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volume 4, issue 1, article 5, 2003.

Received 14 April, 2002;

accepted 03 August, 2002.

Communicated by:A.M. Rubinov

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Journal of Inequalities in Pure and Applied Mathematics

NEW CONCEPTS OF WELL-POSEDNESS FOR OPTIMIZATION PROBLEMS WITH VARIATIONAL INEQUALITY CONSTRAINT

IMMA DEL PRETE1, M. BEATRICE LIGNOLA1 AND JACQUELINE MORGAN2

1Università degli Studi di Napoli “Federico II”

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”

Via Claudio, 21 - 80125 Napoli.

EMail:delprete@unina.it EMail:lignola@unina.it

2Università degli Studi di Napoli “Federico II”

Dipartimento di Matematica e Statistica Compl. Universitario Monte S. Angelo, Via Cintia - 80126 Napoli.

EMail:morgan@unina.it

c

2000Victoria University ISSN (electronic): 1443-5756 035-02

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New Concepts of Well-Posedness for Optimization Problems with Variational Inequality Constraint

Imma Del Prete, M. Beatrice Lignola and

Jacqueline Morgan

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Abstract

In this note we present a new concept of well-posedness for Optimization Prob- lems with constraints described by parametric Variational Inequalities or para- metric Minimum Problems. We investigate some classes of operators and func- tions that ensure this type of well-posedness.

2000 Mathematics Subject Classification:49J40, 49J53, 65K10.

Key words: Variational Inequalities, Minimum Problems, Set-Valued Functions, Well- Posedness, Monotonicity, Hemicontinuity.

Contents

1 Introduction. . . 3

2 Definitions and Background. . . 5

3 Parametricallyα−Well-Posed Variational Inequalities. . . 12

4 Parametricallyα−Well-Posed Minimum Problems . . . 27

5 α−Well-Posedness for OPVIC . . . 32 References

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New Concepts of Well-Posedness for Optimization Problems with Variational Inequality Constraint

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1. Introduction

Let E be a reflexive Banach space with dualE, A be an operator from E to E andK ⊆ E be a nonempty, closed, convex set. The Variational Inequality (V I), defined by the pair(A, K), consists of finding a pointu0 such that:

u0 ∈K and hAu0, u0−vi ≤0 ∀v ∈K.

This problem, introduced by G. Stampacchia in [22], has been recently investi- gated by many authors including [2], [4], [8], [9] and [15].

If(X, τ)is a topological space, one can consider the parametric Variational Inequality(V I)(x), defined by the pair(A(x,·), H(x)),where, for all x ∈ X, A(x,·)is an operator fromE toEandHis a set-valued function fromXtoE with nonempty and convex values.

The interest in this study is twofold: one is to study the behavior of pertur- bations of(V I), another is to consider the parameterxas a decision variable in a multilevel optimization problem. More precisely, the solution set to(V I)(x) can be seen as the constraint set T(x)of the following Optimization Problem with Variational Inequality Constraints:

(OPVIC) inf

x∈X inf

u∈T(x)f(x, u), wheref :X×E →R∪ {+∞}.

The problemsOPVIC(often termed Mathematical Programming with Equi- librium Constraints MPEC) have been investigated by many authors (see for example [13], [14], [17], [19] and [21]) since they describe many economic or engineering problems (see for example [18]) such as:

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• The price setting problem

• Price setting of telecommunication networks

• Yield management in airline industry

• Traffic management through link tolls.

Assuming that(V I)(x)has a unique solution, a well-posedness concept for OPVIC, inspired from numerical methods, has been considered in [13]. How- ever, in many applications, the problems(V I)(x)do not always have a unique solution.

So, in this paper, motivated from a numerical method for Variational Inequal- ities (M. Fukushima [7]), we introduce and study, for α ≥ 0,the concepts of α−well-posedness andα−well-posedness in the generalized sense for a family of Variational Inequalities(VI) = {(V I)(x), x∈X}and forOPVIC. The par- ticular case of variational inequalities arising from minimum problems is also considered.

The paper is organized as follows. In Section2we review some basic notions for variational inequalities and present some new results on α−well-posedness for unparametric variational inequalities. Section 3 is devoted to introducing and investigating the concept of α−well-posedness for parametric variational inequalities and Section4to parametric minimum problems. Finally, some new concepts of well-posedness forOPVICis presented and investigated in Section 5.

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New Concepts of Well-Posedness for Optimization Problems with Variational Inequality Constraint

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2. Definitions and Background

In this section, some notions of well-posedness for variational inequalities(V I) introduced in [13] and in [15] and their connections with optimization problems are presented, together with equivalent characterizations.

LetE be a reflexive Banach space with dualE, σbe a convergence on E, andKbe a nonempty, closed and convex subset ofE.

Definition 2.1. [5, 23]. Leth : K → R∪ {+∞}.The minimization problem (2.1):

(2.1) min

v∈Kh(v)

is Tikhonov well-posed (resp. well-posed in the generalized sense) with respect toσif there exists a unique solutionu0 to (2.1) and every minimizing sequence σ−converges to u0 (resp. if (2.1) has at least a solution and every minimizing sequence has a subsequenceσ−converging to a minimum point).

For an operatorA fromE toE, we consider the following Variational In- equality(V I)defined by the pair (A, K):

findu0 ∈K such that hAu0, u0−vi ≤0 ∀v ∈K.

Definition 2.2. [13, 15] Letα≥ 0. A sequence(un)n isα−approximating for (V I)if:

i) un∈K ∀n ∈N;

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ii) there exists a sequencen)n,εn>0, decreasing to 0 such that hAun, un−vi − α

2 kun−vk2 ≤εn ∀v ∈K ∀n ∈N.

A variational inequality (V I)is termed α−well-posed with respect to σ, if it has a unique solutionu0 and everyα−approximating sequence(un)n σ− converges to u0. If σ is the strong convergence s (resp. the weak convergencew) on E, (V I)will be termed strongly α−well-posed (resp.

weakly α−well-posed).

The above concept originated from the notion of Tikhonov well-posedness for the following minimization problem (2.2):

(2.2) min

u∈Kgα(u), where

gα(u) = sup

v∈K

hAu, u−vi − α

2 ku−vk2 . Indeed, the following result holds:

Proposition 2.1. Letα ≥0.The variational inequality problem(VI)isα−well- posed if and only if the minimization problem (2.2) is Tikhonov well-posed.

Proof. If(V I)isα−well-posed there exists a unique solutionu0for(VI),that is:

u0 ∈K and g0(u0) = sup

v∈K

hAu0, u0−vi ≤0

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and, consequently, gα(u0) ≤ g0(u0) ≤ 0. Since gα(u) ≥ 0for everyu ∈ K, gα(u0) = 0and u0 is a minimum point forgα. In order to prove that (2.2) has a unique solution, consideru0 ∈ K such that gα(u0) = gα(u0) = 0. For every v ∈K consider the pointw =λu0+ (1−λ)v,λ∈[0,1], which belongs toK.

Sincegα(u0) = 0one has:

hAu0, u0−wi −α

2 ku0−wk2 = (1−λ)hAu0, u0−vi −α

2(1−λ)2ku0 −vk2 ≤0 which implies:

hAu0, u0 −vi −α

2(1−λ)ku0−vk2 ≤0 ∀λ ∈[0,1].

So, whenλconverges to 1, one gets:

hAu0, u0−vi ≤0 ∀v ∈K.

Then alsou0 solves(V I)and it must coincide withu0.

As the family of minimizing sequences for (2.2) coincides with the family of α−approximating sequence for(V I),the first part is proved.

Now, assume that (2.2) is well-posed anduαis the unique solution for (2.2), that isuα ∈K andgα(uα) = 0.

With the same arguments used in the first part of this proof it can be proved thatuαsolves also the variational inequality(V I)(this has been already proved in [7] with other arguments). In order to prove thatuαis the unique solution to (V I), letu0 be another solution to(V I). Sincegα(u0) ≤ g0(u0) = 0, the point u0 should be a solution to (2.2), thus it has to coincide withuα.

Then the result follows as in the first part.

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The gap functiongα, which provides an optimization problem formulation for (V I), is, forα = 0, the gap function introduced by Auslender in [1], and, for α > 0, the merit function introduced by Fukushima in [7] for numerical purposes.

As it is well known, when the setKis not bounded, the setT of the solutions to(V I)may be empty, even in finite dimensional spaces. This does not happen when the operatorAsatisfies some of the following well known properties.

Definition 2.3. The operatorAis said to be:

monotone onK ifhAu−Av, u−vi ≥0for everyuandv ∈K,

pseudomonotone on K if for every u and v ∈ K hAu, u− vi ≤ 0 ⇒ hAv, u−vi ≤0;

strongly monotone onK(with modulusβ)ifhAu−Av, u−vi ≥βku−vk2 for everyuandv ∈K;

hemicontinuous on K if it is continuous from every segment of K to E endowed with the weak topology.

It is well known (see for example [2]) that the variational inequality(V I)has a unique solution if the operator Ais strongly monotone and hemicontinuous, while there exists at least a solution for(V I)if the operatorAis pseudomono- tone and hemicontinuous and some coerciveness condition is satisfied (see for example [8]).

We recall some continuity properties for set-valued functions that will be used later on:

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Definition 2.4. A set-valued function F from a topological space (X, τ) to a convergence space(Y, σ)(see [11]) is:

sequentially σ−lower semicontinuous at x ∈ X if, for every sequence (xn)n τ−converging to x and every y ∈ F(x), there exists a sequence (yn)nσ−converging toysuch thatyn∈F(xn) ∀n ∈N;

sequentially σ−subcontinuous at x ∈ X if, for every sequence (xn)n τ−converging to x, every sequence (yn)n, yn ∈ F(xn) ∀n ∈ N, has a σ−convergent subsequence;

sequentiallyσ−closed atx∈Xif for every sequence(xn)nτ−converging tox, for every sequence(yn)n σ−converging toy, yn ∈ F(xn)∀n ∈ N, one hasy∈F(x).

We have chosen to deal with sequential continuity notions for set-valued functions since our well-posedness concepts are defined in a sequential way.

However, for brevity, from now on the term sequentially will be omitted.

Letε >0. The following approximate solutions set, introduced in [15], Tα,ε=n

u∈K :hAu, u−vi ≤ε+ α

2 ku−vk2 ∀v ∈Ko

forε >0 can be used to provide a characterization ofα−well-posedness in line with [13, Prop. 2.3 bis] and [5].

Proposition 2.2. Letα ≥0and assume that the operatorAis hemicontinuous and monotone on K and that (V I) has a unique solution. The variational inequality(V I)is stronglyα−well-posed if and only if

Tα,ε6=∅ ∀ε >0 and lim

ε→0 diam(Tα,ε) = 0.

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Proof. Assume that(V I)is stronglyα−well-posed and limε→0diamTα(ε)>0.

Then there exists a positive number β such that, for every sequence (εn)nde- creasing to 0, εn > 0, there exist two sequences (yn)n and (vn)n in K such that

yn∈ Tα,εn, vn∈ Tα,εn and kyn−vnk> βfornsufficiently large.

Since(V I)is stronglyα−well-posed, the sequences(yn)nand(vn)nmust con- verge to the unique solutionu0,so

limn kyn−vnk= 0 which gives a contradiction.

Conversely, let(yn)nbe anα−approximating sequence for(V I),that isyn∈ Tα,εn for a sequence(εn)n, εn >0, decreasing to 0. Beinglim

n diamTα,εn = 0, for every positive numberβthere exists a positive integermsuch thatkyn−ypk

< β ∀n≥mandp≥m.

Therefore(yn)nis a Cauchy sequence and has to converge to a pointu0∈K.

SinceAis monotone one has:

hAv, u0−vi= lim

n hAv, yn−vi

≤lim inf

n hAyn, yn−vi

≤lim

n

α

2 kyn−vk2 = α

2 ku0 −vk2 ∀v ∈K.

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SinceAis monotone and hemicontinuous, the following equivalence holds:

hAv, u0 −vi −α

2ku0−vk2 ≤0 ∀v ∈K

⇔ hAu0, u0−vi − α

2 ku0−vk2 ≤0 ∀v ∈K.

In fact, assume that

hAv, u0−vi −α

2ku0−vk2 ≤0 ∀v ∈K.

Ifv is a point ofK, for every numbert ∈ [0,1]the pointvt = tv+ (1−t)u0

belongs toK, so:

hAvt, u0−vti−α

2 ku0−vtk2 =thAvt, u0−vi−t2α

2 ku0−vk2 ≤0 ∀t∈[0,1].

So one has:

limt→0

hAvt, u0−vi − α

2tku0−vk2

≤0 and, in light of the hemicontinuity ofA:

hAu0, u0−vi − α

2 ku0−vk2 ≤ hAu0, u0−vi ≤0 ∀v ∈K.

The converse is an easy consequence of the monotonicity ofA.

Sogα(u0) = 0and, arguing as in Proposition 2.1, it can be proved that u0 coincides with the unique solution to(V I). This completes the proof.

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3. Parametrically α−Well-Posed Variational Inequalities

In what follows we shall consider a topological space(X, τ), a convergenceσ onE and, for everyx∈X, a parametric variational inequality onE,(V I)(x), defined by the pair(A(x,·), H(x)), whereAis an operator fromX×EtoE andH is a set-valued function fromXtoE which is assumed to be nonempty, convex and closed-valued. In many situations H(x) is described by a finite number of inequalities: H(x) = {u∈E :gi(x, u)≤0, ∀i= 1, . . . , n}, where giis a real-valued function, fori= 1, . . . , n, satisfying suitable assumptions.

Throughout this section we will consider the following family of variational inequalities:

(VI) = {(V I)(x), x∈X}.

Letα ≥ 0andε > 0. In the sequel, we shall denote by T (resp. Tα,ε) the map which associates to every x ∈ X the solution set (resp. the approximate solution set) to(V I)(x) :

T(x) = {u∈H(x) :hA(x, u), u−vi ≤0 ∀ v ∈ H(x)}

(resp.Tα,ε(x) = n

u∈H(x) :hA(x, u), u−vi ≤ε+α

2ku−vk2 ∀v ∈H(x)o ).

Now, we introduce the notion of parametricα−well-posedness for the fam- ily(VI).

Definition 3.1. Letx∈Xand(xn)nbe a sequence converging tox.A sequence (un)nis said to beα−approximating for(V I)(x)(with respect to(xn)n) if:

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i) un∈H(xn) ∀n ∈N,

ii) there exists a sequencen)n,εn>0, decreasing to 0 such that hA(xn, un), un−vi −α

2 kun−vk2 ≤εn ∀v ∈H(xn) ∀n ∈N. Definition 3.2. The family of variational inequalities(VI)is termed paramet- ricallyα−well-posed with respect toσif:

for everyx∈X,(V I)(x)has a unique solutionux;

for every sequence (xn)n converging to x, every α− approximating se- quence(un)nfor(V I)(x)(with respect to(xn)n)σ−converges toux. If σ is the strong convergence s (resp. the weak convergence w) on E, (VI) will be termed parametrically strongly α−well-posed (resp. parametrically weakly α−well-posed).

Observe that for α = 0 the above definition amounts to Definition 2.3 in [13].

Definition 3.3. The family of variational inequalities(VI)is termed paramet- rically α−well-posed in the generalized sense with respect to σ if, for every x ∈ X,(V I)(x)has at least a solution and for every sequence (xn)nconverg- ing to x,everyα−approximating sequence(un)nfor(V I)(x)(with respect to (xn)n) has a subsequenceσ−convergent to a solution to(V I)(x).

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For a parametric variational inequality it is natural to consider the following parametric gap functiongα(x, u):

gα(x, u) = sup

v∈H(x)

hA(x, u), u−vi − α

2 ku−vk2

and with the same arguments as in Proposition2.1one can prove the following two propositions:

Proposition 3.1. Let α ≥ 0 and x ∈ X. A point ux solves the variational inequality(V I)(x)if and only if :

ux ∈H(x)andgα(x, ux) = inf

u∈H(x)gα(x, u) = 0, that is:

hA(x, u), u−vi − α

2 ku−vk2 ≤0 ∀v ∈H(x).

Proposition 3.2. The family of variational inequality (VI) is parametrically α−well-posed (resp. parametrically-α−well-posed in the generalized sense) with respect toσif and only if, for everyx∈X, the minimization problem

(3.1) min

u∈H(x)gα(x, u)

is parametrically Tikhonov well-posed (resp. parametrically Tikhonov well- posed in the generalized sense) with respect to σ, that is: gα is bounded from below, (3.1) has a unique solution (resp. has at least a solution) ux and for every sequence(xn)nconverging tox,every sequence(un)nsuch that

inf

u∈H(x)gα(x, u)≥lim

n infgα(xn, un)

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σ−converges (resp. has a subsequenceσ−convergent) toux(see Definition2.3 in [13]).

The connection between parametricα−well-posedness and the convergence to 0 of the diameters ofTα,ε(x)is given by the following result.

Proposition 3.3. Let α ≥ 0. If the family of variational inequalities (V I) is strongly parametricallyα−well-posed, then, for every x∈X , every sequence (xn)n converging to x and every sequencen)n of positive real numbers de- creasing to 0, one has:

Tα,ε(x)6=∅ ∀ε >0 and lim

n diam(Tα,εn(xn)) = 0.

Proof. In light of the assumption, the set Tα,ε(x) is nonempty since {ux} = T(x)⊆ Tα,ε(x). Assume thatlim

n diam(Tα,εn(xn) >0.Then there existη > 0 and two sequences (un)n and(yn)n such thatun ∈ Tα,εn(xn), yn ∈ Tα,εn(xn) and kyn−unk > η, for n sufficiently large. But, being (un)n and (yn)n se- quencesα−approximating for(V I)(x)(with respect to(xn)n), they must con- verge toux,and this gives a contradiction.

In order to achieve a similar result for generalizedα−well-posedness, one can consider the non compactness measureµ, introduced by Kuratowski in [11]:

if (S, d) is a metric space andB is a bounded subset of S, µ(B)is defined as the infimum of ε > 0 such that B can be covered by a finite number of open sets having diameter less than ε. The following proposition, whose proof is in line with previous results concerning generalized well-posedness for mini- mum problems (see [5]), gives the link between the noncompactness measure

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of Tα,εn(x)and the generalized α−well-posedness, when the set-valued func- tionHis constant:

Proposition 3.4. Letα ≥0.Assume that for everyu∈ E the operatorA(·, u) is continuous fromXto(E, w)and the set-valued functionHis constant, that isH(x) =K, whereK is a nonempty, closed convex subset ofE. If the family of variational inequalities(VI)is parametrically stronglyα−well-posed in the generalized sense, then, for every x ∈ X, every sequence(xn)n converging to xand every sequencen)nof positive real numbers decreasing to 0, one has:

Tα,ε(x)6=∅ ∀ε >0 and lim

n µ(Tα,εn(xn)) = 0.

Proof. Letn)nbe a sequence of positive real numbers, let x ∈ X and(xn)n be a sequence converging tox.

We start by proving thatlim

n h(Tα,εn(xn), T(x)) = 0, whereh(Tα,εn(xn), T(x))

= hnis the Hausdorff distance [11] betweenTα,εn(xn)and the set of solutions to(V I)(x),that is:

hn= max (

sup

u∈Tα,εn(xn)

d(u, T(x)), sup

v∈T(x)

d(Tα,εn(xn), v) )

.

By the assumptions, every u ∈ T(x) belongs to Tα,εn(xn), for n sufficiently large.

Indeedu ∈ T(x)if and only ifhA(x, u), u−vi ≤ 0 ∀ v ∈ K and, conse- quently:

hA(x, u), u−vi − α

2 ku−vk2 ≤0 ∀v ∈K.

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If

v 6=u, hA(x, u), u−vi − α

2 ku−vk2 <0 = lim

n εn

and in light of continuity ofA(·, u)one gets hA(xn, u), u−vi − α

2 ku−vk2 < εn fornsufficiently large.

Ifv =u, the result is obvious since hA(xn, u), u−vi −α

2 ku−vk2 = 0 < εnfor everyn∈N. So, iflim sup

n

h(Tα,εn(xn), T(x))> c >0, there exists a sequence(un)n:

un∈Tα,εn(xn)andd(un, T(x))> cfornsufficiently large.

Since (un)n isα−approximating, there is a subsequence (unk)k converging to ux ∈T(x)and one gets:

0 =d(ux, T(x))≥lim sup

k

d(unk, T(x))> c,

which gives a contradiction.

In order to complete the proof, it takes only to observe that Tα,εn(xn) ⊆ B(T(x), hn)(the ball of radius hnaroundT(x)) andµ(T(x)) = 0, so the fol- lowing inequality holds (see, for example [5]):

µ(Tα,εn(xn))≤2hn+µ(T(x)) = 2hn.

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The next lemma is in the spirit of the Minty’s Lemma and will be used to characterize α−well-posedness for parametric variational inequalities. The proof is omitted since it is similar to the proof given in Proposition2.2 for un- parametric variational inequalities.

Lemma 3.5. Letα ≥ 0.If, for everyx ∈ X, the operator A(x,·)is hemicon- tinuous and monotone onH(x),then the following conditions are equivalent:

i) u0 ∈H(x)andhA(x, u0), u0−vi −α2 ku0−vk2 ≤0 for everyv ∈H(x), ii) u0 ∈H(x)andhA(x, v), u0−vi −α2 ku0−vk2 ≤0 for everyv ∈H(x).

The next proposition proves that in finite dimensional spaces the parametric α−well-posedness is equivalent to the uniqueness of solutions to(V I)(x), for everyα≥0.

Proposition 3.6. Letα≥0andE =Rk. If the following conditions hold:

i) the set-valued functionH is lower semicontinuous, closed and subcontin- uous;

ii) for everyx∈X, A(x,·)is monotone and hemicontinuous;

iii) for everyu∈Rk, A(·, u)is continuous onX;

iv) Ais uniformly bounded onX×Rk,that is there existsk > 0such that for every converging sequence(xn, un)n one haskA(xn, un)k ≤ k for every n∈N;

then(VI)is parametricallyα−well-posed if and only if, for everyx ∈ X, (V I)(x)has a unique solutionux.

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Proof. For x ∈ X, let(xn)n be a sequence converging toxand (un)n be an α−approximating sequence (with respect to(xn)n), that is:

un∈H(xn) and hA(xn, un), un−vi ≤εn+ α

2 kun−vk2 ∀v ∈H(xn), where(εn)nn>0, is a sequence decreasing to 0.

SinceH is closed and subcontinuous there exists a subsequence (unk)k of (un)nconverging to a point uex ∈ H(x).Moreover, in light of the lower semi- continuity of H,for everyv ∈ H(x)there exists a sequence(vn)n converging tovsuch thatvn∈H(xn)for everyn∈N.

The monotonicity ofA(xnk,·)implies:

hA(xnk, v), unk −vi ≤ hA(xnk, unk), unk −vnki+hA(xnk, unk), vnk−vi

≤εnk+ α

2 kunk −vnkk2+kA(xnk, unk)k kvnk−vk for everyk ∈N.

SinceA(·, v)is continuous atxandAis uniformly bounded one has:

hA(x, v),uex−vi ≤ α

2 kuex−vk2 and applying the previous lemma:

hA(x,uex),uex−vi ≤ α

2 keux−vk2. But, from Proposition3.1, this inequality is equivalent to:

hA(x,eux),uex−vi ≤0 ∀v ∈H(x)

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that isuex solves(V I)(x).

Since(V I)(x)has a unique solution, the pointeuxmust coincide withuxand the whole sequence(un)nhas to converge toux.

A similar result could be obtained in infinite dimensional spaces if one mod- ifies the assumptions: in iii) A(·, u)should be continuous from X to (E, s), but in i) H should be assumed to be s−lower semicontinuous, w−closed and s−subcontinuous, which unfortunately lead to the strong compactness ofH(x) for everyx∈X.

Remark 3.1. If the set-valued functionHis constant, that isH(x) =K ∀x∈ X,the same result holds assuming that the setKis compact and convex,A(x,·) is monotone and hemicontinuous onK for everyx ∈X, andA(·, u)is contin- uous on X for every u ∈ K. Indeed, arguing as in Proposition3.6, for every v ∈K one has:

hA(xnk, v),ue−vi=hA(xnk, v),eu−unki+hA(xnk, v), unk −vi

≤ hA(xnk, v),ue−unki+hA(xnk, unk), unk−vi

≤ hA(xnk, v),ue−unki+εnk

2 kunk−vk2, and forkconverging to+∞the result follows.

Example 3.1. IfEis an infinite dimensional space, the previous result may fail to be true whenKis only weakly compact, that is: there are variational inequal- ities with a unique solution which are notα−well-posed. Indeed, the following example (already considered in [5]) holds: letE be a separable Hilbert space with an ortonormal basis(en)n, B be the unitary closed ball ofE. Consider the

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operator 5h(u),whereh(u) = P

n hu,eni

n2 and the variational inequality(V I) defined by: v ∈B andh5h(u), u−vi ≤0 ∀v ∈B.

It has as unique solutionu0 = 0, but(en)n is an approximating (and con- sequentlyα−approximating for every α >0)sequence that does not strongly converge to0.

The next result and the following remark, concerning α−well-posedness in the generalized sense, can be easily proved with the same arguments as in Proposition3.6and Remark3.1.

Proposition 3.7. LetE =Rkandα≥0.If the assumptions of Proposition3.6 hold, then the family (VI)is parametrically α−well-posed in the generalized sense.

Proof. Since under assumption i) the setH(x)is compact, (V I)(x)has at least a solution for every x ∈X (see for example [10] or [2]), so the result can be easily proved as in Proposition3.6.

The previous proposition says nothing else that, under conditions i) to iv), in finite dimensional spaces, the parametricα−well-posedness in the generalized sense is equivalent to the existence of solutions.

Remark 3.2. If the set-valued functionKis constant, that isH(x) =K ∀x∈ X, the same result holds assuming that the set K is compact and convex, for every x ∈ X A(x,·) is monotone and hemicontinuous on H, and, for every u∈K A(·, u)is continuous onX.

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The following propositions furnish classes of operators for which the corre- sponding variational inequalities are parametricallyα−well-posed or paramet- ricallyα−well-posed in the generalized sense.

Proposition 3.8. Assume that the following conditions are satisfied:

i) the operatorAis strongly monotone onEin the variableu, uniformly with respect tox, that is:

∃β >0 such that hA(x, u)−A(x, v), u−vi

≥βku−vk2 ∀x∈X, ∀u∈E, ∀v ∈E;

ii) for everyu∈E, A(·, u)is continuous from(X, τ)to(E, s);

iii) for everyx∈X, A(x,·)is hemicontinuous onH(x);

iv) Ais uniformly bounded onX×E;

v) the set-valued function H is w−closed, w−subcontinuous ands−lower semicontinuous.

Then (VI) is parametrically strongly α−well-posed for everyα such that 0≤α≤2β.

Proof. First of all, for every x ∈ X,the variational inequality (V I)(x) has a unique solutionux(see, for example, [10] or [2]).

To prove that, for0≤α≤2β, everyα−approximating sequence is strongly convergent, let x ∈ X, (xn)n be a sequence converging tox and (un)n be an α−approximating sequence for(VI)with respect to(xn)n.

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SinceH isw−closed andw−subcontinuous, the sequence(un)nhas a sub- sequence, still denoted by (un)n, which weakly converges to uex ∈ H(x). To prove that eux = ux,consider a pointv ∈ H(x)and a sequence(vn)n strongly converging tovsuch thatvn ∈H(xn)for everyn ∈N(such sequence exists in virtue of the lower semicontinuity ofH).One has, for everyn ∈N:

hA(xn, v), un−vi

≤ hA(xn, un), un−vi −βkun−vk2

=hA(xn, un), un−vni+hA(xn, un), vn−vi −βkun−vk2

≤εn

2 kun−vnk2−βkun−vk2+kA(xn, un)k kvn−vk. Since α2 ≤β,one gets:

hA(xn, v), un−vi

≤εn+β kvn−vk2+ 2kun−vk kvn−vk

+kA(xn, un)k kvn−vk and in light of assumptions ii) and iv):

hA(x, v),uex−vi ≤0.

The last inequality, for the arbitrarity ofv, implies, by Minty’s Lemma (see, for example, [2]), thatuexsolves(V I)(x), souex =ux.

To prove that the sequence (un)n strongly converges to ux, let(wn)n be a sequence strongly converging to ux, wn ∈ H(xn) ∀n ∈ N(such a sequence

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exists sinceHiss−lower semicontinuous). Observe that:

βkun−uxk2

≤ hA(xn, un)−A(xn, ux), un−uxi

=hA(xn, un), un−wni+hA(xn, un), wn−uxi − hA(xn, ux), un−uxi

≤εn

2 kun−wnk2+kA(xn, un)k kwn−uxk

− hA(xn, ux), un−uxi ∀n∈N.

Sincekwn−unk2 ≤(kwn−uxk+kun−uxk)2,one gets, for everyn∈N: 0≤

β− α 2

kun−uxk2

≤εn

2kux−wnk2+αkun−uxk kux−wnk +kA(xn, un)k kwn−uxk − hA(xn, ux), un−uxi and this implies thatlim

n kun−uxk= 0.So, we have proved that every weakly converging subsequence of(un)nis also strongly converging to the unique solu- tion for(V I)(x). Then the whole sequence(un)nstrongly converges toux. Remark 3.3. If the set-valued functionH is constant, that isH(x) = K ∀x∈ X,the same result can be established assuming that:

i) the operatorAis strongly monotone in the variableuonE(with modulus β), uniformly with respect tox;

ii) for everyu∈K, A(·, u)is continuous from(X, τ)to(E, s);

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iii) for everyx∈X, A(x,·)is hemicontinuous onH(x);

iv) the setKis convex, closed and bounded.

For what concerning parametricα−well-posedness in the generalized sense, we have the following result forα = 0 :

Proposition 3.9. Assume that the following conditions are satisfied:

i) for everyx∈X,A(x,·)is monotone onH(x);

ii) for everyu∈H, A(·, u)is continuous from(X, τ)to(E, s);

iii) for everyx∈X, A(x,·)is hemicontinuous onH(x);

iv) Ais uniformly bounded onX×E;

v) the set-valued function H is w−closed, w−subcontinuous ands−lower semicontinuous.

Then(VI)is parametrically weakly well-posed in the generalized sense.

Proof. First of all, for every x ∈ X,the variational inequality (V I)(x)has at least a solution (see, for example, [10] or [2]), since under our assumptions the setH(x)is compact with respect to the weak convergence.

Letx ∈ X,(xn)nbe a sequence converging tox,and(un)n be an approxi- mating sequence for(VI)with respect to(xn)n.

SinceH isw−closed andw−subcontinuous, the sequence(un)nhas a sub- sequence, still denoted by (un)n, which weakly converges to ux ∈ H(x). To prove that ux ∈ T(x),consider a point v ∈ H(x), a sequence (vn)n strongly

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converging tovsuch thatvn ∈H(xn)for everyn ∈N(such sequence exists in virtue of the lower semicontinuity ofH).Since:

hA(xn, v), un−vi ≤ hA(xn, un), un−vi

=hA(xn, un), un−vni+hA(xn, un), vn−vi

≤εn+hA(xn, un), vn−vi

≤εn+kA(xn, un)k kvn−vk ∀n ∈N and assumptions ii) and iv) hold, one gets:

hA(x, v), ux−vi ≤0 ∀v ∈H(x),

that, for the Minty’s Lemma, is equivalent to say thatuxsolves(V I)(x).

Remark 3.4. If the set-valued functionHis constant, that isH(x) = K,∀x∈ X,the same result can be established assuming that:

i) the operatorA(x,·)is hemicontinuous onH;

ii) the operatorA(x,·)is monotone;

iii) for everyu∈K, A(·, u)is continuous onX;

iv) the setKis convex, closed and bounded.

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4. Parametrically α−Well-Posed Minimum Problems

In this section we consider variational inequalities arising from parametric min- imum problems and we investigate, for α > 0, the links between parametric α−well-posedness of such problems and parametricα−well-posedness of the corresponding variational inequalities. The caseα= 0can be found in [13].

Lethbe a function fromX×EtoR∪{+∞}andHbe a set-valued function fromX toE, which is assumed to be nonempty, convex and closed-valued. If, for every x ∈ X, the function h(x,·)is Gâteaux differentiable, bounded from below and convex onH(x), the minimum problem:

((P) (x)) inf

u∈H(x)h(x, u)

is equivalent to the following variational inequality problem:

((V I)(x)) findu∈H(x)such that

∂h

∂u(x, u), u−v

≤0 ∀v ∈H(x),

where ∂h∂u is the derivative of the function hwith respect to the variableu (see [2]). Then, it is natural to introduce the notion of parametricα−well-posedness for a family of minimization problems P ={ (P) (x), x ∈ X}and compare it with the parametricα−well-posedness for the familyVI={(V I)(x), x∈ X}.

Definition 4.1. Letx∈ X,(xn)nbe a sequence converging tox; the sequence (un)nis termedα−minimizing for(P) (x)(with respect to(xn)n) if:

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i) un∈H(xn)∀n ∈N,

ii) there exists a sequencen)n,εn>0, decreasing to 0 such that:

h(xn, un)≤h(xn, v) + α

2 kun−vk2n ∀v ∈H(xn) and∀n ∈N. Definition 4.2. The family of minimum problems P is called parametrically α−well-posed, with respect toσ,if:

i) for everyx∈X,h(x,·)is bounded from below, ii) for everyx∈X,(P) (x)has a unique solutionux,

iii) for every sequence (xn)n converging to a point x, every α−minimizing sequence(un)nfor(P) (x)(with respect to(xn)n) σ−converges toux. Definition 4.3. The family of minimum problems P is called parametrically α−well-posed in the generalized sense, with respect toσ,if:

i) for everyx∈X,h(x,·)is bounded from below, ii) for everyx∈X,(P) (x)has at least a solutionux,

iii) for every sequence (xn)n converging to a point x, every α−minimizing sequence (un)n for (P) (x) (with respect to (xn)n) has a subsequence σ−convergent to a solution for(P) (x).

The following two propositions give, under suitable assumptions, the equiv- alence between parametric α−well-posedness for a minimization problem and the corresponding variational inequality.

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Proposition 4.1. Assume that, for all x ∈ X, the functionh(x,·) is bounded from below, convex and Gâteaux differentiable onH(x)and the family of prob- lems P is parametrically α−well-posed (resp. in the generalized sense) with respect toσ. Then the family of variational inequalities defined by

((V I)(x)) findu∈H(x)such that

∂h

∂u(x, u), u−v

≤0 ∀v ∈H(x), is parametricallyα−well-posed (resp. in the generalized sense) with respect to σ.

Proof. Under the above assumptions, for allx∈X, the problems(V I)(x)and (P) (x)have the same solutions. Consider a point x ∈ X, a sequence (xn)n converging to x and an α−approximating sequence (un)n for (V I)(x), with respect to(xn)n, that is:

un ∈H(xn) and ∂h

∂u(xn, un), un−v

− α

2 kun−vk2 ≤εn

∀v ∈H(xn) ∀n ∈N, where(εn)nn>0, decreases to 0. Sinceh(xn,·)is convex one has:

h(xn, un)−h(xn, v)≤ ∂h

∂u(xn, un), un−v

≤ α

2 kun−vk2n ∀v ∈H(xn) ∀n ∈N, that is(un)nisα−minimizing for(P) (x)(with respect to(xn)n) and the result then follows.

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