expressed by means of neither equalities

nor inequalities). The partial ordering in the image of the objective

func-tion is given by a closed convexpointed cone

C

with nonempty interior

(3)

(that is a solid cone, not necessarily the Paretian one), while the inequality

constraints are expressed by means of a partial ordering given by a closed

convexpointed cone

V

with nonempty interior.

Problems of this kind have been studied in the literature in finite

di-mensional spaces with a scalar objective function and under differentiability

hypothesis (

1

), obtaining (with some additional hypothesis) necessary

opti-mality conditions of the so called “maximum/minimum principle” type (also

called “generalized Lagrange multiplier rule”) [3, 23, 21]; these optimality

conditions are stated in the decision space, that is to say that they are based

on the use of derivatives and multipliers.

The aim of this paper is twofold; first it is to state some optimality

con-ditions by means of the so called image space approach [5, 6, 7, 8, 9, 10, 11],

first suggested in [20], then it is to generalize the minimum/maximum

princi-ple conditions to multiobjective problems having nondifferentiable functions.

In particular, in Section 3 a characterization of the efficiency of a point is

first stated in the image space without any assumptions on the functions of

the problem, then some more necessary optimality conditions in the image

space are given assuming the Hadamard directional differentiability of the

functions.

In Section 4, the existence of necessary optimality conditions in the

decision space is studied, still assuming that the functions are Hadamard

directionally differentiable; a characterization in the image space of such

conditions is provided thus making possible a comparison with the

previ-ously stated conditions in the image space. The conditions in the decision

space result to be stronger than the image space ones, hence a new

regu-larity concept, called “

U

-regularity”, is introduced in order to commute the

conditions in the image space to the ones in the decision space.

Finally, in Section 5 the previously obtained results are specified

assum-ing the differentiability of the functions, it is also pointed out that the given

conditions generalize some of the results known in the literature.

2

Statement of the problem

The vector optimization problem studied in this paper has both inequality

and equality constraints as well as a set constraint, covering the constraints

which cannot be expressed by means of neither equalities nor inequalities:

P

:

        

C

max

f

(

x

)

g

(

x

)

V

inequality constraints

h

(

x

) = 0

equality constraints

x

X

set constraint

1

Minimum/maximum principle optimality conditions are used also in infinite dimen-sional spaces, for instance in optimal control theory [15, 19, 24, 26].

(4)

where

f

:

A

s

,

g

:

A

m

and

h

:

A

p

are vector valued functions,

with

A

n

open set,

C

s

and

V

m

are closed convexcones with

nonempty interior (that is to say solid cones), and

X

A

is a set verifying

no particular topological properties, that is to say that

X

is not required to

be open or convexor with nonempty interior. For the sake of convenience,

note that problem

P

can be rewritten in the following form:

P

:

        

C

max

f

(

x

)

g

(

x

)

V

x

(

X

S

)

S

=

{

x

A

:

h

(

x

) = 0

}

The aim of this paper is to study optimality conditions for a feasible

point

x

0

X

which is assumed, without loss of generality, to bind all the

inequality constraints, so that

g

(

x

0

) = 0. Note that it is not known whether

or not

x

0

belongs to the boundary of

X

. The feasible point

x

0

X

is said

to be a

local efficient point

if there exists a suitable neighbourhood

I

x0

of

x

0

such that:

y

I

x0

X

such that

f

(

y

)

f

(

x

0

) +

C

0

, g

(

y

)

V, h

(

y

) = 0

(2.1)

where

C

0

=

C

\ {

0

}

. For the sake of simplicity the following function is also

used:

F

:

A

s+m+p

such that

F

(

x

) = (

f

(

x

)

, g

(

x

)

, h

(

x

))

By means of function

F

,

x

0

X

is a local efficient point if and only if there

exists a suitable neighbourhood

I

x0

of

x

0

such that:

y

I

x0

X

such that

F

(

y

)

F

(

x

0

) + (

C

0

×

V

×

0)

(2.2)

The study of optimality conditions is based on the so called image space

approach, originally suggested by Hestenes [20]; with this aim a key tool

results to be the

Bouligand Tangent cone to

X

at

x

0

, denoted with

T

(

X, x

0

),

which is a closed cone defined as follows:

T

(

X, x

0

)

=

{

x

n

:

∃{

x

k

} ⊂

X, x

k

x

0

,

∃{

λ

k

} ⊂

++

, λ

k

+

,

x

= lim

k→+

λ

k

(

x

k

x

0

)

}

.

Also subcones of

T

(

X, x

0

) are fundamental in this paper; with this aim just

recall that particular subcones of

T

(

X, x

0

) are the well known

cone of feasible

directions to

X

at

x

0

(

2

), denoted with

F

(

X, x

0

), and the

cone of interior

directions to

X

at

x

0

, denoted with

I

(

X, x

0

) (see for example [3, 17, 18]).

2

LetX⊆ nbe a nonempty set and letx0Cl(X). Thecone of feasible directions to

X atx0 F(X, x0) and thecone of interior directions to X at x0 I(X, x0) are defined as

follows:

(5)

Note finally that the results stated in this paper deal also with problems

having no equality and/or no inequality constraints. With this regard, it

is important to note that the absence of equality constraints is extremely

relevant in the optimality conditions expressed in the decision space; for this

reason, when necessary, the absence of equality constraints is specified with

the condition

p

= 0 (remind that

h

:

A

p

) and in this case

S

=

A

is

assumed.

3

Optimality conditions in the Image Space

The aim of this section is to state in the image space necessary and/or

sufficient optimality conditions for problem

P

.

By means of an approach similar to the one used in [5, 6, 7, 8, 9, 10, 11],

the following subset of the Bouligand tangent cone at

F

(

x

0

) in the image

space is introduced:

T

1

=

{

t

s+m+p

:

∃{

x

k

} ⊂

X, x

k

x

0

, h

(

x

k

) = 0

,

∃{

λ

k

} ⊂

,

λ

k

>

0

, λ

k

+

, t

= lim

k→+

λ

k

(

F

(

x

k

)

F

(

x

0

))

}

.

(3.1)

The cone

T

1

plays a key role in stating optimality conditions in the image

space. The forthcoming results extend the ones stated in [6, 7, 8, 10], which

can be seen as the particular cases of problem

P

where

X

is an open set or

where

x

0

Int(

X

).

3.1

The nonsmooth case

The aim of this subsection in to characterize in the image space the efficiency

of

x

0

, this allows also to determine a necessary optimality condition as well

as a sufficient one. Note that no hypothesis on the functions

f

,

g

and

h

are assumed, that is to say that they may be not only nondifferentiable but

even noncontinuous.

Theorem3.1

Consider problem

P

. If

x

0

X

is a local efficient point

then:

T

1

(Int(

C

)

×

Int(

V

)

×

0) =

(3.2)

I(X, x0) = {x∈ n:∃ >0,∃δ >0 such thatλ∈(0, δ),y−x< imply

x0+λy∈X}.

It is very well known that for any setX:

(6)

Proof

The result is proved by contradiction. Suppose that

t

T

1

(Int(

C

)

×

Int(

V

)

×

0); then

∃{

x

k

} ⊂

X

,

x

k

x

0

,

h

(

x

k

) = 0,

∃{

λ

k

} ⊂

,

λ

k

>

0,

λ

k

+

, such that

t

= lim

k+

λ

k

(

F

(

x

k

)

F

(

x

0

)).

Being

t

(Int(

C

)

×

Int(

V

)

×

0) and being

h

(

x

k

) = 0

k

then for a known

limit theorem:

¯

k >

0 such that

λ

k

(

F

(

x

k

)

F

(

x

0

))

(Int(

C

)

×

Int(

V

)

×

0)

k >

¯

k

so that, being

λ

k

>

0,

F

(

x

k

)

F

(

x

0

) + (Int(

C

)

×

Int(

V

)

×

0)

k >

k

¯

and

this contradicts the local efficiency of

x

0

.

The next theorem shows that it is possible to characterize in the image

space the optimality of

x

0

.

Theorem3.2

Consider problem

P

. The point

x

0

X

is a local efficient

point if and only if the following condition holds:

t

T

1

(

C

×

V

×

0)

,

t

= 0

, and

∀{

x

k

} ⊂

X

,

x

k

x

0

,

h

(

x

k

) = 0

, such that

∃{

λ

k

} ⊂

,

λ

k

>

0

,

λ

k

+

, with

t

= lim

k→+

λ

k

(

F

(

x

k

)

F

(

x

0

))

, there exists an integer

¯

k >

0

such that:

F

(

x

k

)

/

F

(

x

0

) + (

C

0

×

V

×

0)

k >

¯

k

.

Proof

) If

x

0

is a local efficient point then, for (2.2),

∀{

x

k

} ⊂

X

,

x

k

x

0

,

h

(

x

k

) = 0, there exists an integer ¯

k >

0 such that

F

(

x

k

)

/

F

(

x

0

) + (

C

0

×

V

×

0)

k >

k

¯

, and this is true also for particular sequences

such that

t

= lim

k→+

λ

k

(

F

(

x

k

)

F

(

x

0

)) with

t

T

1

(

C

×

V

×

0).

) The result is proved by contradiction. Suppose that

x

0

X

is not a

local efficient point, then by means of (2.2)

∃{

x

k

} ⊂

X

,

x

k

x

0

, such

that

F

(

x

k

)

F

(

x

0

) + (

C

0

×

V

×

0)

k

, so that in particular

h

(

x

k

) = 0

k

.

Let us consider now the sequence

{

d

k

} ⊂

s+m+p

with

d

k

=

FF((xkx )−F(x0)

k)−F(x0)

;

since the unit ball is a compact set, we can suppose (substituting

{

d

k

}

with

a suitable subsequence, if necessary) that lim

k→+

d

k

=

t

= 0,

t

T

1

.

On the other hand,

d

k

=

FF((xkxk))FF((xx00))

(

C

0

×

V

×

0) so that its limit

t

(

C

×

V

×

0).

It then results that

t

T

1

(

C

×

V

×

0),

t

= 0,

and this contradicts the assumptions since

t

= lim

k→+FF((xkxk))FF((xx00))

and

F

(

x

k

)

F

(

x

0

) + (

C

0

×

V

×

0)

k

.

Directly from Theorem 3.2 we can state the following sufficient

optimal-ity condition.

Corollary 3.1

Consider problem

P

. If the following condition holds then

x

0

X

is a local efficient point:

(7)

3.2

The nondifferentiable case

The previously stated optimality conditions are extremely general since no

properties are assumed regarding to functions

f

,

g

and

h

. On the other

hand, those conditions are not easy to be verified, since the cone

T

1

is not

trivial to be determined.

Some more “easy to use” necessary optimality conditions, still based on

the image space approach, can be proved with the following assumption.

(

H

N

)

Nondifferentiability Assumptions

Functions

f

,

g

and

h

are Hadamard directionally differentiable at the

point

x

0

X

(

3

).

A complete study of Hadamard directionally differentiable functions can

be found for example in [16] (see also [1, 2, 25, 28]). The nondifferentiability

hypothesis (

H

N

) allows to define the following cones, which play a key role

in stating further necessary optimality conditions in the image space.

Definition 3.1

Consider problem

P

, suppose (

H

N

) holds and let

U

n

be a cone. The following sets are defined:

Ker

∂h

=

{

0

} ∪ {

v

n

\ {

0

}

:

∂h

∂v

(

x

0

) = 0

}

Ker

∂hC

=

n

\

Ker

∂h

=

{

v

n

\ {

0

}

:

∂h

∂v

(

x

0

)

= 0

}

Im

∂h

(

U

)

=

{

0

} ∪ {

t

p

:

t

=

∂h

∂v

(

x

0

)

, v

= 0

, v

U

}

L

(

X, S, x

0

)

=

T

(

X

S, x

0

)

Ker

∂hC

=

n

\

(

Ker

∂h

\

T

(

X

S, x

0

)) =

L

K

L

=

{

t

m+s+p

:

t

= (

∂f

∂v

(

x

0

)

,

∂g

∂v

(

x

0

)

,

∂h

∂v

(

x

0

))

, v

= 0

, v

L

}

3Letf:A, withAnopen set. The limit:

lim

λ→0+,h→v

f(x0+λh)−f(x0)

λ

is called theHadamard directional derivative of f(x)atx0 ∈A in the directionv; if this

derivative exists and is finite for allvthenf(x) isHadamard directionally differentiable at x0∈A. In order to verify the Hadamard directional derivability, remind that a function

f(x) is Hadamard directionally differentiable atx0 (see [16]) if and only if its derivative

∂f ∂v(x0)

def

= limλ→0+f(x0+λvλ)−f(x0)is continuous as a function of direction and the function

itself isDini uniformly directionally differentiable atx0(hence directionally differentiable

atx0), that is to say that:

lim v→0 f(x0+v)−f(x0) ∂f ∂v(x0) = 0

Recall also that if a functionf(x) is Hadamard directionally differentiable atx0 then it is

also continuous atx0. A vector valued function F :A→ m is Hadamard directionally

(8)

K

U

=

{

t

m+s+p

:

t

= (

∂f

∂v

(

x

0

)

,

∂g

∂v

(

x

0

)

,

∂h

∂v

(

x

0

))

, v

= 0

, v

U

}

Note that

Ker

∂h

,

Ker

∂hC

,

Im

∂h

(

U

),

K

L

and

K

U

are cones, since

∂f∂v

(

x

0

),

∂g

∂v

(

x

0

) and

∂h

∂v

(

x

0

) are positively homogeneous (of the first degree) as

func-tions of direction

v

, due to the Hadamard directional differentiability of

f

,

g

and

h

(

4

). In the rest of the paper, cones

U

L

(

X, S, x

0

) will be very

used, with this aim note that:

U

L

(

X, S, x

0

)

U

Ker

∂h

T

(

X

S, x

0

)

Remark 3.1

Since

L

(

X, S, x

0

) =

T

(

X

S, x

0

)

Ker

C∂h

it is worth noticing

that if

h

is Hadamard directionally differentiable at

x

0

X

then (

5

):

T

(

X

S, x

0

)

T

(

S, x

0

)

Ker

∂h

In order to verify this property, firstly note that

T

(

X

S, x

0

)

T

(

S, x

0

)

being

X

S

S

. Being

t

= 0

T

(

S, x

0

)

Ker

∂h

just the case

t

T

(

S, x

0

),

t

= 0, has to be considered. By means of the definition of tangent cone,

∃{

x

k

} ⊂

S

,

x

k

x

0

,

∃{

λ

k

} ⊂

,

λ

k

>

0,

λ

k

+

, such that

t

=

lim

k→+

λ

k

(

x

k

x

0

); it can be supposed also (eventually substituting

{

x

k

}

with a proper subsequence) that

v

= lim

k→+xxkk−xx00

. Since

{

x

k

} ⊂

S

it yields

h

(

x

0

) =

h

(

x

k

) = 0

k >

0 so that, by means of the Hadamard

directional differentiability of

h

(

x

), it is:

0 = lim

k→+

h

(

x

k

)

h

(

x

0

)

x

k

x

0

=

lim

γk→0+,dk→v

h

(

x

0

+

γ

k

d

k

)

h

(

x

0

)

γ

k

=

∂h

∂v

(

x

0

)

where

γ

k

=

x

k

x

0

and

d

k

=

xxk−x0 k−x0

, so that

v

Ker

∂h

.

By means of the definition it results:

t

= lim

k+

λ

k

(

x

k

x

0

) = lim

k+

λ

k

x

k

x

0

k

lim

+

x

k

x

0

x

k

x

0

=

µv

where

µ

= lim

k→+

λ

k

x

k

x

0

0 and

v

= 1. Being

Ker

∂h

a cone

and being

v

Ker

∂h

it follows that

t

Ker

∂h

.

By means of these cones the following necessary optimality conditions in

the image space can be stated.

4

Note also that the given definition ofKL generalizes the one given in [5, 6, 7, 8, 9,

10, 11] for differentiable problems having no set constraints; in particular these papers considerKL={t∈ s+m:t= [Jf(x0), Jg(x0)]v, v∈ n}, which is nothing but the image

of [Jf(x0), Jg(x0)]. 5

It is also known, see for instance [3], that ifhis differentiable atx0∈X it is:

T(S, x0)Cl(Co(T(S, x0)))⊆Ker∂h

(9)

Theorem3.3

Consider Problem

P

and suppose

(

H

N

)

holds; if the feasible

point

x

0

X

is a local efficient point then the two following equivalent

conditions hold:

K

L

(Int(

C

)

×

Int(

V

)

×

0) =

(3.4)

(

K

L

(

C

×

V

×

0))

(Int(

C

)

×

Int(

V

)

×

0) =

(3.5)

In addiction, for any cone

U

n

such that

U

Ker

∂h

T

(

X

S, x

0

)

the

two following further equivalent conditions hold:

K

U

(Int(

C

)

×

Int(

V

)

×

0) =

(3.6)

(

K

U

(

C

×

V

×

0))

(Int(

C

)

×

Int(

V

)

×

0) =

(3.7)

Proof

Condition (3.4) is proved by contradiction. Suppose that there

ex-ists

t

= (

t

f

, t

g

, t

h

)

K

L

(Int(

C

)

×

Int(

V

)

×

0), so that

µ >

0,

v

L

(

X, S, x

0

),

v

= 1, such that

t

=

µ

(

∂f

∂v

(

x

0

)

,

∂g

∂v

(

x

0

)

,

∂h

∂v

(

x

0

))

(Int(

C

)

×

Int(

V

)

×

0)

.

Being

∂h∂v

(

x

0

) = 0 then

v

Ker

∂h

which implies that

v /

Ker

C∂h

and

v

T

(

X

S, x

0

). By means of the definition of

T

(

X

S, x

0

) it yields

that

∃{

x

k

} ⊂

(

X

S

),

x

k

x

0

,

∃{

λ

k

} ⊂

,

λ

k

>

0,

λ

k

+

, such that

v

= lim

k→+

v

k

where

v

k

=

λ

k

(

x

k

x

0

). Being functions

f

and

g

Hadamard

directionally differentiable it results:

lim

k→+

f

(

x

k

)

f

(

x

0

)

1 λk

= lim

k→+

f

(

x

0

+

λk1

v

k

)

f

(

x

0

)

1 λk

=

∂f

∂v

(

x

0

)

Int(

C

)

and, in the same way:

lim

k+

g

(

x

k

)

g

(

x

0

)

1 λk

=

∂g

∂v

(

x

0

)

Int(

V

)

By means of a well known limit theorem it then exists ¯

k >

0 such that

f

(

x

k

)

f

(

x

0

)

Int(

C

) and

g

(

x

k

)

g

(

x

0

)

Int(

V

) for any

k >

k

¯

; this

means that the sequence

{

x

k

} ⊂

(

X

S

),

x

k

x

0

, is feasible for

k >

k

¯

and

that

x

0

is not a local efficient point, which is a contradiction.

The equivalence of (3.4) and (3.5) can be easily verified; the whole result

then follows noticing that

U

L

(

X, S, x

0

) implies

K

U

K

L

.

Remark 3.2

For the sake of completeness, note that (3.4) can be stated as

a corollary of Theorem 3.1.

Denoting with

B

=

{

t

= (

t

f

, t

g

, t

h

)

s+m+p

:

t

h

= 0

}

, directly from

Theorem 3.1 it follows that the efficiency of

x

0

implies:

(10)

It is now just needed to verify that

K

L

(

T

1

B

). Let

t

=

µ

∂F∂v

(

x

0

)

K

L

,

v

L

(

X, S, x

0

),

v

= 1,

µ

0; if

µ

= 0 then

t

=

µ

∂F∂v

(

x

0

) = 0

T

1

while if

µ

= 0 and

v

Ker

C∂h

then

∂v∂h

(

x

0

)

= 0 and

t

B

. Suppose now

µ

= 0 and

v

T

(

X

S, x

0

), then

∃{

x

k

} ⊂

X

,

x

k

x

0

,

h

(

x

k

) = 0, such

that

v

= lim

k→+xxkk−xx00

; let also

λ

k

=

x

k

x

0

1

. By means of the

Hadamard directional differentiability of

F

(

x

) at

x

0

it is:

∂F

∂v

(

x

0

) = lim

k→+

F

(

x

k

)

F

(

x

0

)

x

k

x

0

= lim

k→+

λ

k

(

F

(

x

k

)

F

(

x

0

))

T

1

;

being

T

1

a cone it then follows that

t

=

µ

∂F∂v

(

x

0

)

T

1

too.

4

Optimality conditions in the Decision Space:

the nondifferentiable case

In the literature some necessary optimality conditions expressed in the

de-cision space are stated for particular problems

P

having a scalar objective

function and assuming the differentiability of functions

f

,

g

and

h

[3, 21, 23].

These conditions are useful in the applications (consider for all the optimal

control theory) and are known as “maximum/minimum principle”

condi-tions.

The aim of this section is to generalize those conditions for Hadamard

directionally differentiable functions and for multiobjective problems. In

other words, the necessary optimality conditions in the decision space (hence

involving the directional derivatives and some multipliers) which are going

to be studied in this section are the followings:

(

C

N

)

α

f

C

+

,

α

g

V

+

,

α

h

p

, (

α

f

, α

g

, α

h

)

= 0, such that:

α

Tf

∂f

∂v

(

x

0

) +

α

T g

∂g

∂v

(

x

0

) +

α

T h

∂h

∂v

(

x

0

)

0

v

Cl(

U

)

\ {

0

}

where

U

n

is a cone and (

H

N

) is assumed.

It is important to note that conditions (

C

N

), depending on the particular

chose cone

U

, do not hold in general even if

x

0

is an efficient point. This is

shown in the following example, which implicitly points out that condition

(3.4) is more general than (

C

N

) ones.

Example 4.1

Consider the following problem:

(11)

where

X

=

X

1

X

2

X

3

with:

X

1

=

{

(

x

1

, x

2

)

2

:

x

1

+

x

2

0

,

2

x

1

+

x

2

0

}

,

X

2

=

{

(

x

1

, x

2

)

2

:

x

1

0

, x

2

0

}

,

X

3

=

{

(

x

1

, x

2

)

2

:

x

1

+

x

2

0

, x

1

+ 2

x

2

0

}

and

x

0

= (0

,

0); since the problem has no equality constraints it is

p

= 0 and

S

=

2

. Note that (Int(

C

)

×

Int(

V

)) =

2

++

and

X

=

T

(

X

S, x

0

) =

K

L

since [

J

f

(

x

0

)

, J

g

(

x

0

)] is equal to the identity matrix. The point

x

0

is the

global efficient point of the problem and the necessary optimality condition

(3.4) is verified being

X

2

++

=

; on the other hand the sets

X

,

I

(

X, x

0

),

T

(

X

S, x

0

) and

K

L

are not convex.

Assume now

U

=

T

(

X

S, x

0

); even if

x

0

X

is a global efficient point it

can be easily verified that (

C

N

) does not hold; this points out that condition

(3.4) is more general than (

C

N

) one.

In this section it is going to be proved that the additional assumption,

needed in order to state the necessary optimality conditions in the

deci-sion space, is the existence of a separation hyperplane between the cone

(Int(

C

)

×

Int(

V

)

×

0) and

K

U

or

K

L

. This result is stated by means of

sep-arating theorems and the use of multipliers, hence a key tool of this approach

is the positive polar of a cone

K

, denoted with

K

+

.

4.1

Characterization in the image space

The aim of this subsection is to characterize conditions (

C

N

) in the image

space, thus making possible a complete comparison with condition (3.4).

With this aim, the following preliminary results are needed.

Lemma 4.1

Consider problem

P

with

p

1

, suppose

(

H

N

)

holds and let

U

n

be a cone such that

Co(

Im

∂h

(

U

))

=

p

. Then

α

h

p

,

α

h

= 0

,

such that:

α

Th

∂h

∂v

(

x

0

)

0

v

Cl(

U

)

\ {

0

}

and hence

(

C

N

)

is verified.

Proof

Since Co(

Im

∂h

(

U

))

=

p

there exists a support hyperplane for the

convexcone Co(

Im

∂h

(

U

)), so that

α

h

p

,

α

h

= 0, such that

α

Th

t

0

t

Co(

Im

∂h

(

U

)); this implies that

α

Th∂v∂h

(

x

0

)

0

v

U

,

v

= 0. Being

∂h

∂v

(

x

0

) continuous as a function of direction

v

due to the Hadamard

direc-tional differentiability of

h

, it then follows that

α

hT∂h∂v

(

x

0

)

0

v

Cl(

U

),

v

= 0. The whole result is then proved just assuming

α

f

= 0 and

α

g

= 0.

Note that Lemma 4.1 points out that the case Co(

Im

∂h

(

U

))

=

p

is

trivial, since a support hyperplane for Co(

Im

∂h

(

U

)) exists without the need

(12)

of any additional hypothesis, such as convexity ones, optimality assumptions

on

x

0

, regularity conditions for the problem.

Lemma 4.2

Consider problem

P

with

p

1

, suppose

(

H

N

)

holds and

let

U

n

be a cone. If

(

C

N

)

is verified and

Co(

Im

∂h

(

U

)) =

p

then

(

α

f

, α

g

)

= 0

.

Proof

Suppose by contradiction that

α

f

= 0 and

α

g

= 0, so that

α

h

= 0.

Then

α

Th

∂h

∂v

(

x

0

)

0

v

Cl(

U

)

\ {

0

}

,

and this yields

α

Th

t

0

t

Im

∂h

(

U

). Consequently it results

α

Th

t

0

t

Co(

Im

∂h

(

U

)) =

p

which implies

α

h

= 0, and this is a contradiction

since (

α

f

, α

g

, α

h

)

= 0.

It is now possible to fully characterize condition (

C

N

) in the image space.

Theorem4.1

Consider problem

P

, suppose

(

H

N

)

holds and let

U

n

be

a cone. Then condition

(

C

N

)

is verified if and only if the following

implica-tion holds:

p

= 0

or

Co(

Im

∂h

(

U

)) =

p

Co(

K

U

)

(Int(

C

)

×

Int(

V

)

×

0) =

In particular, if

p

= 0

or

Co(

Im

∂h

(

U

)) =

p

then

(

α

f

, α

g

)

= 0

.

Proof

)

Suppose (

C

N

) holds and first consider the case

p

1 and

Co(

Im

∂h

(

U

)) =

p

. By means of Lemma 4.2 it is (

α

f

, α

g

)

= 0. Suppose now

by contradiction that

(

t

f

, t

g

, t

h

)

Co(

K

U

)

(Int(

C

)

×

Int(

V

)

×

0)

=

;

be-ing

α

f

C

+

,

α

g

V

+

, (

α

f

, α

g

)

= 0,

t

f

Int(

C

),

t

g

Int(

V

) and

t

h

= 0 it

is:

α

Tf

t

f

+

α

gT

t

g

+

α

Th

t

h

>

0

(4.1)

Since (

t

f

, t

g

, t

h

)

Co(

K

U

)

q

N

,

q >

0,

v

1

, . . . , v

q

U

, such that

(

t

f

, t

g

, t

h

) =

q i=1

∂f

∂v

i

(

x

0

)

,

∂g

∂v

i

(

x

0

)

,

∂h

∂v

i

(

x

0

)

hence

α

Tf

t

f

+

α

Tg

t

g

+

α

Th

t

h

=

q i=1

α

Tf

∂f

∂v

i

(

x

0

) +

α

Tg

∂g

∂v

i

(

x

0

) +

α

Th

∂h

∂v

i

(

x

0

)

0

and this contradicts (4.1). The proof for the case

p

= 0 is analogous.

)

If

p

1 and Co(

Im

∂h

(

U

))

=

p

the result follows from Lemma 4.1.

Consider now the case

p

1 and Co(

Im

∂h

(

U

)) =

p

, so that Co(

K

U

)

(Int(

C

)

×

Int(

V

)

×

0) =

; by means of a well known separation theorem

(13)

between convexsets,

(

α

f

, α

g

, α

h

)

(Int(

C

)

×

Int(

V

)

×

0)

+

, (

α

f

, α

g

, α

h

)

=

0, such that (

α

f

, α

g

, α

h

)

T

t

0

t

Co(

K

U

)

K

U

. A known result on polar

cones (

6

) implies that (Int(

C

)

×

Int(

V

)

×

0)

+

= Int(

C

)

+

×

Int(

V

)

+

×

p

and

hence, being

C

and

V

convexcones (

7

),

α

f

C

+

,

α

g

V

+

,

α

h

p

,

(

α

f

, α

g

, α

h

)

= 0, such that:

α

Tf

∂f

∂v

(

x

0

) +

α

T g

∂g

∂v

(

x

0

) +

α

T h

∂h

∂v

(

x

0

)

0

v

U, v

= 0

.

The directional derivatives

∂f∂v

(

x

0

),

∂g∂v

(

x

0

) and

∂h∂v

(

x

0

) are continuous as

functions of direction, since

f

,

g

and

h

Hadamard directionally differentiable

at

x

0

, hence (

C

N

) is verified. In particular for Lemma 4.2 it is (

α

f

, α

g

)

= 0.

The proof for the case

p

= 0 is analogous.

It is now worth making a comparison between conditions (

C

N

) and (3.4)

one. Condition (3.4) states that

K

L

(Int(

C

)

×

Int(

V

)

×

0) =

while, for a given cone

U

, (

C

N

) implies

Co(

K

U

)

(Int(

C

)

×

Int(

V

)

×

0) =

.

It is then clear that, even when

K

U

K

L

, (

C

N

) condition is stronger than

(3.4) since it requires the existence of a separating hyperplane between

K

U

and (Int(

C

)

×

Int(

V

)

×

0), while

K

L

in (3.4) is not convexin general and

hence a separation hyperplane may not exists.

Note finally that in Example 4.1, where

U

=

T

(

X

S, x

0

) is assumed

and (

C

N

) does not hold, it results that Co(

K

U

) =

2

and hence no

separat-ing hyperplane exists; note also that in Example 4.1 condition (3.4) holds

without any convexity assumption regarding to the cones

U

,

T

(

X

S, x

0

),

K

U

or

K

L

.

4.2

U-regularity conditions

As it has been pointed out in the previous subsection, condition

K

L

(Int(

C

)

×

Int(

V

)

×

0) =

6

LetC1, . . . , Cnbe cones, then (C1×. . .×Cn)+= (C+1 ×. . .×C +

n).

To prove this property it is sufficient to consider just the casen = 2. First verify that (C+ 1 ×C + 2)(C1×C2)+; assuming (α1, α2)(C1+×C + 2) it yields thatα T 1c+αT2v≥0 ∀c C1 and ∀v C2 so that (α1, α2) (C1×C2)+. Verify now that (C1 ×C2)+

(C+ 1 ×C

+

2); assume (α1, α2) (C1×C2)+ and suppose by contradiction thatα1 ∈C1+

[α2 ∈C2+], then∃c¯∈C1 [¯v∈C2] such thatαT1c <¯ 0 [α2Tv <¯ 0]; sinceC1 [C2] is a cone

thenλc¯∈C1 [λv¯∈C2]∀λ >0 so that, givenv∈C2 [c∈C1], forλ >0 great enough we

haveαT1(λ¯c)+α2Tv <0 [αT1c+αT2(λ¯v)<0] and this contradicts that (α1, α2)(C1×C2)+. 7LetC be a cone; it is known (see for all [27]) thatC+ = Cl(C)+ so that Int(C)+ =

Cl(Int(C))+too. IfCis a convex cone we also have (see for instance [4]) that Cl(Int(C)) =

(14)

does not guarantee (

C

N

), since

p

= 0 or

Co(

Im

∂h

(

U

)) =

p

Co(

K

U

)

(Int(

C

)

×

Int(

V

)

×

0) =

is needed. This behaviour suggests the introduction of the following

regu-larity condition (

8

).

Definition 4.1

Consider Problem

P

and suppose (

H

N

) holds. A cone

U

n

verifies an

U

-regularity condition

if the following implication holds:

KL∩(Int(C)×Int(V)×0) =and

[p= 0 or Co(Im∂h(U)) =p ]

Co(KU)(Int(C)×Int(V)×0) =

(4.2)

The use of

U

-regularity conditions is focused on in the next theorem

which follows directly from (4.2) and Theorem 4.1.

Theorem4.2

Consider Problem

P

and suppose

(

H

N

)

holds; the following

properties hold:

i)

U

verifies an

U

-regularity condition if and only if

K

L

(Int(

C

)

×

Int(

V

)

×

0) =

(

C

N

)

holds

;

ii) if

x

0

X

is a feasible local efficient point and

U

n

is a cone then:

U

verifies an

U

-regularity condition

(

C

N

)

holds.

In other words, an

U

-regularity condition is nothing but the additional

hypothesis needed in order to commute condition (3.4) in the image space

to condition (

C

N

) in the decision space. Hence, from now on, the study of

(

C

N

) optimality conditions can be equivalently done in the image space by

means of

U

-regularity conditions.

Theorem4.3

Consider Problem

P

, suppose

(

H

N

)

holds and let

x

0

X

be a feasible local efficient point. Then for every cone

U

n

verifying an

U

-regularity condition

α

f

C

+

,

α

g

V

+

,

α

h

p

,

(

α

f

, α

g

, α

h

)

= 0

,

such that:

α

Tf

∂f

∂v

(

x

0

) +

α

T g

∂g

∂v

(

x

0

) +

α

T h

∂h

∂v

(

x

0

)

0

v

Cl(

U

)

\ {

0

}

.

In particular, if

p

= 0

or

Co(

Im

∂h

(

U

)) =

p

then

(

α

f

, α

g

)

= 0

.

8A different definition ofU-regularity condition, not characterizing conditions (C

N),

References

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