JJ II

(1)

volume 4, issue 4, article 67, 2003.

Received 29 May, 2002;

accepted 16 March, 2003.

Communicated by:Z. Nashed

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON GENERALIZED MONOTONE MULTIFUNCTIONS WITH APPLICATIONS TO OPTIMALITY CONDITIONS IN

GENERALIZED CONVEX PROGRAMMING

A. HASSOUNI AND A. JADDAR

Département de Mathématiques et d’Informatique, Faculté des Sciences,

B.P. 1014, Rue Ibn Batouta, Rabat, Maroc.

EMail:[email protected]

c

2000Victoria University ISSN (electronic): 1443-5756 062-02

(2)

On Generalized Monotone Multifunctions with Applications to Optimality Conditions in Generalized

Convex Programming A. Hassouni and A. Jaddar

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of26

J. Ineq. Pure and Appl. Math. 4(4) Art. 67, 2003

Abstract

Characterization of quasiconvexity and pseudoconvexity of lower semicontinuous functions on Banach spaces are presented in terms of abstract subdifferentials relying on a Mean Value Theorem. We give some properties of the normal cone to the lower level set off. We also obtain necessary and sufficient optimality conditions in quasiconvex and pseudoconvex programming via variational inequalities.

2000 Mathematics Subject Classification:90C26, 26B25, 47H04.

Key words: Generalized monotone multifunction, Generalized convex function, Qua- siconvex, Pseudoconvex, Generalized subdifferentials, Normal cone, Level set, Local minimum, Global minimum, Variational inequalities.

1. Introduction

It is natural in convex analysis to search for characterizations of generalized convex functions in terms of some kind of generalized derivatives or subdifferentials. Several contributions to this question has been made recently. The reader may consult for example [3,5,11,13,16,20] for quasiconvex functions and [2,8,21,23,25] for pseudoconvex functions.

In this paper, we shall define an abstract subdifferential as in [1,23] which allows us to extend some results in [1,2, 8,23] and to give some properties of the normal cone to lower level sets of a given functionf.

Notice that the condition0 ∈∂f(¯x)forx¯∈ X, is known to be a necessary but not a sufficient optimality condition in quasiconvex programming for some subdifferentials. We give, using some variational inequalities, a necessary and sufficient condition for a point to be either a local or a global minimum.

After the introduction of some notations and definitions in Section 2, we present in Section3some properties of the abstract subdifferential and normal cone to lower level sets of quasiconvex and pseudoconvex functions. Then, in Section4, we give some optimality conditions involving variational inequalities.

This should extend our previous results stated for quasiconvex lower semicontinuous functions on Banach spaces with the Clarke-Rockafellar subdifferential in [13].

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of26

2. Preliminaries

LetXbe a real Banach space,X^∗ its dual andh·,·ithe duality pairing between X^∗ andX. The segment[a, b]is the set{a+t(b−a); t∈[0,1]}while[a, b[

is the set [a, b]\ {b}. The open ball with centerxand radiusrinX is denoted byB(x, r), and the polar cone of a nonempty subsetAofXis

A^◦ ={x^∗ ∈X^∗; hx^∗, ai ≤0, ∀a ∈A}.

For an extended real valued functionf :X 7→R∪ {+∞}, the effective domain is defined by

dom(f) ={x∈X; f(x)<∞}.

We write l.s.c. for lower semicontinuous, and x_n→_fx when x_n → x and f(x_n)→f(x).

The abstract subdifferential we consider here is defined as follows:

Definition 2.1. An operator ∂ that associates to any l.s.c. function f : X 7→

R∪ {+∞}and a pointx ∈ X a subset∂f(x)ofX^∗ is a subdifferential if the following assertions hold:

(P1) ∂f(x) = {x^∗ ∈X^∗; f(y)≥ f(x) +hx^∗, y−xi ∀y ∈X} whenf is convex.

(P2) Ifx∈domf is a local minimum off, then0∈∂f(x).

(P3) ∂f(x) = ∂g(x), for anyg : X 7→R∪ {+∞}such thatf −g is constant in a neighborhood ofx.

(P4) ∂f(x) = ∅, for anyx∈X such thatf(x) = +∞.

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of26

It is well known that the Clarke-Rockafellar subdifferential∂^CRf satisfies Zagrodny’s Mean value theorem [27]. In order to extend this theorem to our subdifferential, we shall deal with a particular space associated with ∂ called

∂-reliable.

Definition 2.2. [23]. A Banach spaceXis∂-reliable if for each l.s.c. function f : X 7→R∪ {+∞}, for any Lipschitz convex functiong and anyx ∈ domf such thatf+g achieves its minimum inX and eachε >0we have:

0∈∂f(u) +∂g(v) +εB₁^∗(0), for someu, v ∈B_ε(x)such that|f(u)−f(v)|< ε.

In the case of the Clarke-Rockafellar subdifferential∂^CR[26] or Iofee subdifferential∂^I [7], any Banach space is∂-reliable.

In the sequel, we will restrict ourselves to subdifferentials that are included in the dag subdifferential

∂^†f(x) ={x^∗ ∈X^∗; hx^∗, vi ≤f^†(x, v) ∀v ∈X}, where

f^†(x, v) = lim sup

(t,y)→(0₊,x)

t⁻¹(f(y+t(v+x−y)−f(y)).

This subdifferential was introduced by Penot (see [22]), it is large enough to contain the Clarke-Rockafellar ∂^CR and Upper Dini ∂^D⁺ subdifferentials and still has good properties.

Our results rely on the following mean value theorem.

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of26

Theorem 2.1. [23]. LetX be a ∂-reliable space and f : X 7→ R∪ {+∞}a l.s.c. function. For anya∈domf,b∈X\ {a},β≤b, there exists a sequence c_ninXconverging to somec∈[a, b)and a sequencec^∗_n∈∂f(c_n)such that for anyb⁰ =c+t(b−a), witht >0we have:

i) lim inf_nhc^∗_n, b−ai ≥β−f(a), ii) lim infnhc^∗_n, c−cni ≥0, iii) lim inf_nD

c^∗_n,_||b^||b−a||0−c||(b⁰−c_n)E

≥β−f(a).

Following the methods of [1,16,20], we get a similar lemma for our abstract subdifferential, which is immediate by Theorem2.1.

Lemma 2.2. Let X be a Banach ∂-reliable space, f a l.s.c. function. Let a, b ∈ X with f(a) < f(b) then there exists c ∈ [a, b[ and two sequences cn→c,c^∗_n ∈∂f(cn)with

hc^∗_n, x−c_ni>0, for anyx=c+t(b−a)witht >0.

Proof. Let a, b ∈ X with f(a) < f(b), then we can find by Theorem 2.1, c∈[a, b[and two sequencesc_n→c, c^∗_n ∈∂f(c_n)with

lim inf

n hc^∗_n, c−c_ni ≥0, and

lim inf

n hc^∗_n, b−ai ≥f(b)−f(a)>0.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of26

Forx=c+t(b−a)witht >0, we have

hc^∗_n, x−c_ni=hc^∗_n, c−c_ni+thc^∗_n, b−ai.

It follows that

lim inf

n hc^∗_n, x−c_ni>0.

Hence, fornlarge enough, we have that

hc^∗_n, x−c_ni>0.

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of26

3. Generalized Convex Functions and Generalized Monotone Multifunctions

3.1. Quasiconvex Functions and Quasimonotone Multifunc- tions

We recall the characterization of quasiconvex functions of [22,23]. It will allow us to extend and generalize some properties of the normal cone to the lower level set given in [12,13] to a more general setting.

Indeed, forf :X 7→R∪ {+∞}a l.s.c. function,fis said to be quasiconvex if for everyx, y ∈Xandλ∈[0,1]one has

f(λx+ (1−λ)y)≤max{f(x), f(y)}.

And denoting by

S_f(λ) ={x∈X; f(x)≤λ}.

Quasiconvexity is geometrically equivalent to the fact thatS_f(λ)is a convex set for allλ∈R. In the above one could use the strict level sets as well.

Recall that a multifunctionA :X →X^∗ is said to be quasimonotone if for every pair of distinct pointsx, y ∈X:

∃x^∗ ∈A(x), such that hx^∗, y−xi>0 then, ∀y^∗ ∈A(y), hy^∗, y−xi ≥ 0.

Theorem 3.1. [22, 23] LetX be a Banach space andf : X 7→ R∪ {+∞}a l.s.c. function. And consider the following assertions

i) f is quasiconvex.

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of26

ii) ∂f is quasimonotone.

Then i) implies ii) if∂f ⊂∂^†f. And ii) implies i) ifXis∂-reliable.

Forx₀ ∈X, set

L(x₀) ={x∈X; f(x) = f(x₀)}.

Then we have

Proposition 3.2. Let Xbe a Banach∂-reliable space, andf a l.s.c. quasicon- vex function such that∂f ⊂∂^†f. If forx0 ∈Xthere existsr >0with

06∈∂f(x), for allx∈B(x₀, r)∩L(x₀), then we have

[∂f(x₀)]^◦◦ ⊂N(S_f(f(x₀));x₀),

whereN(S_f(f(x₀));x₀)is the normal cone to the lower level setS_f(f(x₀))at the pointx0.

Proof. Suppose by contradiction that there existsvsuch that v ∈[∂f(x₀)]^◦◦ and v 6∈N(S_f(f(x₀));x₀).

We can check that

Cl(R⁺co(∂f(x₀))) = [∂f(x₀)]^◦◦.

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of26

So, we can suppose without loss of generality thatv = x^∗₀ ∈∂f(x₀).Then, we can find somex1 ∈Sf(x0)such that

(3.1) hx^∗₀, x₁−x₀i>0.

We claim that f(x₀) = f(x₁). Otherwise by Lemma 2.2, there exists c ∈ [x₁, x₀[and two sequencesc_n→_fcandc^∗_n∈∂f(c_n)with

hc^∗_n, x₀−c_ni>0.

By using the quasimonotonicity of∂f we have:

hx^∗₀, x₀−c_ni ≥0.

Then, lettingn→+∞we get

hx^∗₀, x₀−ci ≥0.

It follows that

hx^∗₀, x₀−x₁i ≥0.

A contradiction with (3.1), thusf(x₀) = f(x₁).

Now, set V_x₁ ={x∈X : hx^∗₀, x−x₀i>0}.

Vx1 is an open neighborhood ofx1and using the same argument as above we can check thatx₁ is a minimum off onV_x₁, and that

xλ =x0+λ(x1−x0)∈Vx1 andf(xλ) = f(x0)for anyλ∈]0,1[.

Then there existsr >0and¯λ∈]0,1[such thatxλ¯ is a global minimum offon B(x0, r)∩Vx1. Therefore0∈∂f(xλ¯), which is impossible.

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of26

The former proposition extends some already known results for differentiable functions (see for instance [5]). If we denote byT(Sf(f(x);x), the tangent cone of the lower level convex setS_f(f(x))at the pointx∈X, then

T(S_f(f(x));x) = [N(S_f(f(x));x)]^◦.

A sufficient condition that allows us to obtain the equality in Proposition3.2is stated in the following proposition

Proposition 3.3. Under the hypothesis of Proposition3.2and if [∂f(x)]^◦ ⊂T(S_f(f(x));x).

Then we have

N(S_f(f(x));x) = [∂f(x)]^◦◦. Proof. By the bipolar theorem [4] one has

[∂f(x)]^◦◦⊃N(S_f(f(x));x).

And from Proposition3.2, the equality immediately holds.

The following condition

N(S_f(f(x));x) = [∂f(x)]^◦◦,

is in fact a certain kind of regularity condition, which holds only for a sub- class of quasiconvex functions. Another abstract aproach was developed in [15]

based on Crouzeix’s representation theorem [6] who obtained a similar equality for his quasi-subdifferential.

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of26

Consider the multifunctionΓfromX toX^∗ defined by Γ(x) =N(S_f(f(x));x), forx∈X.

Then by using Proposition3.3, we obtain

Proposition 3.4. Let X be a Banach ∂-reliable space, f a l.s.c. quasiconvex function. If for anyx∈X,∂f(x)is nonempty such that

(∂f(x))^◦ ⊂T(S_f(f(x));x).

Then, the multifunctionΓis quasimonotone.

Proof. Since f is quasiconvex, by Theorem 3.1 ∂f is quasimonotone. Using Proposition 2.8 of [12], it follows easily that the multifunctionx 7→ [∂f(x)]^◦◦

is quasimonotone. Then by Proposition3.3,Γis also quasimonotone.

It follows thatΓis quasimonotone.

A particular case of this proposition when ∂ coincides with the Clarke- Rockafellar subdifferential ∂^CR, was treated in [13], whose exact statement is the following.

Proposition 3.5. Let X be a Banach space,f a l.s.c. function from X toR∪ {+∞}such that∂^CRf(x)is nonempty and06∈∂^CRf(x)for allx∈X.

Iff is quasiconvex then the multifunctionΓis quasimonotone.

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of26

3.2. Pseudoconvexity and Subdifferential Properties

The original definition of pseudoconvexity was introduced by Mangazarian in [21] for differentiable functions. This concept was exended later by many au- thors (see for instance [17,22,24]) for arbitrary functions. We will here use the following form:

A functionf is said to be pseudoconvex for the subdifferential∂ if for any x, y ∈X:

∃x^∗ ∈∂f(x) : hx^∗, y−xi ≥0 =⇒ f(x)≤f(y).

A multifunctionA:X →X^∗is said to be pseudomonotone if for every pair of distinct pointsx, y ∈X

∃x^∗ ∈A(x) :hx^∗, y−xi>0 then, ∀y^∗ ∈A(y), hy^∗, y−xi>0.

As in the differentiable case, every pseudoconvex function satisfies the funda- mental properties:

• every local minimum off is global.

• 0∈∂f(x)implies thatxis a global minimum off.

Another interesting property extending a result of [8] where it was stated for the Clarke-Rockafellar subdifferential is the following.

Proposition 3.6. LetX be a Banach ∂-reliable and f : X 7→ R∪ {+∞}be a l.s.c. function and pseudoconvex function such that∂f ⊂ ∂^†f, letx, y ∈ X.

Then the existence ofx^∗ ∈∂f(x)verifyinghx^∗, y−xi>0impliesf(x)< f(y).

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of26

Proof. Letx, y ∈Xsuch thathx^∗, y−xi>0for somex^∗ ∈∂f(x), then there existsε >0such that

hx^∗, y⁰−xi>0, ∀y⁰ ∈B(y, ε).

By the pseudoconvexity off, we havef(y⁰)≥f(x).

In particular,f(y)≥f(x). If we suppose by contradiction thatf(y) = f(x), then ymust be a global minimum. On the other hand, since f^†(x, y −x) > 0 then, there exist two sequencesx_n→x, t_n→0⁺such that

t_n⁻¹

f(x_n+t_n(y−x_n)−f(x_n))

>0.

By the quasiconvexity of the functionf (see for instance the proof of Proposi- tion 2.2 in [8]), we getf(y)> f(x_n)which is impossible.

We use this proposition to prove a known result for the Clarke-Rockafellar subdifferential for bigger subdifferentials

Theorem 3.7. Let X be a ∂-reliable space and f : X 7→ R∪ {+∞} a l.s.c.

function such that∂f ⊂∂^†f. And consider the following assertions i) f is pseudoconvex.

ii) ∂f is pseudomonotone.

Then, i) implies ii). And ii) implies i) iff is radially continuous.

Proof. The implication ii)=⇒i) is in [23]. For i)=⇒ii), suppose by contradic- tion that there existx, y ∈X, such that there existx^∗ ∈∂f(x)andy^∗ ∈ ∂f(y) verifying

hx^∗, y−xi>0 and hy^∗, y−xi ≤0.

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of26

Then, from Proposition3.5and the pseudoconvexity off we have f(x)< f(y) and f(y)≤f(x).

A contradiction.

Now, we state a similar result to Proposition3.2for pseudoconvex functions.

Proposition 3.8. Let X be a Banach∂-reliable space with ∂ ⊂ ∂^†, f a l.s.c.

and pseudoconvex function fromXtoR∪ {+∞}. Then we have [∂f(x)]^◦◦⊂N(S_f(f(x));x).

Proof. Letx^∗ ∈∂f(x)and suppose by contradiction thatx^∗ 6∈N(S_f(f(x));x).

Then, there existsy∈S_f(f(x))such thathx^∗, y−xi>0for somex^∗ ∈∂f(x).

It follows then by Proposition3.6thatf(y)> f(x), which is impossible.

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of26

4. Optimality Conditions and Variational Inequalities

4.1. Quasiconvex Programming

We recall the Minty variational inequality (we use the terminology of Gian- nessi [9]) that we shall use for our subdifferential. It will be exploited to give some conditions of optimality in nonlinear programming and necessary and sufficient conditions for optimality in quasiconvex programming.

LetΓbe a multifunction fromXtoX^∗,S ⊂Xandx¯∈S.

A pointx¯is a Minty equilibrium ofΓif the following variational inequality holds

(D) ∀x∈S, hγ(x), x−xi ≥¯ 0, ∀γ(x)∈Γ(x).

Suppose that f is a l.s.c. function from X to R ∪ {+∞} and consider the following minimisation problem

(4.1) minimizef(x), subject tox∈C.

Then we have

Proposition 4.1. Let X be a Banach∂-reliable space. Ifx¯is a Minty equilib- rium point of∂f, then we have

i) IfS =X, thenx¯is a global minimum off.

ii) If S = N, whereN is a convex open neighborhood ofx¯thenx¯is a local minimum off.

(17)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of26

Proof. It is enough to prove (ii). Suppose by contradiction thatx¯is not a solution of the program (4.1), then there existsx ∈ S such thatf(x) < f(¯x). By Lemma 2.2, there exists c ∈ [x,x[¯ and two sequencesc_n →_f c, c^∗_n ∈ ∂f(c_n) with

hc^∗_n, d−cni>0, for anyd=c+t(¯x−x)wheret >0.

SinceS is a convex open neighborhood ofx¯then[x,x]¯ ⊂ S. Furthermore, fornlarge enoughc_n∈S.

In the particular case whered = ¯x, we have:

hc^∗_n,x¯−c_ni>0.

A contradiction with the variational inequality (D), thus x¯is a local minimum off.

This proposition extends Theorem 2.2 of [18] for nondifferentiable optimization problems.

If in the problem (4.1), the function f to be minimized is l.s.c. and quasiconvex, then we have

Theorem 4.2. LetX be a Banach∂-reliable, andf be a l.s.c. and quasiconvex function such that ∂f ⊂ ∂^†f, and x¯ ∈ S. If S = N, where N is an open and convex neighborhood of x¯ or S = X, then the following assertions are equivalent

i) x¯is an optimal solution of (4.1).

ii) x¯is a Minty equilibrium point of∂f.

(18)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of26

Proof. ii)=⇒i) is obtained from Proposition4.1. Let us show that

i)=⇒ii). Assume thatx¯is a strict minimum of (4.1), then for allx ∈ S such thatx6= ¯xwe havef(x)> f(¯x).

According to Lemma2.2, there existc ∈ [¯x, x[, c_n →_f candc^∗_n ∈ ∂f(c_n) such that

hc^∗_n, d−c_ni>0, for alld=c+t(x−x)¯ wheret >0.

Whend=x,we obtain that

hc^∗_n, x−c_ni>0.

f being quasiconvex, by Theorem 2.1, ∂f is quasimonotone. It follows then that

for allx^∗ ∈∂f(x), hx^∗, x−xi ≥¯ 0.

Hence,∂f satisfies the variational inequality (D).

Suppose that we are in the case wherex¯is not a strict minimum of (4.1) and let us consider the functionfx¯defined by

f_x_¯(x) = max{f(x), f(¯x)}, and definehby

(4.2) h(x) =

( f_x_¯(x) forx6= ¯x ν forx= ¯x

(19)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of26

whereν < f(¯x). We see easily that his l.s.c. and quasiconvex and thatx¯is a strict local minimum ofh. Then, we have

∀x6= ¯x hx^∗, x−xi ≥¯ 0, ∀x^∗ ∈∂h(x).

From(P3), we get∂f(x) = ∂h(x).

In the case when0is in the interior of∂f(¯x), i.e. 0 ∈ int(∂f(¯x)), we have the more precise result

Proposition 4.3. LetX be a∂-reliable space andf :X 7→R∪ {+∞}a l.s.c.

and quasiconvex function. If0∈int(∂f(¯x))thenx¯is a Minty equilibrium point of∂f. Moreoverx¯is a global minimum off.

Proof. Assume that0∈int(∂f(x))then

there existsε >0such thatB_X^∗(0, ε)⊂∂f(x), where

B_X^∗(0, ε) ={x^∗ ∈X^∗ : kx^∗k< ε}.

Letd∈Xsuch thatd6= 0and consider the linear mapping`_ddefined by

`_d(x^∗) =hx^∗, di, forx^∗ ∈X^∗. By the open mapping Theorem [4] one has

hB_X^∗(0, ε), di ⊂ h∂f(x), di.

Sincef is quasiconvex, then∂f is quasimonotone.

(20)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of26

We already know by Definition 2.1 of [12] that the multifunction∂f_x,d defined by

∂f_x,d(λ) =h∂f(x+λd), di, is quasimonotone, and we can see easily that

hλd, ∂f(x+λd)i ⊂R⁺, for allλ∈Randd∈X\ {0}, thus (D) holds for∂f.

4.2. Pseudoconvex Programming

For the pseudoconvex function f, we shall get necessary and sufficient conditions for a pointx¯to be a global extremum off over a convex setC.

First consider the problem (4.1), withf is pseudoconvex, l.s.c. and radially continuous, then we have

Theorem 4.4. LetXbe a Banach space∂-reliable, andfa pseudoconvex l.s.c.

such that ∂f ⊂ ∂^†f, and let x¯∈ C. Then the following assertions are equiva- lent

i) x¯is an optimal solution of (4.1).

ii) (D) holds.

Proof. Suppose thatx¯is a solution of (4.1), then by Proposition3.6, if f(¯x)≤ f(x), then we must have

∀x^∗ ∈∂f(x), hx^∗,x¯−xi ≤0.

(21)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of26

This means that the variational inequality(D)holds.

Converesly, letx∈Csuch thatx6= ¯xthen for somey∈(¯x, x), we have

∀y^∗ ∈∂f(y), hy^∗,x¯−yi ≤0.

It follows that

∀y^∗ ∈∂f(y), hy^∗, x−yi ≤0.

Since∂f(y)is nonempty and from the pseudoconvexity off we have f(y)≤f(x), ∀y∈(¯x, x).

But sincef is s.c.i., thenf(¯x)≤f(x).

We now proceed to the maximisation problem

(4.3) maximizef(x), subject tox∈C.

Forz∈C, we denote by

C_z ={x∈C; f(x) =f(z)}.

Then we have

Theorem 4.5. Let X be a ∂-reliable space and f a pseudoconvex, l.s.c. and radially continuous such that for anyxinC,∂f(x)is nonempty and ∂f(x)⊂

∂^†f(x). Letx¯∈Csuch that

−∞ ≤inf

C f < f(¯x).

Thenx¯is a maximum off onC if and only if

for allx∈C_x_¯, ∂f(x)⊂N(C, x).

(22)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of26

Proof. Suppose that

f(y)≤f(¯x); ∀y∈C.

By Proposition3.6we have:

for allx∈C¯x, ∂f(x)⊂N(C, x).

Conversely, by contradiction assume that there existsz¯∈C such that f(¯z)> f(¯x).

Since by hypothesis, we can find somez ∈Cwithf(z)< f(¯x).

By the radial continuity off, there exists somex0 ∈(z,z)¯ such that f(x₀) = f(¯x).

It follows then that

for allx^∗₀ ∈∂f(x₀), hx^∗₀, z−x₀i= 0.

Sincef is pseudoconvex then,f(x₀)≤f(z), a contradiction.

(23)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of26

References

[1] D. AUSSEL, Théorème de la Valeur Moyenne et Convexité Generalisée en Analyse Non Régulière, Université Blaise Pascal, PhD. Thesis, (1994).

[2] D. AUSSEL, Subdifferential properties of quasiconvex and pseudoconvex functions: A unified approach, J. Optim. Th. Appl., 97 (1998), 29–45.

[3] D. AUSSEL, J.-N. CORVELLEC ANDM. LASSONDE, Subdifferential characterization of quasiconvexity and convexity, J. Conv. Analysis, 1 (1994), 195–201.

[4] H. BREZIS, Analyse Fonctionnelle Théorie et Applications, Masson (1980).

[5] J.-P. CROUZEIX, About differentiability of order one of quasiconvex functions onRⁿ, J. Optim. Theory Appl., 36 (1982), 367–385.

[6] J.-P. CROUZEIX, Continuity and differentiability properties of quasiconvex functions on Rⁿ, Generalized Concavity in Optimization and Eco- nomics, Academic Press, New York, 109–130, (1981).

[7] A.D. IOFEE, Proximal analysis and approximate subdifferentials, J. Lon- don Math. Soc., 41 (1990), 261–268.

[8] A. DANIILIDISAND N. HADJISAVVAS, On the subdifferentials of qua- siconvex and pseudoconvex functions and cyclic monotonicity, J. Math.

Anal. Appl., 237 (1999), 30–42.

(24)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page24of26

[9] F. GIANNESSI, On Minty variational principle, in: New Trends in Math- ematical Programming, F.Giannessi, S. Komlósi and T. Rapcsák (Eds.), Kluwer Scientific Publishers, Dordrecht, 93–99, (1998).

[10] N. HADJISAVVAS AND S. SCHAIBLE, Quasimonotone variational in- equalities in Banach spaces, J. Optim. Th. Appl., 90(1) (1996), 95–111.

[11] A. HASSOUNI, Sous-différentiels des fonctions quasiconvexes, Thèse de troisième cycle, Université Paul sabatier, pp. 68, (1983).

[12] A. HASSOUNI, Quasimonotone multifunctions; Applications to optimal- ity conditions in quasiconvex programming, Numer. Funct. Anal. Optim., 13(3-4) (1992), 267–275.

[13] A. HASSOUNI AND A. JADDAR, Quasiconvex functions and applica- tions to optimality conditions in nonlinear programming, Appl. Math. Lett., 14 (2001), 241–244.

[14] J.P. HIRIARTY–URRUTY, Tangent cones generalized gradients and math- ematical programming in Banach spaces, Math. Oper. Res., 4 (1979).

[15] S. KOMLÓSI, Quasiconvex first-order approximations and Kuhn-Tucker optimality conditions, Electron. J. Oper. Res., 65 (1993), 327–335.

[16] S. KOMLÓSI, Monotonicity and quasimonotonicity in nonsmooth analy- sis, Recent advances in Nonsmooth Optimization, World Sci. Publishing, River Edge, NJ, (1995), 193–214.

[17] S. KOMLÓSI, Generalized monotonicity and generalized convexity, J.

Optim. Appl., 84(2) (1995), 361–376.

(25)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page25of26

[18] S. KOMLÓSI, On the Stampacchia and Minty variational inequality, in:

Generalized Convexity and Optimization for Economics and Financial De- cisions, G. Giorgi and F.Rossi (Eds.), Pitarora editrice, Bogota, 231–260, (1999).

[19] I.V. KONNOV AND J.C. YAO, On the generalized vector variational in- equality problem, J. Math. Anal. Appl., 206 (1997), 42–58.

[20] D.T. LUC, Characterizations of quasiconvex functions, Bull. Austral.

Math. Soc., 48 (1993), 393–405.

[21] O.L. MANGAZARIAN, Pseudoconvex functions, SIAM J. Control, 3 (1965), 281–290.

[22] J.-P. PENOT, Generalized convex functions in the light of non smooth analysis, Lecture notes in Economics and Math. Systems, 429 (1995), Springer Verlag, 269–291.

[23] J.-P. PENOT, Are generalized derivatives useful for generalized convex functions? , in Generalized Convexity, Generalized Monotonicity: Recent results, Crouzeix (Eds.), 3–59, (1998).

[24] J.-P. PENOT AND P.H. SACH, Generalized monotonicity of subdifferen- tials and generalized convexity, J. Optim. Th. Appl., 92 (1997).

[25] J.-P. PENOT ANDP.H. QUANG, Generalized convexity of functions and generalized monotonicity of set-valued maps, J. Optim. Th. Appl., 92 (1997), 343–365.

(26)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page26of26

[26] R.T. ROCKAFELLAR, Generalized directional derivatives and subgradi- ents of nonconvex functions, Canad. J. Math., 32 (1980), 257–280.

[27] D. ZAGRODNY, Approximate mean value theorem for upper derivatives, Nonlinear Anal., 12 (1988), 1413–1428.