Necessary Optimality Conditions in Vector

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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 - Stampato in Proprio -

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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: [email protected] September 2001

Abstract This paper deals with vector optimization problems having a vector 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 continuity 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 generalizing 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: inequality constraints, 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 function is given by a closed convex pointed cone C with nonempty interior ∗

This paper has been partially supported by M.U.R.S.T.

1

(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 convex pointed cone V with nonempty interior. Problems of this kind have been studied in the literature in finite dimensional spaces with a scalar objective function and under differentiability hypothesis (1 ), obtaining (with some additional hypothesis) necessary optimality 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 conditions 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 principle 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 previously stated conditions in the image space. The conditions in the decision space result to be stronger than the image space ones, hence a new regularity 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 assuming 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:   C max f (x)   

P :

   

g(x) ∈ V h(x) = 0 x∈X

inequality constraints equality constraints set constraint

1

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

2

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 convex cones 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 convex or with nonempty interior. For the sake of convenience, note that problem P can be rewritten in the following form:     

C max f (x) g(x) ∈ V P :  x ∈ (X ∩ S)    S = {x ∈ A : h(x) = 0} The aim of this paper is to study optimality conditions for a feasible point x0 ∈ X which is assumed, without loss of generality, to bind all the inequality constraints, so that g(x0 ) = 0. Note that it is not known whether or not x0 belongs to the boundary of X. The feasible point x0 ∈ X is said to be a local efficient point if there exists a suitable neighbourhood Ix0 of x0 such that: ∃y ∈ Ix0 ∩ X such that f (y) ∈ f (x0 ) + 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 , x0 ∈ X is a local efficient point if and only if there exists a suitable neighbourhood Ix0 of x0 such that: ∃y ∈ Ix0 ∩ X such that F (y) ∈ F (x0 ) + (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 x0 , denoted with T (X, x0 ), which is a closed cone defined as follows: T (X, x0 ) = {x ∈ n : ∃{xk } ⊂ X, xk → x0 , ∃{λk } ⊂ ++ , λk → +∞, x = lim λk (xk − x0 )}. k→+∞

Also subcones of T (X, x0 ) are fundamental in this paper; with this aim just recall that particular subcones of T (X, x0 ) are the well known cone of feasible directions to X at x0 (2 ), denoted with F (X, x0 ), and the cone of interior directions to X at x0 , denoted with I(X, x0 ) (see for example [3, 17, 18]). 2 Let X ⊆ n be a nonempty set and let x0 ∈ Cl(X). The cone of feasible directions to X at x0 F (X, x0 ) and the cone of interior directions to X at x0 I(X, x0 ) are defined as follows:

F (X, x0 )

=

{x ∈ n : ∃δ > 0 such that x0 + λx ∈ X

3

∀λ ∈ (0, δ]};

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 (x0 ) in the image space is introduced: T1 = {t ∈ s+m+p : ∃{xk } ⊂ X, xk → x0 , h(xk ) = 0, ∃{λk } ⊂ , λk > 0, λk → +∞, t = lim λk (F (xk ) − F (x0 ))}. k→+∞

(3.1)

The cone T1 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 x0 ∈ Int(X).

3.1

The nonsmooth case

The aim of this subsection in to characterize in the image space the efficiency of x0 , 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. Theorem 3.1 Consider problem P . If x0 ∈ X is a local efficient point then: T1 ∩ (Int(C) × Int(V ) × 0) = ∅ (3.2) I(X, x0 )

=

{x ∈ n : ∃ > 0, ∃δ > 0 such that λ ∈ (0, δ), y − x < imply x0 + λy ∈ X}.

It is very well known that for any set X: I(X, x0 ) ⊆ Int(F (X, x0 )) ⊆ F (X, x0 ) ⊆ F (Cl(X), x0 ) ⊆ Cl(F (X, x0 )) ⊆ T (X, x0 ).

4

Proof The result is proved by contradiction. Suppose that ∃t∗ ∈ T1 ∩ (Int(C) × Int(V ) × 0); then ∃{xk } ⊂ X, xk → x0 , h(xk ) = 0, ∃{λk } ⊂ , λk > 0, λk → +∞, such that t∗ = limk→+∞ λk (F (xk ) − F (x0 )). Being t∗ ∈ (Int(C) × Int(V ) × 0) and being h(xk ) = 0 ∀k then for a known limit theorem: ∃k¯ > 0 such that λk (F (xk ) − F (x0 )) ∈ (Int(C) × Int(V ) × 0)

∀k > k¯

so that, being λk > 0, F (xk ) ∈ F (x0 ) + (Int(C) × Int(V ) × 0) ∀k > k¯ and this contradicts the local efficiency of x0 . The next theorem shows that it is possible to characterize in the image space the optimality of x0 . Theorem 3.2 Consider problem P . The point x0 ∈ X is a local efficient point if and only if the following condition holds: ∀t ∈ T1 ∩ (C × V × 0), t = 0, and ∀{xk } ⊂ X, xk → x0 , h(xk ) = 0, such that ∃{λk } ⊂ , λk > 0, λk → +∞, with t = limk→+∞ λk (F (xk ) − F (x0 )), there exists an integer k¯ > 0 such that: ¯ F (xk ) ∈ / F (x0 ) + (C 0 × V × 0) ∀k > k. Proof ⇒) If x0 is a local efficient point then, for (2.2), ∀{xk } ⊂ X, xk → x0 , h(xk ) = 0, there exists an integer k¯ > 0 such that F (xk ) ∈ / 0 ¯ F (x0 ) + (C × V × 0) ∀k > k, and this is true also for particular sequences such that t = limk→+∞ λk (F (xk ) − F (x0 )) with t ∈ T1 ∩ (C × V × 0). ⇐) The result is proved by contradiction. Suppose that x0 ∈ X is not a local efficient point, then by means of (2.2) ∃{xk } ⊂ X, xk → x0 , such that F (xk ) ∈ F (x0 ) + (C 0 × V × 0) ∀k, so that in particular h(xk ) = 0 ∀k. k )−F (x0 ) Let us consider now the sequence {dk } ⊂ s+m+p with dk = FF (x (xk )−F (x0 ) ; since the unit ball is a compact set, we can suppose (substituting {dk } with a suitable subsequence, if necessary) that limk→+∞ dk = t∗ = 0, t∗ ∈ T1 . (xk )−F (x0 ) On the other hand, dk = FF (x ∈ (C 0 × V × 0) so that its limit k )−F (x0 ) t∗ ∈ (C × V × 0). It then results that t∗ ∈ T1 ∩ (C × V × 0), t = 0, (xk )−F (x0 ) and this contradicts the assumptions since t∗ = limk→+∞ FF (x and k )−F (x0 ) 0 F (xk ) ∈ F (x0 ) + (C × V × 0) ∀k. Directly from Theorem 3.2 we can state the following sufficient optimality condition. Corollary 3.1 Consider problem P . If the following condition holds then x0 ∈ X is a local efficient point: T1 ∩ (C × V × 0) = {0} 5

(3.3)

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 T1 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. (HN ) Nondifferentiability Assumptions • Functions f , g and h are Hadamard directionally differentiable at the point x0 ∈ 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 (HN ) 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 (HN ) holds and let U ⊆ n be a cone. The following sets are defined: ∂h (x0 ) = 0} ∂v ∂h n \ Ker∂h = {v ∈ n \ {0} : (x0 ) = 0} ∂v ∂h {0} ∪ {t ∈ p : t = (x0 ), v = 0, v ∈ U } ∂v C = n \ (Ker T (X ∩ S, x0 ) ∪ Ker∂h ∂h \ T (X ∩ S, x0 )) = L ∂f ∂g ∂h {t ∈ m+s+p : t = ( (x0 ), (x0 ), (x0 )), v = 0, v ∈ L} ∂v ∂v ∂v

Ker∂h = {0} ∪ {v ∈ n \ {0} : C = Ker∂h

Im∂h (U ) = L(X, S, x0 ) = KL = 3

Let f : A → , with A ⊆ n open set. The limit: lim

λ→0+ ,h→v

f (x0 + λh) − f (x0 ) λ

is called the Hadamard directional derivative of f (x) at x0 ∈ A in the direction v; if this derivative exists and is finite for all v then f (x) is Hadamard directionally differentiable at x0 ∈ A. In order to verify the Hadamard directional derivability, remind that a function f (x) is Hadamard directionally differentiable at x0 (see [16]) if and only if its derivative def ∂f (x0 ) = ∂v

(x0 ) limλ→0+ f (x0 +λv)−f is continuous as a function of direction and the function λ itself is Dini uniformly directionally differentiable at x0 (hence directionally differentiable at x0 ), that is to say that:

 

lim f (x0 + v) − f (x0 ) −

v→0



∂f  (x0 ) = 0 ∂v

Recall also that if a function f (x) is Hadamard directionally differentiable at x0 then it is also continuous at x0 . A vector valued function F : A → m is Hadamard directionally differentiable at x0 if all its components verify this property.

6

= {t ∈ m+s+p : t = (

KU

∂f ∂g ∂h (x0 ), (x0 ), (x0 )), v = 0, v ∈ U } ∂v ∂v ∂v

C , Im (U ), K and K are cones, since ∂f (x ), Note that Ker∂h , Ker∂h 0 L U ∂h ∂v ∂g ∂h ∂v (x0 ) and ∂v (x0 ) are positively homogeneous (of the first degree) as functions 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, x0 ) will be very used, with this aim note that:

U ⊆ L(X, S, x0 )

⇔

U ∩ Ker∂h ⊆ T (X ∩ S, x0 )

C it is worth noticing Remark 3.1 Since L(X, S, x0 ) = T (X ∩ S, x0 ) ∪ Ker∂h that if h is Hadamard directionally differentiable at x0 ∈ X then (5 ):

T (X ∩ S, x0 ) ⊆ T (S, x0 ) ⊆ Ker∂h In order to verify this property, firstly note that T (X ∩ S, x0 ) ⊆ T (S, x0 ) being X ∩ S ⊆ S. Being t = 0 ∈ T (S, x0 ) ∩ Ker∂h just the case t ∈ T (S, x0 ), t = 0, has to be considered. By means of the definition of tangent cone, ∃{xk } ⊂ S, xk → x0 , ∃{λk } ⊂ , λk > 0, λk → +∞, such that t = limk→+∞ λk (xk − x0 ); it can be supposed also (eventually substituting {xk } 0 with a proper subsequence) that v = limk→+∞ xxkk −x −x0  . Since {xk } ⊂ S it yields h(x0 ) = h(xk ) = 0 ∀k > 0 so that, by means of the Hadamard directional differentiability of h(x), it is: 0 = lim

k→+∞

h(xk ) − h(x0 ) h(x0 + γk dk ) − h(x0 ) ∂h = = lim (x0 ) xk − x0  γk ∂v γk →0+ ,dk →v

0 where γk = xk − x0  and dk = xxkk −x −x0  , so that v ∈ Ker∂h . By means of the definition it results:

xk − x0 = µv k→+∞ xk − x0 

t = lim λk (xk − x0 ) = lim λk xk − x0  lim k→+∞

k→+∞

where µ = limk→+∞ λk xk − x0  ≥ 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 of KL generalizes the one given in [5, 6, 7, 8, 9, 10, 11] for differentiable problems having no set constraints; in particular these papers consider KL = {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 if h is differentiable at x0 ∈ X it is:

T (S, x0 ) ⊆ Cl(Co(T (S, x0 ))) ⊆ Ker∂h where Co(X) denotes the convex hull of the set X.

7

Theorem 3.3 Consider Problem P and suppose (HN ) holds; if the feasible point x0 ∈ X is a local efficient point then the two following equivalent conditions hold: KL ∩ (Int(C) × Int(V ) × 0) = ∅

(3.4)

(KL − (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, x0 ) the two following further equivalent conditions hold: KU ∩ (Int(C) × Int(V ) × 0) = ∅

(3.6)

(KU − (C × V × 0)) ∩ (Int(C) × Int(V ) × 0) = ∅

(3.7)

Proof Condition (3.4) is proved by contradiction. Suppose that there exists t = (tf , tg , th ) ∈ KL ∩ (Int(C) × Int(V ) × 0), so that ∃µ > 0, ∃v ∈ L(X, S, x0 ), v = 1, such that t = µ(

∂f ∂g ∂h (x0 ), (x0 ), (x0 )) ∈ (Int(C) × Int(V ) × 0). ∂v ∂v ∂v

C and / Ker∂h Being ∂h ∂v (x0 ) = 0 then v ∈ Ker∂h which implies that v ∈ v ∈ T (X ∩ S, x0 ). By means of the definition of T (X ∩ S, x0 ) it yields that ∃{xk } ⊂ (X ∩ S), xk → x0 , ∃{λk } ⊂ , λk > 0, λk → +∞, such that v = limk→+∞ vk where vk = λk (xk −x0 ). Being functions f and g Hadamard directionally differentiable it results:

lim

k→+∞

f (xk ) − f (x0 ) 1 λk

= lim

f (x0 +

k→+∞

1 λk vk ) 1 λk

− f (x0 )

=

∂f (x0 ) ∈ Int(C) ∂v

and, in the same way: lim

k→+∞

g(xk ) − g(x0 ) 1 λk

=

∂g (x0 ) ∈ Int(V ) ∂v

By means of a well known limit theorem it then exists k¯ > 0 such that ¯ this f (xk ) − f (x0 ) ∈ Int(C) and g(xk ) − g(x0 ) ∈ Int(V ) for any k > k; means that the sequence {xk } ⊂ (X ∩ S), xk → x0 , is feasible for k > k¯ and that x0 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, x0 ) implies KU ⊆ KL . 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 = (tf , tg , th ) ∈ s+m+p : th = 0}, directly from Theorem 3.1 it follows that the efficiency of x0 implies: (T1 ∪ B) ∩ (Int(C) × Int(V ) × 0) = ∅. 8

It is now just needed to verify that KL ⊆ (T1 ∪ B). Let t = µ ∂F ∂v (x0 ) ∈ KL , ∂F v ∈ L(X, S, x0 ), v = 1, µ ≥ 0; if µ = 0 then t = µ ∂v (x0 ) = 0 ∈ T1 C then ∂h (x ) = 0 and t ∈ B. Suppose now while if µ = 0 and v ∈ Ker∂h 0 ∂v µ = 0 and v ∈ T (X ∩ S, x0 ), then ∃{xk } ⊂ X, xk → x0 , h(xk ) = 0, such −1 0 that v = limk→+∞ xxkk −x −x0  ; let also λk = xk − x0  . By means of the Hadamard directional differentiability of F (x) at x0 it is: F (xk ) − F (x0 ) ∂F (x0 ) = lim = lim λk (F (xk ) − F (x0 )) ∈ T1 ; k→+∞ k→+∞ ∂v xk − x0  being T1 a cone it then follows that t = µ ∂F ∂v (x0 ) ∈ T1 too.

4

Optimality conditions in the Decision Space: the nondifferentiable case

In the literature some necessary optimality conditions expressed in the decision 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” conditions. 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: (CN ) ∃αf ∈ C + , ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: αfT

∂f ∂g ∂h (x0 ) + αgT (x0 ) + αhT (x0 ) ≤ 0 ∀v ∈ Cl(U ) \ {0} ∂v ∂v ∂v

where U ⊆ n is a cone and (HN ) is assumed. It is important to note that conditions (CN ), depending on the particular chose cone U , do not hold in general even if x0 is an efficient point. This is shown in the following example, which implicitly points out that condition (3.4) is more general than (CN ) ones. Example 4.1 Consider the following problem: P : {max f (x1 , x2 ) = x1 , g(x1 , x2 ) = x2 ≥ 0, x ∈ X}

9

where X = X1 ∪ X2 ∪ X3 with: X1 = {(x1 , x2 ) ∈ 2 : x1 + x2 ≥ 0, 2x1 + x2 ≤ 0}, X2 = {(x1 , x2 ) ∈ 2 : x1 ≤ 0, x2 ≤ 0}, X3 = {(x1 , x2 ) ∈ 2 : x1 + x2 ≥ 0, x1 + 2x2 ≤ 0} and x0 = (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, x0 ) = KL since [Jf (x0 ), Jg (x0 )] is equal to the identity matrix. The point x0 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, x0 ), T (X ∩ S, x0 ) and KL are not convex. Assume now U = T (X ∩S, x0 ); even if x0 ∈ X is a global efficient point it can be easily verified that (CN ) does not hold; this points out that condition (3.4) is more general than (CN ) 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 decision space, is the existence of a separation hyperplane between the cone (Int(C) × Int(V ) × 0) and KU or KL . This result is stated by means of separating 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 (CN ) 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 (HN ) holds and let U ⊆ n be a cone such that Co(Im∂h (U )) = p . Then ∃αh ∈ p , αh = 0, such that: ∂h αhT (x0 ) ≤ 0 ∀v ∈ Cl(U ) \ {0} ∂v and hence (CN ) is verified. Proof Since Co(Im∂h (U )) = p there exists a support hyperplane for the convex cone Co(Im∂h (U )), so that ∃αh ∈ p , αh = 0, such that αhT t ≤ 0 ∀t ∈ Co(Im∂h (U )); this implies that αhT ∂h ∂v (x0 ) ≤ 0 ∀v ∈ U , v = 0. Being ∂h ∂v (x0 ) continuous as a function of direction v due to the Hadamard directional differentiability of h, it then follows that αhT ∂h ∂v (x0 ) ≤ 0 ∀v ∈ Cl(U ), v = 0. The whole result is then proved just assuming αf = 0 and αg = 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 10

of any additional hypothesis, such as convexity ones, optimality assumptions on x0 , regularity conditions for the problem. Lemma 4.2 Consider problem P with p ≥ 1, suppose (HN ) holds and let U ⊆ n be a cone. If (CN ) is verified and Co(Im∂h (U )) = p then (αf , αg ) = 0. Proof Suppose by contradiction that αf = 0 and αg = 0, so that αh = 0. Then ∂h αhT (x0 ) ≤ 0 ∀v ∈ Cl(U ) \ {0}, ∂v and this yields αhT t ≤ 0 ∀t ∈ Im∂h (U ). Consequently it results αhT t ≤ 0 ∀t ∈ Co(Im∂h (U )) = p which implies αh = 0, and this is a contradiction since (αf , αg , αh ) = 0. It is now possible to fully characterize condition (CN ) in the image space. Theorem 4.1 Consider problem P , suppose (HN ) holds and let U ⊆ n be a cone. Then condition (CN ) is verified if and only if the following implication holds: p = 0 or Co(Im∂h (U )) = p



⇒ Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅

In particular, if p = 0 or Co(Im∂h (U )) = p then (αf , αg ) = 0. Proof ⇒) Suppose (CN ) holds and first consider the case p ≥ 1 and Co(Im∂h (U )) = p . By means of Lemma 4.2 it is (αf , αg ) = 0. Suppose now by contradiction that ∃(tf , tg , th ) ∈ Co(KU )∩(Int(C) × Int(V ) × 0) = ∅; being αf ∈ C + , αg ∈ V + , (αf , αg ) = 0, tf ∈ Int(C), tg ∈ Int(V ) and th = 0 it is: αfT tf + αgT tg + αhT th > 0 (4.1) Since (tf , tg , th ) ∈ Co(KU ) ∃q ∈ N, q > 0, ∃v1 , . . . , vq ∈ U , such that (tf , tg , th ) =

q  ∂f i=1



∂g ∂h (x0 ), (x0 ), (x0 ) ∂vi ∂vi ∂vi

hence αfT tf + αgT tg + αhT th =

q  i=1



αfT

∂f ∂g ∂h (x0 ) + αgT (x0 ) + αhT (x0 ) ≤ 0 ∂vi ∂vi ∂vi

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(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅; by means of a well known separation theorem 11

between convex sets, ∃(αf , αg , αh ) ∈ (Int(C) × Int(V ) × 0)+ , (αf , αg , αh ) = 0, such that (αf , αg , αh )T t ≤ 0 ∀t ∈ Co(KU ) ⊇ KU . 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 convex cones (7 ), ∃αf ∈ C + , ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: αfT

∂f ∂g ∂h (x0 ) + αgT (x0 ) + αhT (x0 ) ≤ 0 ∀v ∈ U, v = 0. ∂v ∂v ∂v

∂g ∂h The directional derivatives ∂f ∂v (x0 ), ∂v (x0 ) and ∂v (x0 ) are continuous as functions of direction, since f , g and h Hadamard directionally differentiable at x0 , hence (CN ) is verified. In particular for Lemma 4.2 it is (αf , αg ) = 0. The proof for the case p = 0 is analogous.

It is now worth making a comparison between conditions (CN ) and (3.4) one. Condition (3.4) states that KL ∩ (Int(C) × Int(V ) × 0) = ∅ while, for a given cone U , (CN ) implies Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅. It is then clear that, even when KU ⊆ KL , (CN ) condition is stronger than (3.4) since it requires the existence of a separating hyperplane between KU and (Int(C) × Int(V ) × 0), while KL in (3.4) is not convex in general and hence a separation hyperplane may not exists. Note finally that in Example 4.1, where U = T (X ∩ S, x0 ) is assumed and (CN ) does not hold, it results that Co(KU ) = 2 and hence no separating 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, x0 ), KU or KL .

4.2

U -regularity conditions

As it has been pointed out in the previous subsection, condition KL ∩ (Int(C) × Int(V ) × 0) = ∅ Let C1 , . . . , Cn be cones, then (C1 × . . . × Cn )+ = (C1+ × . . . × Cn+ ). To prove this property it is sufficient to consider just the case n = 2. First verify that (C1+ × C2+ ) ⊆ (C1 × C2 )+ ; assuming (α1 , α2 ) ∈ (C1+ × C2+ ) it yields that α1T c + α2T v ≥ 0 ∀c ∈ C1 and ∀v ∈ C2 so that (α1 , α2 ) ∈ (C1 × C2 )+ . Verify now that (C1 × C2 )+ ⊆ (C1+ × C2+ ); assume (α1 , α2 ) ∈ (C1 × C2 )+ and suppose by contradiction that α1 ∈ C1+ [α2 ∈ C2+ ], then ∃¯ c ∈ C1 [∃¯ v ∈ C2 ] such that α1T c¯ < 0 [α2T v¯ < 0]; since C1 [C2 ] is a cone then λ¯ c ∈ C1 [λ¯ v ∈ C2 ] ∀λ > 0 so that, given v ∈ C2 [c ∈ C1 ], for λ > 0 great enough we have α1T (λ¯ c)+α2T v < 0 [α1T c+α2T (λ¯ v ) < 0] and this contradicts that (α1 , α2 ) ∈ (C1 ×C2 )+ . 7 Let C be a cone; it is known (see for all [27]) that C + = Cl(C)+ so that Int(C)+ = Cl(Int(C))+ too. If C is a convex cone we also have (see for instance [4]) that Cl(Int(C)) = Cl(C) so that Int(C)+ = C + . 6

12

does not guarantee (CN ), since p = 0 or Co(Im∂h (U )) = p



⇒ Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅

is needed. This behaviour suggests the introduction of the following regularity condition (8 ). Definition 4.1 Consider Problem P and suppose (HN ) 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. Theorem 4.2 Consider Problem P and suppose (HN ) holds; the following properties hold: i) U verifies an U -regularity condition if and only if KL ∩ (Int(C) × Int(V ) × 0) = ∅

⇒

(CN ) holds;

ii) if x0 ∈ X is a feasible local efficient point and U ⊆ n is a cone then: U verifies an U -regularity condition

⇔

(CN ) 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 (CN ) in the decision space. Hence, from now on, the study of (CN ) optimality conditions can be equivalently done in the image space by means of U -regularity conditions. Theorem 4.3 Consider Problem P , suppose (HN ) holds and let x0 ∈ X be a feasible local efficient point. Then for every cone U ⊆ n verifying an U -regularity condition ∃αf ∈ C + , ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: αfT

∂f ∂g ∂h (x0 ) + αgT (x0 ) + αhT (x0 ) ≤ 0 ∀v ∈ Cl(U ) \ {0}. ∂v ∂v ∂v

In particular, if p = 0 or Co(Im∂h (U )) = p then (αf , αg ) = 0. 8 A different definition of U -regularity condition, not characterizing conditions (CN ), has been already introduced in [13, 14].

13

Remark 4.1 Note that Theorem 4.3 cannot be applied to Example 4.1 when U = T (X ∩ S, x0 ) is assumed, since being Co(KU ) = 2 the cone U verifies no U -regularity condition. The aim of this paper is now moved in stating U -regularity conditions; the following trivial ones, which do not need of the optimality of x0 in order to guarantee (CN ), can be obtained directly from (4.2): i)

p = 0 or Co(Im∂h (U )) = p

ii)

p = 0 or Co(Im∂h (n )) = p



⇒ Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅, 

⇒ Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅,

iii) Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅, where iii)⇒ii)⇒i). More interesting U -regularity conditions, based on the optimality of x0 , are stated in the next theorem; with this aim it is interesting to preliminary point out the following property. Lemma 4.3 Consider problem P , suppose (HN ) holds and let U ⊆ n be a cone. The following conditions are equivalent: Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅

(4.3)

Co(KU − (C × V × 0)) ∩ (Int(C) × Int(V ) × 0) = ∅

(4.4)

Proof Being Co(KU ) ⊆ Co(KU − (C × V × 0)), to prove the equivalence among (4.3) and (4.4) it must be shown that (4.3)⇒(4.4). It can be easily verified that (4.3) implies: (Co(KU ) − (C × V × 0)) ∩ (Int(C) × Int(V ) × 0) = ∅ so that the result follows since the convexity of C and V implies: (Co(KU ) − (C × V × 0)) = Co(KU − (C × V × 0)). Theorem 4.4 Consider problem P , suppose (HN ) holds and consider also a cone U ⊆ n . The following conditions are U -regularity ones: i)

p = 0 or Co(Im∂h (U )) = p

ii)

p = 0 or Co(Im∂h (U )) = p



⇒ Co(KU ) ⊆ (KL − (C × V × 0)), 



⇒

U ∩ Ker∂h ⊆ T (X ∩ S, x0 ) and (KU − (C × V × 0)) is a convex cone

iii) p = 0, U ⊆ T (X, x0 ) and (KU − (C × V × 0)) is a convex cone (9 ), 9

Note that when p = 0 it is S = A and Ker∂h = n .

14

where iii)⇒ii)⇒i). Proof Let us first prove that i) is an U -regularity condition. Suppose that KL ∩ (Int(C) × Int(V ) × 0) = ∅ and [ p = 0 or Co(Im∂h (U )) = p ]; then for i) it is Co(KU ) ⊆ (KL − (C × V × 0)) while for Theorem 3.3 it is (KL − (C × V × 0)) ∩ (Int(C) × Int(V ) × 0) = ∅ so that Co(KU ) ∩ (Int(C) × Int(V ) × 0) = ∅ and hence i) is an U -regularity condition. Let us now prove that ii)⇒i). Condition U ∩ Ker∂h ⊆ T (X ∩ S, x0 ) implies KU ⊆ KL , so that (KU − (C × V × 0)) ⊆ (KL − (C × V × 0)); being (KU − (C × V × 0)) a convex cone it results Co(KU ) ⊆ Co(KU − (C × V × 0)) = (KU − (C × V × 0)) ⊆ (KL − (C × V × 0))

and hence ii)⇒i). The whole result is then proved since iii)⇒ii) trivially.

4.3

Subcones of L(X, S, x0 )

The U -regularity condition ii) stated in Theorem 4.4 points out the importance of the cones U ⊆ n such that U ∩ Ker∂h ⊆ T (X ∩ S, x0 ); recall that this happens if and only if U ⊆ L(X, S, x0 ). The study of U -regularity conditions can then be deepened on looking for particular subcones of T (X ∩ S, x0 ) (10 ). For the sake of simplicity, from now on we will use the following notations: IX = I(X, x0 ),

TX = T (X, x0 ),

IS = I(S, x0 ),

TS = T (S, x0 ),

FX = F (X, x0 ) FS = F (S, x0 ).

Lemma 4.4 Let us consider Problem P and suppose (HN ) holds. It results: Cl(IX ∩ TS ) ∪ Cl(TX ∩ IS ) ∪ Cl(FX ∩ FS ) ⊆ T (X ∩ S, x0 ).

(4.5)

Proof We firstly prove that I(X, x0 )∩T (S, x0 ) ⊆ T (X∩S, x0 ). If IX ∩TS = ∅ the result is trivial, otherwise let t ∈ I(X, x0 ) ∩ T (S, x0 ), t = 0 (note that if t = 0 then t ∈ T (X ∩ S, x0 ) trivially), so that ∃{xk } ⊂ S, xk → x0 , ∃{λk } ⊂ ++ , λk → +∞, such that t = limk→+∞ λk (xk − x0 ). Since t ∈ I(X, x0 ) then ∃k¯ > 0, ∃δ > 0 such that µ ∈ (0, δ), k > k¯ imply x0 + µ(λk (xk − x0 )) ∈ X. Being xk = x0 + λ1k (λk (xk − x0 )) and λk → +∞, then ∃k˜ > k¯ such that ∀k > k˜ it results λ1k < δ and xk = x0 + λ1k (λk (xk − x0 )) ∈ X. This means that ∀k > k˜ > k¯ > 0 we have xk ∈ X ∩ S so that t ∈ T (X ∩ S, x0 ) and hence I(X, x0 ) ∩ T (S, x0 ) ⊆ T (X ∩ S, x0 ). Being T (X ∩ S, x0 ) a closed cone we finally have Cl(IX ∩ TS ) ⊆ T (X ∩ S, x0 ). In the same way we can also prove that Cl(TX ∩ IS ) ⊆ T (X ∩ S, x0 ). Since 10

It is known (see for all [3]) that: F (X, x0 ) ∩ F (S, x0 ) = F (X ∩ S, x0 ) ⊆ T (X ∩ S, x0 ) ⊆ T (X, x0 ) ∩ T (S, x0 ).

15

F (X, x0 ) ∩ F (S, x0 ) = F (X ∩ S, x0 ) ⊆ T (X ∩ S, x0 ) (see for example [3]) it results Cl(F (X, x0 ) ∩ F (S, x0 )) ⊆ T (X ∩ S, x0 ) being T (X ∩ S, x0 ) a closed cone. Corollary 4.1 Consider problem P , suppose (HN ) holds and let U ⊆ n be a cone such that: C , U ⊆ Cl(IX ∩ TS ) ∪ Cl(TX ∩ IS ) ∪ Cl(FX ∩ FS ) ∪ Ker∂h

or such that: C with T (S, x ) = Ker . U ⊆ IX ∪ Cl(TX ∩ IS ) ∪ Cl(FX ∩ FS ) ∪ Ker∂h 0 ∂h

An U -regularity condition is verified if one of the following properties holds: i) (KU − (C × V × 0)) is a convex cone, ii) KU is a convex cone. Proof For ii) of Theorem 4.4 we must verify that U ∩Ker∂h ⊆ T (X ∩S, x0 ). In the first case the result follows from Lemma 4.4 since U ∩ Ker∂h ⊆ Cl(IX ∩ TS ) ∪ Cl(TX ∩ IS ) ∪ Cl(FX ∩ FS ) while in the second one we have: U ∩ Ker∂h ⊆ (IX ∩ Ker∂h ) ∪ Cl(TX ∩ IS ) ∪ Cl(FX ∩ FS ) and the result is proved being T (S, x0 ) = Ker∂h . Recall that the above hypothesis T (S, x0 ) = Ker∂h is not trivial since in general it is just T (S, x0 ) ⊆ Ker∂h , as it has been proved in Remark 3.1 and it is pointed out in the next Example 4.2. Example 4.2 points out also that assuming (HN ) it is possible to have I(S, x0 ) = ∅ even when Co(Im∂h (n )) = p . Example 4.2 Let us consider the point x0 = (0, 0) and the following function h : 2 → :   0    min(x , x ) 1 2 h(x) =  x x 1 2   

if if if max(x1 , x2 ) if

It results: ∂h (x0 ) = ∂v



x1 x1 x1 x1

≥ 0, x2 ≥ 0, x2 < 0, x2 < 0, x2

0 if x1 x2 ≤ 0 h(v) if x1 x2 > 0 16

≤0 >0 ≥0 0, x2 < 0}, and I(S, x0 ) = {(x1 , x2 ) : x1 > 0, x2 < 0}. Even if Co(Im∂h (n )) = Im∂h (n ) = , we then have I(S, x0 ) = ∅ and T (S, x0 ) ⊂ Ker∂h but T (S, x0 ) = Ker∂h , since for example d = (−1, 1)T ∈ Ker∂h but d ∈ / T (S, x0 ). Remark 4.2 Note that in Lemma 4.4 no particular properties at all are required for the sets X and S. Note also the difficulty of stating a subcone of T (X ∩ S, x0 ) greater than Cl(IX ∩ TS ) ∪ Cl(TX ∩ IS ) ∪ Cl(FX ∩ FS ) since in general it results (see Example 4.3): Int(F (X, x0 )) ∩ T (S, x0 ) ⊆ T (X ∩ S, x0 ), and even if X and S are convex sets it results (see Example 4.4): Cl(I(X, x0 )) ∩ T (S, x0 ) ⊆ T (X ∩ S, x0 ), Cl(F (X, x0 )) ∩ Cl(F (S, x0 )) ⊆ T (X ∩ S, x0 ).

Example 4.3 Let X = X1 ∪ X2 ⊂ 2 , X1 = {(x1 , x2 ) : 0 ≤ x2 ≤ |x1 |} and X2 = {(x1 , x2 ) : x1 = 0}, let x0 = (0, 0) and let S = {(x1 , x2 ) : T h(x1 , x2 ) = x22 − 4x1 = 0} so that X ∩ S = {x0 } and ∂h ∂v (x0 ) = ∇h(x0 ) v = −4v1 . It then results T (X ∩ S, x0 ) = {0} and T (S, x0 ) = X2 so that: Int(FX ) ∩ TS = Cl(IX ) ∩ TS = TS = X2 ⊆ {0} = T (X ∩ S, x0 ) Example 4.4 Consider the convex set with nonempty interior X = {(x1 , x2 ) ∈ 2 : x2 ≥ x21 }, let x0 = (0, 0) and let h(x1 , x2 ) = x2 , so that S = {(x1 , x2 ) ∈ 2 : x2 = 0}, T X ∩ S = {x0 }, ∂h ∂v (x0 ) = ∇h(x0 ) v = v2 and Ker∂h = S. It results: Cl(I(X, x0 )) = {(x1 , x2 ) ∈ 2 : x2 ≥ 0 }, T (X ∩ S, x0 ) = {0}, L(X, S, x0 ) = {0} ∪ {(x1 , x2 ) ∈ 2 : x2 = 0}, hence Cl(IX ) ∩ TS = Cl(FX ) ∩ Cl(FS ) = TS = S ⊆ {0} = T (X ∩ S, x0 ).

17

5

Optimality conditions in the differentiable case

In this section the optimality conditions in the decision space are deepened on assuming functions f , g and h to be differentiable. ateaux Differentiability Assumptions (HG ) Gˆ • Functions f , g and h are Gˆ ateaux differentiable at x0 ∈ X (11 ). Note that (HG ) implies that the directional derivatives are linear with respect to the direction, hence the following properties hold (12 ): Co(KU ) = KCo(U ) = Co(KCo(U ) ) and Co(Im∂h (U )) = Jh (x0 )[Co(U )]. It results also that: Ker∂h = Ker(Jh (x0 )) and Im∂h (n ) = Img(Jh (x0 )) = Jh (x0 )[n ], that is to say that Ker∂h is the kernel of the Jacobian matrix Jh (x0 ) while Im∂h (n ) is its image. Note finally that when p ≥ 1: Co(Im∂h (n )) = Jh (x0 )[n ] = p ⇐⇒ Jh (x0 ) is surjective, hence when p ≥ 1 and Jh (x0 ) is not surjective the trivial case already discussed in Lemma 4.1 occurs with U = n . It is worth noticing that when Jh (x0 ) is surjective then assumptions (HG ) imply that I(S, x0 ) = ∅. Theorem 5.1 Let h : X → p , X ⊆ n , be a given mapping, let x0 ∈ S = ateaux differentiable at x0 . {x ∈ n : h(x) = 0} and let h(x) be Gˆ i) If ∃d ∈ n such that Jh (x0 )d = 0 then I(S, x0 ) = ∅, ii) if I(S, x0 ) = ∅ then Img(Jh (x0 )) = {0} and Ker(Jh (x0 )) = n , iii) if Jh (x0 ) is surjective then I(S, x0 ) = ∅. Proof i) Let d ∈ I(S, x0 ) = ∅; if d = 0 then x0 ∈ Int(S) (13 ), so that there exists a suitable neighbourhood of x0 , say Ix0 , such that h(x) = 0 ∀x ∈ Ix0 and this implies that Jh (x0 ) = 0 which contradicts Jh (x0 )d = 0. Suppose now d = 0; then there exists a suitable neighbourhood of d, say Id , such that all the directions v ∈ Id are feasible for the set S, this implies that h(x0 + tv) = 0 in a neighbourhood of t = 0 ∀v ∈ Id and hence Jh (x0 )v = 0 Let F : A → m , with A ⊆ n open set, and let JF (x0 ) be the Jacobian matrix of F at x0 . Recall that F (x) is called Gˆ ateaux differentiable at x0 ∈ A if for all directions v it (x0 ) = J (x0 )T v, while F (x) is called Fr´echet differentiable at yields limλ→0+ F (x0 +λv)−F F λ 11

T

0 )−JF (x0 ) v x0 ∈ A if for all directions v it yields limv→0+ F (x0 +v)−F (x = 0. v 12 Let U ∈ n be any cone; from now on the following notation is used:

Jh (x0 )[U ] = {t ∈ p : t = Jh (x0 )v, v ∈ U }. 13

It is known that the following conditions i), ii) and iii) are equivalent for any set S ∈ n (see [18]): i) 0 ∈ I(S, x0 ) ii) x0 ∈ Int(S) iii) I(S, x0 ) = n

18

∀v ∈ Id ; since n linearly independent directions di exist in Id we then have that Jh (x0 )v = 0 ∀v ∈ n which is a contradiction. ii),iii) Follow directly from the previous result i). By means of the above discussed properties it can be easily proved that, given a cone U ⊆ n and assuming (HG ), condition (CN ) is equivalent to the following one: (CG ) ∃αf ∈ C + , ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: [αfT Jf (x0 ) + αgT Jg (x0 ) + αhT Jh (x0 )]v ≤ 0 ∀v ∈ Cl(Co(U )). Note that (CG ) refers to any direction of Cl(Co(U )), while (CN ) considers just the directions of Cl(U ). Conditions like (CG ) are known in the literature as “maximum principle conditions”. Theorem 4.3 can now be specified in the differentiable case as follows. Theorem 5.2 Consider Problem P , suppose (HG ) holds and let x0 ∈ X be a feasible local efficient point. Then for every cone U ⊆ n verifying an U -regularity condition ∃αf ∈ C + , ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: [αfT Jf (x0 ) + αgT Jg (x0 ) + αhT Jh (x0 )]v ≤ 0 ∀v ∈ Cl(Co(U )). In particular, if p = 0 or Jh (x0 )[Co(U )] = p then (αf , αg ) = 0. Let us now determine some more U -regularity conditions based on the differentiability of the functions f , g and h. The next trivial ones, not needing of the optimality of x0 , follows directly from (4.2): i)

p = 0 or Jh (x0 )[Co(U )] = p

ii)

p = 0 or Jh (x0 ) is surjective



⇒ KCo(U ) ∩ (Int(C) × Int(V ) × 0) = ∅, 

⇒ KCo(U ) ∩ (Int(C) × Int(V ) × 0) = ∅,

iii) KCo(U ) ∩ (Int(C) × Int(V ) × 0) = ∅, Further U -regularity conditions, based on the optimality of x0 , are stated in the next theorem. Theorem 5.3 Consider problem P , suppose (HG ) holds and consider also a cone U ⊆ n . The following conditions are U -regularity conditions: i)

p = 0 or Jh (x0 )[Co(U )] = p



⇒ KCo(U ) ⊆ (KL − (C × V × 0)),

19

ii)

p = 0 or Jh (x0 )[Co(U )] = p



⇒ Co(U ) ∩ Ker(Jh (x0 )) ⊆ T (X ∩ S, x0 ),

iii) U is a convex cone and U ∩ Ker(Jh (x0 )) ⊆ T (X ∩ S, x0 ), iv) U = I(X, x0 ) is a convex cone and T (S, x0 ) = Ker(Jh (x0 )), where iv)⇒iii)⇒ii)⇒i). Proof i) is an U -regularity condition since it is equivalent to i) of Theorem 4.4; ii)⇒i) since Co(U ) ∩ Ker(Jh (x0 )) ⊆ T (X ∩ S, x0 ) implies that KCo(U ) ⊆ KL ; iii)⇒ii) trivially; iv)⇒iii) for Lemma 4.4. Some more U -regularity conditions can be found when T (S, x0 ) = Ker∂h ; with this aim let us recall the following result, which is a generalization of the well known Lyusternik theorem (see for all [21, 22]). Theorem 5.4 [21] Let h : X → p , X ⊆ n , be a given mapping and let x0 ∈ S = {x ∈ n : h(x) = 0}. Let also h be locally Fr´echet differentiable on a neighbourhood of x0 , let Jh (x) be continuous at x0 and let Jh (x0 ) be surjective. Then it follows: T (S, x0 ) = Ker(Jh (x0 )) = {d ∈ n : Jh (x0 )d = 0} = Ker∂h From ii) of Theorem 5.3 we obtain the following U -regularity conditions. Theorem 5.5 Consider problem P , suppose (HG ) holds and consider a cone U ⊆ n . Suppose also that h is locally Fr´echet differentiable on a neighbourhood of x0 and the Jacobian matrix Jh (x) is continuous at x0 . The following conditions are U -regularity conditions: i)

p = 0 or Jh (x0 )[Co(U )] = p



⇒ Co(U ) ∩ T (S, x0 ) ⊆ T (X ∩ S, x0 ),

ii) U is a convex cone and U ∩ T (S, x0 ) ⊆ T (X ∩ S, x0 ), iii) U = I(X, x0 ) is a convex cone, iv) U = I(X, x0 ), X is locally convex at x0 (14 ), v) U = I(X, x0 ), X is convex with Int(X) = ∅, where v)⇒iv)⇒iii)⇒ii)⇒i). X ⊆ n is a locally convex set at x0 if ∃Ix0 , arbitrary open ball about x0 , such that X ∩ Ix0 is convex 14

20

Proof Being v)⇒iv)⇒iii)⇒ii)⇒i) it must be proved just that i) is an U -regularity condition. If Jh (x0 ) is not surjective, that is Jh (x0 )[n ] = Co(Im∂h (n )) = p , the result follows from (4.2); if Jh (x0 ) is surjective then for Theorem 5.4 it is T (S, x0 ) = Ker∂h and hence the results follows from ii) of Theorem 5.3. Finally, it is now shown that some “maximum principle conditions” given in the literature are nothing but particular cases of the results stated in this paper. First it is necessary to recall the following known results. Theorem 5.6 Consider problem P with a scalar objective function f , suppose (HG ) holds and assume that x0 ∈ X is a local maximizer. Suppose also that h is locally Fr´echet differentiable on a neighbourhood of x0 and the Jacobian matrix Jh (x) is continuous at x0 . i) [21, 23] If the following condition holds: X is convex, with Int(X) = ∅ then ∃αf ≥ 0, ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: [αf ∇f (x0 ) + αgT Jg (x0 ) + αhT Jh (x0 )](x − x0 ) ≤ 0

∀x ∈ X

and hence: [αf ∇f (x0 ) + αgT Jg (x0 ) + αhT Jh (x0 )]v ≤ 0

∀v ∈ F (X, x0 )

ii) [3] If the following condition holds (15 ): I(X, x0 ) is a convex cone then ∃αf ≥ 0, ∃αg ∈ V + , ∃αh ∈ p , (αf , αg , αh ) = 0, such that: [αf ∇f (x0 ) + αgT Jg (x0 ) + αhT Jh (x0 )]v ≤ 0

∀v ∈ I(X, x0 )

Note that both the previous results are based on a sort of convexity hypothesis regarding to problem P , since the convexity of the set X or of the cone I(X, x0 ) is required. In the light of the stated results, these assumptions are nothing but U -regularity conditions, needed in order to have necessary optimality conditions in the decision space. It is now worth comparing Theorem 5.2 with the results recalled in Theorem 5.6. • Theorem 5.2 refers to a multiobjective problem, while the results in Theorem 5.6 deal with a scalar objective function; 15

As it has been pointed out in [3] by its author, if I(X, x0 ) = ∅ the result is trivial.

21

• case ii) of Theorem 5.6 [3] can be obtained by means of Theorem 5.2 using the U -regularity condition iii) of Theorem 5.5; • assuming the U -regularity condition v) of Theorem 5.5, the maximum principle condition of Theorem 5.2 holds ∀v ∈ T (X, x0 ) = Cl(I(X, x0 )) (16 ), generalizing case i) of Theorem 5.6 where the thesis is verified for a scalar optimization problem ∀v ∈ F (X, x0 ). Note also that this is the first time that we require Int(X) = ∅.

References [1] Averbukh V.I. and O.G. Smolyanov, The theory of differentiation in linear topological spaces, Russian Mathematical Surveys, vol.22, pp.201258, 1967. [2] Averbukh V.I. and O.G. Smolyanov, The various definitions of the derivative in linear topological spaces, Russian Mathematical Surveys, vol.23, pp.67-113, 1968. [3] Bazaraa M.S. and C.M. Shetty, Foundations of Optimization, Lecture Notes in Economics and Mathematical Systems, vol.122, SpringerVerlag, 1976. [4] Bazaraa M.S., Sherali H.D. and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, 1993. [5] Cambini A. and L. Martein, Tangent cones in optimization, in “Generalized Concavity for Economic Applications”, edited by P. Mazzoleni, Tecnoprint, Bologna, pp.29-39, 1992. [6] Cambini A. and L. Martein, Generalized concavity and optimality conditions in vector and scalar optimization, in “Generalized Convexity”, edited by S. Koml´ osi, T. Rapcs´ak and S. Schaible, Lecture Notes in Economics and Mathematical Systems, vol.405, Springer-Verlag, Berlin, pp.337-357, 1994. 16

Recall (see for all [17, 18]) that if X is locally convex at x0 then the cones I(X, x0 ), F (X, x0 ) and T (X, x0 ) are convex and: I(X, x0 ) = cone(Int(X), x0 ),

F (X, x0 ) = cone(X, x0 ),

T (X, x0 ) = Cl(F (X, x0 )),

where cone(X, x0 ) = {y : y = λ(x − x0 ), λ ≥ 0, x ∈ X}. If X is locally convex at x0 and Int(X) = ∅ then: I(X, x0 ) = Int(T (X, x0 ))

and

22

T (X, x0 ) = Cl(I(X, x0 )).

[7] Cambini A. and L. Martein, Second order necessary optimality conditions in the image space: preliminary results, in “Scalar and Vector Optimization in Economic and Financial Problems”, edited by E. Castagnoli and G. Giorgi, pp.27-38, 1995. [8] Cambini A., Martein L. and R. Cambini, A new approach to second order optimality conditions in vector optimization, in “Advances in Multiple Objective and Goal Programming”, edited by R. Caballero, F. Ruiz and R. Steuer, Lecture Notes in Economics and Mathematical Systems, vol.455, Springer-Verlag, Berlin, pp.219-227, 1997. [9] Cambini A., Koml´ osi S. and L. Martein, Recent developments in second order necessary optimality conditions, in “Generalized Convexity, Generalized Monotonicity: Recent Results”, edited by J.-P. Crouzeix, J.-E. Martinez-Legaz and M. Volle, Nonconvex Optimization and Its Applications, vol.27, Kluwer Academic Publishers, Dordrecht, pp.347-356, 1998. [10] Cambini R., Second order optimality conditions in Multiobjective Programming, Optimization, vol.44, pp.139-160, 1998. [11] Cambini R., Generalized concavity and optimality conditions in vector optimization, in “Operations research and its applications”, edited by D.Z. Du, X.S. Zhang and K. Cheng, Lecture Notes in Operations Research, vol.2, World Publishing Corporation, Beijing, pp.172-180, 1996. [12] Cambini R., Minimum Principle Type Necessary Optimality Conditions: Preliminary Results, in “Convessit´ a e Calcolo Parallelo”, edited by G. Giorgi and F.A. Rossi, Libreria Universitaria Editrice, Verona, pp.65-79, 1998. [13] Cambini R., Minimum principle type necessary optimality conditions with equality constraints: preliminary results, in “Generalized Convexity and Optimization for Economic and Financial Decisions”, edited by G. Giorgi and F.A. Rossi, Pitagora Editrice, Bologna, pp.63-79, 1999. [14] Cambini R., Minimum Principle Type Optimality Conditions, Report n.180, Department of Statistics and Applied Mathematics, University of Pisa, June 2000. [15] Canon M., Cullum C. and E. Polak, Constrained Minimization Problems in Finite-dimensional Spaces, SIAM Journal on Control, vol.4, pp.528-547, 1966. [16] Demyanov V.F. and A.M. Rubinov, Constructive Nonsmooth Analysis, Peter-Lang, Berlin, 1995.

23

[17] Giorgi G. and A. Guerraggio, On the notion of Tangent Cone in Mathematical Programming, Optimization, vol.25, pp.11-23, 1992. [18] Giorgi G. and A. Guerraggio, Approssimazioni Coniche Locali: Propriet´ a Algebriche e Topologiche, Technical Report n.14, Istituto di Metodi Quantitativi, Bocconi University of Milan (Italy), 1992. [19] Halkin H., A Maximum Principle of the Pontryagin Type for Systems Described by Nonlinear Difference Equations, SIAM Journal on Control, vol.4, pp.90-111, 1966. [20] Hestenes M.R., Optimization Theory: The Finite Dimensional Case, John Wiley & Sons, New York, 1975. [21] Jahn J., Introduction to the Theory of Nonlinear Optimization, Springer-Verlag, Berlin, 1994. [22] Lyusternik L.A., Conditional extrema of functionals, Math. Sb., vol.41, pp.390-401, 1934. [23] Mangasarian O.L., Nonlinear Programming, McGraw-Hill, New York, 1969. [24] Mangasarian O.L. and S. Fromovitz, A Maximum Principle in Mathematical Programming, in “Mathematical Theory of Control”, edited by A.V. Balakrishnan and L.W. Neustadt, Academic Press Inc., New York, pp.85-95, 1967. [25] Massam H. and S. Zlobec, Various definitions of the derivative in mathematical programming, Mathematical Programming, vol.7, pp.144-161, 1974. [26] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V. and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, John Wiley and Sons Inc., New York, 1962. [27] Sawaragi Y., Nakayama H. and T. Tanino, Theory of Multiobjective Optimization, Academic Press, London, 1985. [28] Shapiro A., On concepts of directional differentiability, Journal of Optimization Theory and Applications, vol.66, n.3, pp.477-487, 1990.

24

PUBBLICAZIONI del Dipartimento di Statistica e Matematica Applicata all Economia Report n. 1 - Some Optimality Conditions in Vector Optimization.(A.Cambini- L.Martein), 1987 Report n. 2 - On Maximizing a Sum of Ratios. (A.Cambini-L.Martein-S.Schaible), 1987 Report n.3 - On the Charnes-Cooper Transformation in Linear Fractional Programming. (G.Gasparotto), 1987 Report n. 4 - Non-linear Separation Theorems, Duality and Optimality. (A.Cambini), 1987 Report n. 5 - Indicizzazione parziale: aspetti metodologici e riflessi economici. (G.Boletto), 1987 Report n. 6 - On Parametric Linear Fractional Programming. (A.Cambini-C.Sodini), 1987 Report n. 7 - Alcuni aspetti meno noti delle migrazioni in Italia. (A.Bonaguidi), 1987 Report n. 8 - On Solving a Linear Program with one Quadratic Constraint. (L.Martein-S.Schaible), 1987 Report n. 9 - Alcune osservazioni sull'equazione funzionale φ(x,y,z) = φ(φ(x,y,t),t,z). (E.Lari), 1988 Report n.10 - Une étude par ménage des migrations des personnes âgées: comparaison des résultats pour l'Italie et les Etats-Unis. (F.Bartiaux), 1988 Report n.11 - Metodi di scomposizione del tasso di inflazione (G.Boletto), 1988 Report n.12 - A New Algorithm for the Strictly Convex Quadratic Programming Problem. (C.Sodini), 1988 Report n.13 - On Generating the Set of all Efficient Points of a Bicriteria Fractional Problem. (L.Martein), 1988 Report n.14 - Applicazioni della programmazione frazionaria nel campo economico-finanziario. (L.Martein), 1988 Report n.15 - On the Bicriteria Maximization Problem. (L.Martein), 1988 Report n.16 - Un prototipo di sistema esperto per la consulenza finanziaria rivolta ai piccolirisparmiatori.(P.Manca), 1988 Report n.17 - Operazioni finanziarie di Soper e operazioni di puro investimento secondo Teichroew Robichek-Montalbano. (P.Manca), 1988 Report n.18 - A k-Shortest Path Approach to the Minimum Cost Matching Problem. (P.Carraresi-C.Sodini), 1988 Report n.19 - Sistemi gravitazionali e fasi di transizione della crescita demografica. (O.Barsotti-M.Bottai), 1988 Report n.20 - Metodi di scomposizione dell'inflazione aggregata: recenti sviluppi. (G.Boletto), 1988 Report n.21 - Multiregional Stable Population as a Tool for Short-term Demographic Analysis. (M.Termote-A.Bonaguidi), 1988

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Report n.66 - Soluzioni efficienti e condizioni di ottimalità nell'otti-mizzazione vettoriale. (L.Martein), 1992 Report n.67 - Le rilevazioni ufficiali ISTAT della popolazione universitaria : problemi e definizioni alternative. (M.F. Romano), 1992 Report n.68 - La ricerca " Spazio Utilizzato". Obiettivi e primi risultati. (M. Bottai - O. Barsotti), 1993 Report n.69 - Composizione familiare e mobilità delle persone anziane. Una analisi regionale. (M. Bottai - F. Bartiaux), 1993 Report n.70 - An algorithm for a non-differentiable non-linear fractional programming problem. (A. Marchi - C.Sodini), 1993 Report n.71 - An finite algorithm for generalized linear multiplicative programming. (C.Sodini - S.Schaible), 1993 Report n.72 - An Approach to Optimality Conditions in Vector and Scalar Optimization. (A.Cambini - L. Martein), 1993 Report n.73 - Generalized concavity and optimality conditions in Vector and Scalar Optimization. (A.Cambini - L. Martein), 1993 Report n.74 - Alcune nuove classi di funzioni concavo-generalizzate. (R.Cambini), 1993 Report n.75 - On Nonlinear Scalarization in Vector Optimization (A.Cambini, A.Marchi, L.Martein), 1994 Report n.76 - Analisi delle carriere degli studenti immatricolati dal 1980 al 1982 (M.F. Romano, G. Nencioni), 1994 Report n.77 - Indici statistici della congiuntura. (G.Ghilardi), 1994 Report n.78 - Condizioni di efficienza locale nella ottimizzazione vettoriale. (R.Cambini), 1994 Report n.79 - Funzioni di utilizzazione dello spazio. (O.Barsotti, M.Bottai), 1994 Report n.80 - Alcuni aspetti dinamici della popolazione dei comuni della Toscana, distinti per ampiezza demografica e per classi di urbanità e di ruralità. (V.Bruno), 1994 Report n.81 - I numeri indici del potere d’acquisto della moneta. (G.Boletto), 1994 Report n.82 - Some optimality conditions in multiobjective programming. (A.Cambini, L.Martein, R.Cambini), 1994 Report n.83 - Fractional Programming with Sums of Ratios. (S. Schaible), 1994 Report n.84 - The Minimum-Risk Approach For Continuous Time Linear-Fractional Programming. (S.Tigan, M.Stancu-Minasian),1994 Report n.85 - On Duality for Multiobjective Mathematical Programming of n-Set Functions. (V.Preda, I.M.Stancu-Minasian),1994 Report n.86 - Optimality and Duality in Nonlinear Programming Involving Semilocally Preinvex and Related Functions.(V.Preda, I.M.Stancu-Minasian, A.Batatorescu), 1994

Report n.87 - Una nota storica sulla programmazione lineare: un problema di Kantorovich rivisto alla luce del problema degli zeri. (E. Melis),1995 Report n.88 - Mobilità territoriale dell'Italia e di tre Regioni tipiche:Lombardia, Toscana, Sicilia. (V.Bruno),1995 Report n.89 - Bibliografia sulla presenza straniera in Italia. (A.Cortese),1995 Report n.90 - Funzioni Scalari Affini Generalizzate (R.Cambini), 1995 Report n.91 - Modelli epidemiologici: teoria e simulazione. (I) (P.Manfredi - F. Tarini), 1995 Report n.92 - The "OLIVAR" Survey. Methodology and Quality. (M. Bottai - M. Caputo - L. Lecchini), 1995 Report n.93 - Old People and social network. (L. Lecchini - D. Marsiglia - M. Bottai), 1995 Report n.94 - Uno studio empirico sul confronto tra alcuni indici statistici della congiuntura. (G. Ghilardi), 1995 Report n.95 - Il traffico nei porti italiani negli anni recenti. (V. Bruno) , 1995 Report n.96 - An Analysis of the Falk-Palocsay Algorithm. (A.Cambini,A.Marchi,L.Martein,S.Schaible) , 1995 Report n 97 - Sulla esistenza di elementi massimali. (A.Cambini, L.Carosi) , 1995 Report n.98 - Generalized Concavity and Generalized Monotonicity Concepts for Vector Valued (R.Cambini, S.Komlòsi), Report n.99 - Second Order Optimality Conditions in the Image Space. (R.Cambini) , 1996 Report n.100 - La Stagionalità delle correnti di navigazione marittima. (V.Bruno) , 1996 Report n.101 - A Comparison of Alternative Discrete Approximations of the Cox- Ingersoll - Ross Model. (E.M.Cleur), 1996 Report n.102 - Sul calcolo del rapporto di concentrazione del Gini. (G. Ghilardi), 1996 Report n.103 - A New Approach to Second Order Optimality Conditions in Vector Optimization. (A.Cambini, L.Martein, R.Cambini), 1996 Report n.104 - Alcune osservcazioni sull’immunizzazione semideterministica. (F. Gozzi), 1996. Report n.105 - Innovation and Capital Accumulation in a Vintage Capital Model: an Infinite Dimensional Control Approach. (E. Barucci, F. Gozzi), 1996. Report n.106 - A survey of bicriteria fractional problems. (A.Cambini, L.Martein, I.M.Stancu-Minasian), 1996 Report n.107 - Viscosità dei salari, offerta di lavoro endogena e ciclo (L. Fanti, P. Manfredi), 1996 Report n.108 - Crescita con ciclo, ritardi nei piani di investimento ed effettivi popolazione. (L.Fanti, P. Manfredi), 1996

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Report n.128 - Programmazione frazionaria e problemi bicriteria (A. Cambini - L. Martein - E. Moretti), 1998 Report n.129 - Incentive compatibility constraints and dynamic programming in continuous time (E. Barucci - F. Gozzi A. Swiech), 1998 Report n 130 - Impatto delle immigrazioni sulla popolazione italiana: confronto tra scenari alternativi (A. Valentini), 1 9 9 9 Report n.131 -Recent developement of migrations from Poland to Europe with a special emphasis on Italy (K. Iglicka) Le migrazioni est-ovest : Le unioni miste in Italia - (O.Barsotti – L.Lecchini),1999 Report. n.132 - Proiezioni demografiche multiregionali a due sessi, con immigrazioni internazionale e vincoli di consistenza (A. Valentini), 1999. Report. n 133 - Backward- Forward stochastic differential utility: Existence, consumption and equilibrium analysis (F.Antonelli - E. Barucci - M.E. Mancino), 1999 Report. n.134 - Asset pricing with endogenous Aspirations (E. Barucci - M.E. Mancino), 1999 Report n. 135 - Estimating a class of diffusion models: An evaluation of the effects of sampled discrete observations. ( E. M. Cleur), 1999. Report n. 136 - Labour supply, time delays, and demoeconomic oscillations in a Solow-type growth model (L. Fanti - P. Manfredi), 1999. Report n.137 - Some Results on Partial Differential Equations and Asian Options ( E. Barucci - S.PolidoriV.Vespri), 1999 Report n.138 - Hedging European Conitngent Claims in a Markovian Incomplete Market (E. Barucci M. Elvira Mancino), 1999 Report n.139 - L’applicazione del modello multiregionale-multistato alla popolazione in Italia mediante l’utilizzo del Lipro: procedura di adattamento dei dati e particolarità tecniche del programma. (A. Valentini), 1999. Report n.140 - Optimality Conditions and Duality in Fractional Programming- Involving Semilocally Preinvex and Related Functions ( I.M.Stuncu-Minasian), 1999 Report n.141- Proiezioni demografiche con algoritmi di consistenza per la popolazione in Italia nel periodo 1997 2142: presentazione dei risultati e confronto con metodologie di stima alternative (A. Valentini), 1999 Report n. 142 - Competitive equilibria with money and restricted participation (L. Carosi), 1999 Report n. 143 - Monetary policy and Pareto improvability in a financial economy with restricted Participation (L. Carosi ), 1999 Report n. 144 -Misurare il benessere e lo sviluppo dai paradossi del Pil a misure di benessere economico sostenibile, con uno sguardo allo sviluppo umano (B. Cheli), 1999 Report n.145 - The old people’s perception of well-being: the role of material and non material resources (B.Cheli, L. Lecchini, L. Masserini), 1999

Report n. 146 - Maximum likelihood estimation of one-dimensional stochastic differential equation models from discrete data: some computational results ( Eugene M. Cleur), 1999 Report n. 147- Utilizzi empirici di modelli multistato continui con durate multiple (Alessandro Valentini Francesco Billari Piero Manfredi), 1999 Report n.148- Transition into adulthoold: its macro-demographic consequences in a multistatew stable population framework ( F. Billari , P. Manfredi, A. Valentini , A. Bonaguidi), 1999 Report n.149- Becoming Adult and its Macro-Demographic Impact: Multistate Stable Population Theory And an Application to Italy (F. Billari, P. Manfredi, A. Valentini), 1999 Report n.150 - Le previsioni demografiche in presenza di immigrazioni: confronto tra modelli alternativi e loro utilizzo empirico ai fini della valutazione dell'equilibrio nel sistema pensionistico (A.Valentini), 1999 Report n.151 - Diffusion processes for asset prices under bounded rationality (E. Barucci. R. Monte) Report n.152 - Reti neurali e analisi delle serie storiche: un modello per la previsione del BTP future (E.Barucci, P.Cianchi, L.Landi, A.Lombardi), 1999 Report n.153 - On the supremum in fractional programming (A. Cambini, L. Carosi, L. Martein), 1999 Report n.154 - First and second order characterizations of a class of pseudoconcave vector functions. ( R.Cambini and L. Martein), 1999 Report n. 155 - Embedding population dynamics in macro-economic models. The case of the goodwin's growth cycle ( P. Manfredi and L. Fanti), 1999 Report n. 156 - Migrazioni dei preti dalla Polonia in Italia (Laura Lecchini e Odo Barsotti), 1999 Report n. 157 - Analisi dei prezzi, in Italia, dal 1975 in poi (Vincenzo Bruno), 1999 Report n.158 - Analisi del commercio al minuto in Italia (Vincenzo Bruno), 1999 Report. n.159 -.Aspetti ciclici della liquidità bancaria, dal 1971 in poi (Vincenzo Bruno), 1999 Report n. 160 - A separation theorem in alternative theorems and vector optimization (Anna Marchi), 1999 Report n. 161- Labour suppley, population dynamicics, and persistent oscillations in a Goodwin-type growth cycle model (P.Manfredi and L.Fanti), 2000 Report n. 162- Neo-classical labour market dynamics and chaos (and the Phillips curve revisited) (P. Manfredi and L.Fanti) 2000 Report n.163- Detection of Hopf bifurcations in continuous-time macro-economic models, withan application to reducible delay-systems ( P. Manfredi and Luciano Fanti)2000 Report n.164 – The dynamics of pareto efficient allocations with stochastic differential utility (F. Antonelli E.Barucci), 2000 Report n. 165 - Computing maximum likelihood estimates of a class of One-Dimensional stochastic differential equatin models from discrete Data* (Eugene M. Cleur), 2000

Report n. 166 - Estimating the drift parameter in diffusion processes more efficiently at discrete times: a role of indirect estimation (Eugene M. Cleur), 2000 Report n. 167 – “Forecasting the forecasts of others” e la politica di inflation targeting (E.Barucci – V.Valori), 2000 Report n. 168 – First and second order optimality conditions in vector optimization (A.Cambini –L. Martein), 2000 Report.n. 169 – Theorems of the Alternative by way of separation Theorems (A. Marchi ) 2000 Report. n. 170 – Asset Pricing and Diversification with Partially Exchangeable Random Variables (E.Barucci, M. Elvira Mancino), 2000 Report.n. 171 – Lon Term Effects of the Efficiency Wage Hypothesis in Goodwin-Type Economies (P. Manfredi and L. Fanti), 2000 Report. n.172 - Long term effects of the efficiency wage hypothesis in Goodwin-type economies: a reply (P. Manfredi and L. Fanti), 2000 Report. n.173 – Innovazione Finanziaria e domanda di moneta in un modello dinamico IS-LM con accumulazione (L. Fanti), 2000 Report n. 174 – Social Heterogeneities in Classical New product Diffusion Models. I: “External” and “Internal” Models. ( P.Manfredi, A.Bonaccorsi, A. Secchi), 2000 Report n. 175 – Modelli per formazione di coppie e modelli di Dinamica familiare. (P. Manfredi, E.Salinelli), 2000 Report n. 176 – Long term interference between Demography and Epidemiogy: the case of tuberculosis. (P. Manfredi, E. Salinelli, A. Melegaro, A. Secchi), 2000 Report n. 177 – Toward the development of an age structure teory for family dynamics I: general frame (P. Manfredi – E. Salinelli), 2000 Report n. 178 – Population heterogeneities, nonlinear oscillations and chaos in some goodwin-type demoeconomic models paper to be presented at the: second workshop on “nonlinear demography” max planck institute for demographic research Rostock, Germany , May 31June 2, 2000 - ( P.Manfredi and Luciano Fanti), 2000 Report n. 179 – Volatility estimation via fourier analysis ( E. Barucci, M.E.Mancino, R. Renò), 2000 Report n. 180 – Minimum Principle Type Optimality Conditions (Riccardo Cambini ), 2000 Report n. 181 – Asset Prices under Bounded Rationality and Noise Trading (Emilio Barucci, Massimiliano Giuli, Roberto Monte), 2000 Report n. 182 – Order Preserving Transformations and application (A. Cambini, D.T.Luc and L. Martein), 2000 Report n. 183 – Variazioni dinamiche (1971-1981-1991) dei fenomeni demografici dei comuni urbani e rurali della Sicilia, in relazione ad alcune caratteristiche di mobilità territoriale (V.Bruno), 2000

Report n. 184 – Asset Pricing with a Backward-Forward Stochastic Differential Utility (F. Antonelli, E.Barucci, M.E.Mancino), 2000 Report n. 185 - Coercivity Concepts and Recession Function in Constrained Problems (R. Cambini, L. Carosi), 2000 Report n. 186 - The pre-vaccination dynamics of measles in Italy: estimating levels of under-reporting of measles cases (John R. Williams – Piero Manfredi), 2000.

Report n. 187. - To what extent can inter-regional migration perturb local endemic patterns? Estimating numbers of measles cases in the Italian regions (Piero Manfredi, John R. Williams),2000 Report.n. 188 - On the Connections between semidefinite Optimization and Vector Optimization (L. Carosi, J.Jahn, L. Martein), Report n. 189 - On the Pseudoconvexity of a Quadratic Fractional Function (A. Cambini, J.P. Crouzeix, L. Martein), 2000 Report n. 190 - A finite Algorithm for particular d.c. quadratic programmng problem. (R. Cambini, C. Sodini), 2000 Report n. 191 – Pseudoconvexity of a class of quadratic fractional functions (R. Cambini – L. Carosi), 2000 Report n. 192 – A note on endogenous restricted partecipation opn financial markets: an existence result (L.Carori), 2000 Report n.193 – Asset price anomalies under bounded rationality (E.Barucci, R.Monte, Roberto Renò), 2000 Report n.194 – A note on volatility estimate-forecast with GARCH models (E.Barucci, R. Renò), 2000 Report n.195 – Sulla misura del benessere economico: i paradossi del pil e le possibili correzioni in chiave etica e sostenibile, con uno spunto per l’analisi della povertà. (Bruno Cheli),2000 Report n.196 – Le proiezioni demografiche con il programma nostradamus (applicazione all’area pisana) (M. Bottai, M Bottai, N.Salvati, M.Toigo), 2000 Report n.197 – La misura della povertà multidimensionale: aspetti metodologici e analisi della realtà italiana alla metà degli anni 90 (A.Lemmi, B.Cheli, B.Mazzolli), 2000 Report n.198 – Generalized B-invex Vector Valued Functions ( C.R. Bector – R. Cambini), 2000 Report n.199 – The workers’resistance to wage cuts is not necessarily detrimental for the economy: the case of a Goodwin’s growth model with endogenous population (L.Fanti – P.Manfredi),2000 Report n.200 – On Measuring volatility of diffusion processes with high frequecy data (E.Barucci – R.Renò),2000 Report n. 201 – Demographic transition and balanced growth (P. Manfredi – L. Fanti), 2000 Report n. 202 – Asset pricing, diversification and Risk Ordering with partially exchangeable random variables (Emilio Barucci, Maria E. Mancino, Emanuele Vannucci), 2001 Report n. 203 – Executive stock options evaluation (E. Barucci, R.Renò, E. Vannucci), 2001

Report n. 204 – Dimensioni delle rimesse e variabili esplicative: un’indagine sulla collettività marocchina immigrata nella Toscana Occidentale (Odo Barsotti, Moreno Toigo), 2001 Report n. 205 – I consumi voluttuari, nell’ultimo trentennio, in Italia (Vincenzo Bruno), 2001 Report n. 206 – The monopolist choice of innovation adoption: A regular-singular stochastic control problem (Michele Longo), 2001 Report n. 207 – The competitive choice of innovation adoption: A finite-fuel singular stochastic control problem (Michele Longo), 2001 Report n. 208 – On the pseudoaffinity of a class of quadratic fractional functions (R. Cambini, L. Carosi), 2001 Report n. 209 – A Finite Algorithm for a Class of Non Linear Multiplicative Programs (R. Cambini, C. Sodini), 2001 Report n. 210 – A method for calculating subdifferential of convex vector functions ( Alberto Cambini, Dinh The Luc, Laura Martein), 2001 Report n. 211 – Pseudolinearity in scalar and vector optimization (Alberto Cambini – Laura Martein), 2001 Report n. 212 – Necessary Optimality Conditions in Vector Optimizations in Vector Optimization (Riccardo Cambini), 2001