Nonlinear Analysis 63 (2005) e1655–e1664www.elsevier.com/locate/na

Duality theory and optimality conditions forGeneralized Complementarity Problems

S. Giuffrèa,∗, G. Idonea, A. MaugeribaD.I.M.E.T. Faculty of Engineering, University of Reggio Calabria, via Graziella, Loc. Feo di Vito,

89060 Reggio Calabria, ItalybDepartment of Mathematics and Computer Science, University of Catania, Viale A. Doria, 6,

95125 Catania, Italy

Abstract

In this paper Generalized Complementarity Problems are expressed in terms of suitable optimizationproblems and some necessary optimality conditions are given. The infinite dimensional Lagrangeanand Duality Theories play an important role in order to achieve the main results.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Generalized complementarity problem; Lagrangean function; Dual problem; Quasi-relative interior;Saddle point

1. Introduction

Let S be a nonempty subset of a real linear space X. Let Y be a partially ordered realnormed space with the ordering cone C. Let Z be the set of nonnegative measurable functionsand let

L : S → Z,

B : S → Z

be two operators. Let g : S → Y be a given constraint mapping and let us set

K = {v ∈ S : g(v) ∈ −C}. (1)

∗ Corresponding author.E-mail address: sofia.giuffre@unirc.it (S. Giuffrè).

0362-546X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2004.12.019

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Let us suppose that

L(v)�0 B(v)�0 ∀v ∈ S

and let us observe that the Generalized Complementarity Problem{B(u)L(u) = 0,

u ∈ K(2)

expresses many economic and physical equilibrium problems. In fact, starting from the clas-sical Signorini problem, it has been observed that the Obstacle problem, the Elastic–PlasticTorsion problem, the Traffic Equilibrium problem both in the discrete and continuous cases,the Spatial Price Equilibrium problem, the Financial Equilibrium problem and many others(see [5,9,8,15]) satisfy the Generalized Complementarity Problem (2). For example, thecontinuous traffic equilibrium problem fits very well with the above scheme assuming

X = L2div(�) = {u ∈ L2(�, R2) : div u ∈ L2(�)}, Y = L2(�), S = L2

div(�),

Bv = v, Lv =(

ci(x, v(x)) − ��

�xi

)i=1,2

.

Problem (2) becomes{(ci(x, u(x)) − ��(x)

�xi

)ui(x) = 0

u ∈ Ki = 1, 2, a.e. in �,

where � ∈ H 1(�) is a given function (potential) and K is given by

K = {u ∈ L2div(�) : ui(x)�0, ui(x)|�� = �i (x), div u + t (x) = 0},

with � a simply connected bounded domain in R2 with Lipschitz boundary ��.In this model, ui(x) i =1, 2 represent the traffic density through a neighbourhood of x in

the direction of the increasing axis xi . ui(x) has nonnegative fixed trace �i (x) on �� whichrepresents the entering flow. If we associate to each point x ∈ � a scalar field t (x) ∈ L2(�),which measures the density of the flow originating or terminating at x, the flow u(x) satisfiesthe conservation law: div u(x) + t (x) = 0 a.e. in �. The function ci(x, v(x)) represents thetravel cost density along the axis xi (i = 1, 2).

The equilibrium condition is the following one:

Definition 1. u(x) ∈ K is an equilibrium distribution flow if there exists a potential � ∈H 1(�) such that⎧⎪⎨

⎪⎩(

ci(x, u(x)) − ��(x)

�xi

)ui(x) = 0,

ci(x, u(x)) − ��(x)

�xi

�0,

i = 1, 2 a.e. in �.

The potential � measures the cost occurred when a user travels from the point x to theboundary �� using the cheapest possible path (see [4,6,11]).

S. Giuffrè et al. / Nonlinear Analysis 63 (2005) e1655–e1664 e1657

The same happens for the Elastic–Plastic Torsion Problem. In this case we have X =H 2(�) (or H 1(�) if we consider the weak formulation);

Y = L2(�); Bv = 1 −n∑

i=1

(�v

�xi

)2

Lv an elliptic operator. The equilibrium condition is;[1 −

n∑i=1

(�u

�xi

)2]Lu = 0

and

K ={

v ∈ H 10 (�) : v�0, 1 −

n∑i=1

(�v

�xi

)2

�0

}

(see [12,13]).The Evolutionary Financial Time Equilibrium has been considered very recently and also

it perfectly agrees with the above scheme. In this case a vector of sector assets, liabilitiesand instrument prices (x∗(t), y∗(t), r∗(t)) ∈ ∏m

i=1 Pi × L2([0, T ], Rn+), where

Pi =⎧⎨⎩(xi(t), yi(t)) ∈ L2([0, T ], R2n) :

n∑j=1

xij (t) = si(t),

n∑j=1

yij (t) = si(t), xij (t), yij (t)�0 a.e. in [0, T ]⎫⎬⎭ ,

with si(t) the total financial volume held by sector i at the time t is an equilibrium of theevolutionary financial model if and only if it satisfies the system of equalities:

x∗ij (t)[2[Qi

11(t)]Tj x∗

i (t) + 2[Qi21(t)]T

j y∗i (t) − r∗

j (t) − �(1)i (t)] = 0,

y∗ij (t)[2[Qi

12(t)]Tj x∗

i (t) + 2[Qi22(t)]T

j y∗i (t) + r∗

j (t) − �(2)i (t)] = 0,

m∑i=1

(x∗ij (t) − y∗

ij (t))r∗j (t) = 0

with all the functions x∗ij (t), y∗

ij (t), 2[Qi11(t)]T

j x∗i (t) + 2[Qi

21(t)]Tj y∗

i (t) − r∗j (t) − �(1)

i (t),

2[Qi12(t)]T

j x∗i (t) + 2[Qi

22(t)]Tj y∗

i (t) + r∗j (t) − �(2)

i (t),∑m

i=1(x∗ij (t) − y∗

ij (t)) nonnegative.The meaning of this definition is the following one:To each financial volume si(t) invested by sector i there are associated two functions

�(1)i (t) and �(2)

i (t) related to the assets and to the liabilities which represent the “EquilibriumUtilities” per unit of the sector i, respectively; 2[Qi

11(t)]Tj x∗

i (t)+2[Qi21(t)]j y∗

i (t)−r∗j (t) is

the personal utility of the investor in the instrument j as an asset. Then if this personal utilityequals the equilibrium utility �(1)

i (t), it results x∗ij (t)�0, whereas if the personal utility is

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greater than the equilibrium utility �(1)i (t), it results x∗

ij (t) = 0. The meaning of the secondcondition is analogous, whereas the third one

∑mi=1 (x∗

ij (t) − y∗ij (t))r

∗j (t) = 0 states that

if the price r∗j of the instrument j is positive, then the amount of the assets is equal to the

amount of liabilities, on the contrary if there is an excess supply of an instrument in theeconomy:

∑mi=1 x∗

ij (t) >∑m

i=1 y∗ij (t), then r∗

j (t) = 0 (see [7]).In this paper we observe that the Generalized Complementarity Problem (2) can be written

as the Optimization Problem{minB(v)L(v) = 0,

v ∈ K(3)

and we investigate how we can associate to Problem (3), by means of Lagrangean and DualityTheories, some optimality conditions. For the sake of simplicity, we confine ourselves to aless general case. Let us suppose that X andY are real Hilbert spaces with the usual inclusionX ⊆ Y ⊆ X∗; let it be C the ordering convex cone of Y and let L, B, g three functionsdefined on X with values in Y . Let us suppose that the set K = {v ∈ X : g(v) ∈ −C} isnonempty and let us assume that the Generalized Complementarity Problem (3) holds in thesense of the scalar product on Y and that 〈Lv,Bv〉�0 ∀v ∈ X. Then Problem (3) becomes

minv∈K

〈Lv,Bv〉 = 0. (4)

The main result of this paper is the following:

Theorem 1. Let the function (〈L(v),B(v)〉, g(v)) be convex-like. Let us assume thatqri[g(X) + C] �= ∅ and cone(qri(g(X) + C)) is not a linear subspace of Y. In additionsuppose that C is closed, C − C = Y and there exists v̄ ∈ X such that g(v̄) ∈ −qri C.Then if the functions L, B, g are Fréchet differentiable and Problem (4) admits a solutionu ∈ K, then there exists an element l̄ ∈ C∗ such that

〈Lu(u)v,B(u)〉 + 〈L(u),Bu(u)v〉 + 〈l̄, gu(u)v〉 = 0 ∀v ∈ X

and

〈l, g(u)〉�0, ∀l ∈ C∗,〈l̄, g(u)〉 = 0.

Note that, in virtue of Proposition 6 in Section 2, qri C �= ∅. Further, it is worth remarkingthat, taking into account that 〈Lu(u)v,B(u)〉, 〈L(u),Bu(u)v〉, 〈l̄, gu(u)v〉 define threecontinuous linear mappings on the Hilbert space X, there exist three elements of X∗, thatwe denote by B(u)Lu(u), L(u)Bu(u), l̄gu(u), such that

〈Lu(u)v,B(u)〉 + 〈L(u),Bu(u)v〉 + 〈l̄, gu(u)v〉= 〈B(u)Lu(u) + L(u)Bu(u) + l̄gu(u), v〉 = 0 ∀v ∈ X.

Hence we derive the equivalent condition

B(u)Lu(u) + L(u)Bu(u) + l̄gu(u) = 0X∗ .

S. Giuffrè et al. / Nonlinear Analysis 63 (2005) e1655–e1664 e1659

We can generalize the result obtained in Theorem 1 assuming that the set of the constraintsis given by K = {v ∈ S : g(v) ∈ −C}, where S is a nonempty convex subset of X and thatL(v), B(v) and g(v) are defined on S. In this case the following result holds:

Theorem 2. Let the function (〈L(v),B(v)〉, g(v)) be convex-like. Let us assume thatqri [g(S) + C] �= ∅ and cone(qri(g(S) + C)) is not a linear subspace of Y. In additionsuppose that C is closed, C − C = Y and there exists v̄ ∈ S such that g(v̄) ∈ −qri C. Thenif the functions L, B, g are Fréchet differentiable and Problem (4) admits a solution u ∈ K,then there exists an element l̄ ∈ C∗ such that

〈L(u),Bu(u)(v − u)〉 + 〈Lu(u)(v − u),B(u)〉 + 〈l̄, gu(u)(v − u)〉�0 ∀v ∈ S

and

〈l, g(u)〉�0, ∀l ∈ C∗,〈l̄, g(u)〉 = 0.

Moreover, we observe that the above technique is complementary to the study of thegeneralized complementarity problems by means of variational inequalities.

Finally we would like to mention that the results of Theorem 1 have been presented at theInternational Conference “Variational Analysis and Applications” and an abridged versionof these results has been published in [10].

2. The Lagrangean and duality theory

Let us introduce the dual cone C∗

C∗ = {l ∈ Y ∗ : l(v)�0 ∀v ∈ C}that, in virtue of the usual identification Y = Y ∗, can be rewritten

C∗ = {l ∈ Y : 〈l, v〉�0, ∀v ∈ C}.Then, using the same technique used by Jhan in [14], it is possible to show the followingresult:

Theorem 3. Let the ordering cone C be closed. Then u is a minimal solution of (4) if andonly if u is a solution of the problem

minv∈X

supl∈C∗

{〈Lv,Bv〉 + 〈l, g(v)〉}. (5)

and the extremal values of the two problems are equal.

Now let us introduce the Dual Problem

maxl∈C∗ inf

v∈X{〈Lv,Bv〉 + 〈l, g(v)〉}. (6)

It is known (see [14, Theorem 6.7]) that if int C is nonempty, if Problem (4) (or 5) issolvable and the generalized Slater condition is satisfied, namely there exists v̄ ∈ X with

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g(v̄) ∈ −int C, then Problem (6) is also solvable and the extremal values of the two problemsare equal

minv∈X

supl∈C∗

{〈Lv,Bv〉 + 〈l, g(v)〉} = maxl∈C∗ inf

v∈X{〈Lv,Bv〉 + 〈l, g(v)〉}. (7)

However, in many concrete situations the request that int C is nonempty is not verified: forexample if X,Y are Lebesgue spaces. For this reason in [2], the authors develop the notationof quasi-relative interior of a convex set that is an extension of the relative interior in finitedimension and that may constitute a first contribute to the search of effective regularityassumptions. Let us recall the definition and some properties of quasi-relative interior of aconvex subset C of a real Hilbert space Y .

Definition 2. Let C be a convex subset of Y . The quasi-relative interior of C, denoted byqri C, is the set of those x ∈ C for which

Cone(C − x) = {�y : ��0, y ∈ C − x}is a subspace.

Proposition 1. Let C be a convex subset of Y and x̄ ∈ C. Then x̄ ∈ qri C if and only if thenormal cone to C at x̄ NC(x̄) = {l ∈ Y : 〈l, x − x̄〉�0 ∀x ∈ C} is a subspace.

Proposition 2. Let C be a convex subset of Y. If qri C �= ∅, then

qri C = C and qri C = qri(qri C).

Proposition 3. Let C be a convex subset of Y and suppose x1 ∈ qri C and x2 ∈ C. Then�x1 + (1 − �)x2 ∈ qri C for all 0 < ��1.

Proposition 4. Let C and D be two convex subsets of Y such that qri C �= ∅, qri D �= ∅ andlet � ∈ R. Then

qri C + qri D ⊆ qri (C + D), � qri C = qri (�C), qri (C × D) = qri C × qri D.

Proposition 5. Let C be a convex subset of Y such that qri C �= ∅ and l ∈ Y . If int〈l, C〉 �=∅, then 〈l, qri C〉 = int 〈l, C〉.

Proposition 6. Let C be a convex closed subset of a separable Banach space. Thenqri C �= ∅.

Proposition 7. Let C be a nontrivial convex cone. If C is in addition acute, namely C̄ ∩(−C̄) = {0Y }, then 0Y /∈ qri C.

The proofs of these propositions can be found in [2,1].Using this concept of quasi-relative interior more general separation theorems can be

proved (see [3]). In fact the following statements hold.

S. Giuffrè et al. / Nonlinear Analysis 63 (2005) e1655–e1664 e1661

Lemma 1. Let A be a convex subset of Y such that qri A �= ∅ and 0Y /∈ qri A. Then thereexists g ∈ Y − {0Y } such that 〈g, v〉�0 for all v ∈ A.

Theorem 4. Let S and T be two nonemtpy convex subsets of Y such that qri S �= ∅ andqri T �= ∅ and such that cone (qri S − qri T ) is not a linear subspace of Y or, alternatively,cone (qri S − qri T ) is acute. Then there exists l ∈ Y − {0Y } such that 〈l, s〉�〈l, t〉, for alls ∈ S, t ∈ T .

Theorem 5. Let S and T be two nonempty disjoint convex subsets of Y such that qri S �= ∅and qri T �= ∅. Suppose that there exists a convex set V ⊆ Y such that V − V = Y ,0Y ∈ qri V and cone (qri (S − T ) − qri V )is not a linear subspace of Y or, alternatively,cone(qri(S − T ) − qri V ) is acute. Then there exists l ∈ Y − {0Y } and � ∈ R such that

〈l, s〉 < � < 〈l, t〉 ∀s ∈ S, t ∈ T .

Using the new separation theorems, we can show that Problem (6) is solvable and thatthe extremal values of the two problems are equal.

At first we recall the definition of a convex-like function.

Definition 3. Let X be a real linear space and let Y be a real linear space partially orderedby a convex cone C. A function f : X → Y is called convex–like if the set f (X) + C isconvex.

Theorem 6. Let the function �(v) = (〈L(v),B(v)〉, g(v)) be convex-like with respect tothe product cone R+ × C in R × Y . Let qri [g(X) + C] �= ∅ and cone(qri (g(X) + C))

be not a linear subspace of Y or, alternatively, cone[qri(g(X) + C)] is acute. In additionsuppose that qri C �= ∅ and C − C = Y . If Problem (4) is solvable and there exists v̄ ∈ X

with g(v̄) ∈ −qri C, then also Problem (6) is solvable and the extremal values of the twoproblems are equal. Moreover, if u is a solution to Problem (4) and l̄ ∈ C∗ of (6), it turnsout to be 〈l̄, g(u)〉 = 0.

3. Proof of Theorem 1

Let us consider the Lagrangean functional L : X × C∗ → R

L(v, l) = 〈L(v),B(v)〉 + 〈l, g(v)〉.

Using the preceding theorems we are able to state the following.

Theorem 7. Let the assumptions of Theorem 6 be fulfilled, with C closed. Then a point(u, l̄) ∈ X × C∗ is a saddle point of L, namely

L(u, l)�L(u, l̄)�L(v, l̄), ∀v ∈ X, ∀l ∈ C∗ (8)

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if and only if u is a solution of Problem (4) (or (5)), l̄ is a solution of Problems (6) and (7)holds, namely

minv∈X

supl∈C∗

{〈Lv,Bv〉 + 〈l, g(v)〉} = maxl∈C∗ inf

v∈X{〈Lv,Bv〉 + 〈l, g(v)〉}

= 〈Lu,Bu〉 + 〈l̄, g(u)〉 = 0.

(see for the proof [5]).From (8) we can derive a lot of consequences. First let us take into account the right-hand

side inequality

L(v, l̄)�L(u, l̄) = 0 ∀v ∈ X, (9)

namely

〈L(v),B(v)〉 + 〈l̄, g(v)〉�0 ∀v ∈ X. (10)

Now bearing in mind that L : X → Y , B : X → Y , g : X → Y are Fréchet differentiablefunctions. Then from (9) we derive

〈Lu(u)v,B(u)〉 + 〈L(u),Bu(u)v〉 + 〈l̄, gu(u)v〉 = 0 ∀v ∈ X. (11)

Taking into account that the three terms of the left-hand side of (11) define three continuouslinear mappings on X and that our setting is the Hilbert one, there exist three elements ofX∗ that for the sake of simplicity we denote by B(u)Lu(u), L(u)Bu(u), l̄gu(u) such that

〈Lu(u)v,B(u)〉 = 〈B(u)Lu(u), v〉 ∀v ∈ X,

〈L(u),Bu(u)v〉 = 〈L(u)Bu(u), v〉 ∀v ∈ X,

〈l̄, gu(u)v〉 = 〈l̄gu(u), v〉 ∀v ∈ X.

Then we get

〈B(u)Lu(u) + L(u)Bu(u) + l̄gu(u), v〉 = 0 ∀v ∈ X

and hence

B(u)Lu(u) + L(u)Bu(u) + l̄gu(u) = 0X∗ . (12)

Now let us consider the inequality at the left-hand side of (8). We get

〈L(u),B(u)〉 + 〈l, g(u)〉�0 ∀l ∈ C∗,

namely

〈l, g(u)〉�0 ∀l ∈ C∗.

So we find that if u is a solution of Problem (4), there exists l̄ ∈ C∗ such that

〈l̄, g(u)〉 = 0 and 〈l, g(u)〉�0, ∀l ∈ C∗,

namely u and l̄ satisfy the Variational Inequality{ 〈g(u), l − l̄〉�0,

∀ l ∈ C∗. (13)

S. Giuffrè et al. / Nonlinear Analysis 63 (2005) e1655–e1664 e1663

4. Proof of Theorem 2

Assume that L, B and g are defined on S and that K = {v ∈ S : g(v) ∈ −C}. Thefollowing version of Theorem 6 holds.

Theorem 8. Let the function (〈L(v),B(v)〉, g(v)) be convex-like with respect to the prod-uct cone R+ ×C in R×Y . Let qri[g(S)+C] �= ∅ and cone(qri(g(S) + C)) be not a linearsubspace ofY. In addition suppose that qri C �= ∅ and C − C=Y . If Problem (4) is solvableand there exists v̄ ∈ S with g(v̄) ∈ −qri C, then also Problem (6), with X replaced by S, issolvable and the extremal values of the two problems are equal. Moreover, if u is a solutionto Problem (4) and l̄ ∈ C∗ of (6), with X replaced by S, it turns out to be 〈l̄, g(u)〉 = 0.

Then Theorem 7 takes the form

Theorem 9. Let the assumptions of Theorem 8 be fulfilled, with C closed. Then a point(u, l̄) ∈ S × C∗ is a saddle point of L, namely

L(u, l)�L(u, l̄)�L(v, l̄), ∀v ∈ S, ∀l ∈ C∗ (14)

if and only if u is a solution of problem (4) (or (5), with X replaced by S), l̄ is a solution ofProblem (6), with X replaced by S, and it results

minv∈S

supl∈C∗

{〈Lv,Bv〉 + 〈l, g(v)〉} = maxl∈C∗ inf

v∈S{〈Lv,Bv〉 + 〈l, g(v)〉}

= 〈Lu,Bu〉 + 〈l̄, g(u)〉 = 0.

Now, in order to prove Theorem 2, from Theorem 9 we get L(v, l̄)�L(u, l̄)= 0 ∀v ∈ S,namely

〈Lv,Bv〉 + 〈l̄, g(v)〉�〈Lu,Bu〉 + 〈l̄, g(u)〉 = 0 ∀v ∈ S.

Taking into account that L, B and g are Fréchet differentiable and that S is a convex subsetof X, from the inequality

〈L(u + �(v − u)),B(u + �(v − u))〉 − 〈L(u),B(u)〉 + 〈l̄, g(u + �(v − u))〉− 〈l̄, g(u)〉 = 〈L(u + �(v − u)) − L(u),B(u + �(v − u))〉 + 〈L(u),

B(u + �(v − u)) − B(u)〉 + 〈l̄, g(u + �(v − u)) − g(u)〉�0 ∀� ∈ [0, 1], ∀v ∈ S,

we get

〈Lu(u)(v − u),B(u)〉 + 〈L(u),Bu(u)(v − u)〉 + 〈l̄, gu(u)(v − u)〉�0 ∀v ∈ S.

The other part of Theorem 2 follows as in the proof of Theorem 1.

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