JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 104, No. 3, pp. 577–588, MARCH 2000

Exceptional Families of Elements,Feasibility and Complementarity

G. ISAC1

Communicated by F. Giannessi

Abstract. Feasibility is an important property for a complementarityproblem. A complementarity problem is solvable if it is feasible andsome supplementary assumptions are satisfied. In this paper, we intro-duce the notion of (α , β )-exceptional family of elements for a continu-ous function and we apply this notion to the study of feasibility ofnonlinear complementarity problems.

Key Words. Complementarity problems, feasibility, exceptional familyof elements, zero-epi mappings.

1. Introduction

Complementarity theory, defined thirty years ago (Refs. 1–2), con-tinues to be developing because of the variety of applications in optimiz-ation, economics, engineering, game theory, etc. (Refs. 1–4). The idea ofcomplementarity is associated with the idea of equilibrium, as it is studiedin physics, engineering, and even in economics. Several kinds of comp-lementarity problems have been defined and studied until now. Given aparticular complementarity problem, its solvability is not evident. Becauseof this fact, many existence theorems have been proved (Refs. 1–3).

Recently, a new investigating method has been introduced. Thismethod is based on the concept of exceptional family of element (EFE) fora continuous function (Refs. 5–6), and many papers are now based on thisnotion (Refs. 5–14). This notion is natural, deep, and related to topologicaldegree. It can be considered as a general coercivity condition.

In this paper, we will show that a variant of the concept of EFE canbe used to study the feasibility of complementarity problems. A comple-mentarity problem can be feasible but unsolvable (Refs. 1–2 and 15–16).

1Professor, Department of Mathematics and Computer Science, Royal Military College ofCanada, STN Forces, Kingston, Ontario, Canada.

5770022-3239y00y0300-0577$18.00y0 2000 Plenum Publishing Corporation

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Feasibility is important, since it is used in the study of existence theorems.There exist now several existence theorems using feasibility as an assump-tion (Refs. 1–2 and 17).

We will introduce the concept of (α , β )-exceptional family of elements[(α , β )-EFE] and we will show that, in the Euclidean space Rn, given aclosed pointed convex cone K⊂Rn such that K* ⊆ K, a pair of real numbers(α , β ) such that βHαX0, and a continuous function f: Rn→Rn, then eitherthe complementarity problem NCP( f, K ) is feasible or there exists an (α , β )-EFE for f.

This paper can be considered as a new kind of applications of the EFEconcept to complementarity theory.

2. Preliminaries

Let (H, ⟨ , ⟩) be a Hilbert space, and let K⊂H be a closed pointed con-vex cone; i.e., K is a nonempty closed set satisfying the following properties:

(i) KCK ⊆ K,(ii) λK ⊆ K, for all λ∈R+ ,(iii) K∩ (−K )G0.

By definition, the dual of K is

K*Gy∈H u ⟨x, y⟩X0, for all x∈K.

We can show that K* is a closed convex cone. It is known that, if K⊂H isa proper closed pointed convex cone, then K∩K*≠0 and three situationsare of particular interest:

(i) K*⊂K (K is superdual),(ii) K*GK (K is selfdual),(iii) K*⊃K (K is subdual).

If K⊂H is a closed pointed convex cone, the projection onto K, denotedby PK , is well defined for every x∈H. PK (x) is the unique element in Ksatisfying

uuxAPK(x)uuGminy∈K

uuxAy uu.

We recall that the projection PK is characterized by the properties below:For every x∈H, PK (x) is the (unique) element in K satisfying the followingconditions:

(p1) ⟨PK(x)Ax, y⟩X0, for all y∈K,(p2) ⟨PK(x)Ax, PK(x)⟩G0.

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Given a closed pointed convex cone K⊂H and a mapping f: H→H, thenonlinear complementarity problem associated to f and K is described by

(NCP( f, K )) find x∏∈K such that f (x∏)∈K* and ⟨x∏ , f (x∏)⟩G0.

Problem NCP( f, K ) is the object of complementarity theory (Refs. 1–4),etc. The existence of solution of this problem is not evident (Refs 1–2).

3. (0, k )-Epi Mappings

The concept of 0-epi (zero-epi) mapping is simpler and more refinedthan the concept of topological degree; it was introduced in 1980 by Furi,Martelli, and Vignoli (Ref. 18). In 1987, this concept was extended fromcompact mappings to k-set contractions by Tarafdar and Thompson (Ref.19). By this extension, the concept of (0, k )-epi mapping was obtained. Werecall now the definitions and the properties of this concept.

Let (E, uu uu ) and (F, uu uu ) be Banach spaces, let D and A be subsets of E,and let f: D→F be a mapping. By definition, the Kuratowski measure ofnoncompactness of A is

α (A)Ginf ((H0 uA can be covered by a finite number of sets

of diameter less than (.

It is known that α (A)G0 if and only if A is relatively compact. A continu-ous mapping f: D→F is said to be a k-set contraction if, for each boundedsubset A of D, we have that

α ( f (A))Ykα (A), where kX0.

Let Ω be a bounded open subset of E.

Definition 3.1. See Ref. 18. A continuous mapping f: Ω→F is said tobe 0-admissible [p-admissible] if 0∉ f (∂Ω ) [p∉ f (∂Ω )].

Definition 3.2. See Ref. 19. A 0-admissible mapping f: Ω→F is saidto be (0, k )-epi if, for each k-set contraction h: Ω→F, with h(x)G0 for eachx∈∂Ω, the equation f (x)Gh(x) has a solution in Ω.

Similarly, a p-admissible mapping f: Ω→F is said to be (p, k )-epi if themapping fAp, defined by ( fAp)(x)Gf (x)Ap (x) for each x∈Ω, is (0, k )-epi.

In Definition 3.2, if we replace the term ‘‘k-set contraction’’ by the term‘‘compact mapping’’, we obtain the concept of 0-epi mapping introducedand studied by Furi, Martelli, and Vignoli (Ref. 18). The concept of 0-epimapping has many applications in nonlinear analysis.

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Properties.

(I) Existence Property. If f: Ω→F is a (p, k)-epi mapping, then theequation f (x)Gp has a solution in Ω.

(II) Normalization Property. The inclusion mapping i: Ω→E is(p, k )-epi for k∈[0, 1[ if and only if p∈Ω.

(III) Localization Property. If f: Ω→F is a (0, k)-epi mapping andif f −1(0) is contained in an open set Ω1⊂Ω, then f restricted toΩ1 is also (0, k )-epi.

(IV) Homotopy Property. Let f: Ω→F be (0, k )-epi, and let h:[0, 1]BΩ→F be a β -set contraction with 0YβYkF1 such thath(0, x)G0 for all x∈Ω. Assume that f (x)Ch(t, x)≠0, for all x∈∂Ω and for all t∈[0, 1]. Then, f ( · )Ch(1, · ): Ω→F is a (0, kAβ )-epi mapping.

(V) Boundary Dependence Property. Let f: Ω→F be (0, k )-epi, andlet g: Ω→F be aβ -set contraction with 0YβYkF1 and g(x)G0 for each x∈∂Ω. Then, fCg: Ω→F is a (0, kAβ )-epi mapping.

For proof of these properties, the reader is referred to Refs. 18–19.When kG0, we obtain the properties of 0-epi mappings introduced andstudied in Ref. 18.

4. Complementarity Problems: Solvability and Feasibility

Let (H, ⟨ , ⟩) be a Hilbert space, let K⊂H be a closed pointed convexcone, and let f: H→H be a mapping. Consider problem NCP( f, K ).

By definition, the feasible set of this problem is

F Gx∈K u f (x)∈K*.

The set F can be empty; when F is nonempty, we say that NCP( f, K ) isfeasible. If the cone K* has a nonempty interior, and if the set

F SGx∈K u f (x)∈int K*

is nonempty, we say that NCP( f, K ) is strictly feasible. The solvability ofNCP( f, K ) implies its feasibility. The converse is not true. In this sense, wecite the following example presented in Ref. 15.

Consider the Euclidean space (R2, ⟨, ⟩), KGR 2+ , and

f (x)G3A1

1

1

143x1

x24C3A2

14 , where xG3x1

x24∈R2.

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In this case, problem NCP( f, K ) is problem LCP( f, R2+ ), and we can show

that this problem is feasible but unsolvable. The relation between feasibilityand solvability has been studied by several authors (Refs. 1–2 and 16–17).

It is interesting to know under what assumptions feasibility of problemNCP( f, K ) implies its solvability. For example, if f: Rn→Rn is continuousand monotone, and if NCP( f, R2

+ ) is strictly feasible, then it is solvable (Ref.17). Also, if f: Rn→Rn is off-diagonally antitone on R2

+ and continuous, thenfeasibility of NCP( f, R2

+) implies its solvability (Ref. 17).Because feasibility implies solvability under some conditions, we will

study in the next section the feasibility of a nonlinear complementarity prob-lem using the concept of exceptional family of element for a continuousfunction.

5. Exceptional Family of Elements, Solvability, and Feasibility

Recently, a new method has been introduced in the study of solvabilityof complementarity problems. This method is based on an alternative the-orem and on the concept of exceptional family of elements for a continuousfunction (Refs. 5–6). Suppose that (Rn, ⟨ , ⟩) is the Euclidean space, thatK⊂Rn is a closed pointed convex cone, and that f: Rn→Rn is a continuousfunction. The notion of exceptional sequence of elements, defined in Ref.20, is more restrictive than our notion and is not related to that of topologi-cal degree.

Definition 5.1. See Refs. 5–6. We say that the family of elementsxrrH0⊂K is an exceptional family of elements for f with respect to coneK if and only if, for every real number rH0, there exists a real number µrH0such that the vector urGf (xr )Cµrxr satisfies the following conditions:

(i) ur∈K*,(ii) ⟨ur , xr ⟩G0,(iii) uuxr uu→CS, as r→CS.

In Refs. 5–6, the following result is proved.

Theorem 5.1. Given a closed pointed convex cone K⊂Rn, then for anycontinuous function f: Rn→Rn, there exists either a solution for problemNCP( f, K ) or an exceptional family of elements for f with respect to K.

An important consequence of Theorem 5.1 is the fact that, if we knowa class of functions without exceptional families of elements, we know also

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a class of functions for which problem NCP( f, K ) is solvable. In this sense,many results are obtained in Refs. 5–8 and 9–14. In Ref. 12, Zhao extendedthis method to variational inequalities in Rn with respect to unboundeddomains.

Now, we extend the concept of exceptional families of elements forfeasibility. Again, we suppose given the Euclidean space (Rn, ⟨ , ⟩), a closedpointed convex cone K⊂Rn, and a continuous function f: Rn→Rn.

Definition 5.2. Given a pair of real numbers (α , β ) such that 0YαFβ ,we say that the family of elements xrrH0⊂Rn is an (α , β )-exceptional fam-ily of elements for f with respect to K if and only if limr→CS uuxr uuGCS

and, for each real number rH0, there exists a scalar tr∈]0, 1[ such that thevector urG(1ytrA1)xrC(βAα ) f (xr ) satisfies the following properties:

(i) ur∈K*,(ii) ⟨ur , xrAα tr f (xr)⟩G0.

The importance of this notion is given by the following result.

Theorem 5.2. Let (α , β ) be a pair of real numbers such that 0YαFβ ,and let K⊂Rn be a closed pointed convex cone such that K*⊂K or K*GK. Then, for any continuous function f: Rn→Rn, either problem NCP( f, K )is feasible or there exists an (α , β )-exceptional family of elements for f withrespect to K.

Proof. For any rH0, we denote

SrGx∈Rn u uuxuuGr, BrGx∈Rn u uuxuuFr.

First, we remark that, by the normalization property of 0-epi-mappings, theidentity mapping (denoted by Id) is a 0-epi mapping on each Br . Considerthe equation

xAα f (x)GPK [xAβ f (x)] (1)

and the mapping h: [0, 1]BBr r→Rn defined by

h(t, x)Gt [−α f (x)APK [xAβ f (x)]].

We observe that h is continuous compact and that h(0, x)G0 for all x∈Br r .We have the following two situations:

(I) There exists rH0 such that xCt [−α f (x)APK [xAβ f (x)]]≠0 for allx∈Sr and all t∈[0, 1].

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In this case, applying the homotopy property of 0-epi mappings, we havethat

xC[−α f (x)APK [xAβ f (x)]]G0

has a solution in Br ; that is, there exists x∏∈Br such that

x∏Aα f (x∏ )GPK [x∏Aβ f (x∏ )]. (2)

Using property (p1) of the projection operator PK , we obtain the followingrelation:

⟨x∏Aα f (x∏ )A[x∏Aβ f (x∏ )], y⟩X0, for all y∈K, (3)

which implies that

⟨(βAα ) f (x∏ ), y⟩X0, for all y∈K; (4)

that is, we have that f (x∏ )∈K*. Since

PK [x∏Aβ f (x∏ )]∈K and K* ⊆ K,

we have that α f (x∏ )∈K, and finally from (2) we deduce that

x∏Gα f (x∏ )CPK [x∏Aβ f (x∏ )]∈K. (5)

Hence, f (x∏ )∈K* and x∏∈K; that is, problem NCP( f, K ) is feasible.

(II) For every rH0, there exists xr∈Sr and tr∈[0, 1] such that

xrCtr [−α f (xr )APK [xrAβ f (xr )]]G0. (6)

If trG0, then xrG0, which is impossible since xr∈Sr . If trG1, then we have

xrC[−α f (xr )APK [xrAβ f (xr )]]G0,

which implies that problem NCP( f, K ) is feasible. Hence, we can say thateither Eq. (2) has a solution or, for any rH0, there exists xr∈Sr and tr∈]0, 1[ such that

xrCtr [−α f (xr )APK [xrAβ f (xr )]]G0. (7)

Because of the fact that solvability of Eq. (2) for some rH0 implies feasibil-ity of problem NCP( f, K ), we have that, if this problem is infeasible, thenfor any rH0 there exists xr∈Sr and t∈]0, 1[ such that Eq. (7) is satisfied.From Eq. (7), we have

xrAα tr f (xr )GtrPK [xrAβ f (xr )], (8)

or

PK [xrAβ f (xr )]G(1ytr )xrAα f (xr ). (9)

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Using (9) and properties (p1), (p2) of the projection operator PK , we deducethat

⟨(1ytr )xrAα f (xr )A[xrAβ f (xr )], y⟩X0, for all y∈K, (10)

⟨(1ytr )xrAα f (xr )A[xrAβ f (xr )], (1ytr )xrAα f (xr )⟩G0. (11)

If we denote

urG(1ytrA1)xrC(βAα ) f (xr ), for every rH0,

we deduce from (10) and (11) that

ur∈K* and ⟨ur , xrAα tr f (xr )⟩G0;

and since, for every rH0, xr∈Sr , we have that uuxr uu→CS as r→CS. Weconclude that xrrH0 is (α , β )-exceptional family of elements for f withrespect to K. Therefore, for any pair (α , β ) of real numbers such that0YαFβ , either problem NCP is feasible, or there exists an (α , β )-excep-tional family of elements for f with respect to K. h

Remark 5.1. From Theorem 5.2, we deduce immediately that, ifK* ⊆ K and αG0 [that is, if we consider the pair (0, β ) with βH0, and iff: Rn→Rn is without (0, β )-exceptional families of elements], then problemNCP( f, K ) is solvable.

6. Generalization to Infinite-Dimensional Hilbert Spaces

Let (H, ⟨ , ⟩) be an infinite-dimensional Hilbert space, and let K⊂H bea closed pointed convex cone. It is interesting to know if it is possible togeneralize Theorem 5.2 to an arbitrary infinite-dimensional Hilbert space.This problem has sense, since in our paper (Ref. 8) we presented an exten-sion of Theorem 5.1 to the infinite-dimensional case.

Suppose that f: H→H is a completely continuous field of the formf (x)G(1yβ )xAT (x), where βH0 and T: H→H is a completely continuousoperator; i.e., T is continuous and, if D⊂H is any bounded set, then T (D)is a relatively compact set.

Definition 6.1. Let f: H→H be a completely continuous field of theform f (x)G(1yβ )xAT (x), for all x∈H. Given a real number α such that0YαFβ , we say that the family of elements xrrH0⊂H is an (α , β )-excep-tional family of elements for f with respect to K if and only iflimr→CS uuxr uuGCS and, for every real number rH0, there exists a scalar

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tr∈]0, 1[ such that the element urG(1yβ tr )xrAT (xr ) satisfies the followingproperties:

(i) ur∈K*,(ii) ⟨ur , [(βAα )ytr]xrCαβT (xr )⟩G0.

We have the following result.

Theorem 6.1. Let (H, ⟨ , ⟩) be an infinite-dimensional Hilbert space, letK⊂H be a closed convex cone such that K* ⊆ K, and let f: H→H be acompletely continuous field of the form f (x)G(1yβ )xAT (x), where βH0.Then, for any real α such that 0YαFβ , either problem NCP( f, K ) is feas-ible or there exists an (α , β )-exceptional family of elements in the sense ofDefinition 6.1 for f with respect to K.

Proof. For any rH0, we denote

SrGx∈H u uuxuuGr, BrGx∈H u uuxuuFr.

By the normalization property of 0-epi mapping, we have that the identitymapping is a 0-epi mapping on each Br . Consider the equation

xAα f (x)GPK [xAβ f (x)]; (12)

since f (x)G(1yβ )xAT (x), we have

xC[βαy(βAα )]T (x)APK[β2y(βAα )]T (x)G0. (13)

The mapping h: [0, 1]BBr r→H, defined by

h(t, x)Gt[βαy(βAα )]T (x)APK [[β2y(βAα )]T (x)],

is completely continuous and h(0, x)G0 for all x∈Br r . Hence, the mappingh satisfies the assumptions asked in the homotopy property of (0, k )-epimappings for kG0, that is, for 0-epi mappings.

We have the following two situations:

(I) There exists rH0 such that

xCt [[βαy(βAα )]T (x)APK[β2y(βAα )]T (x)]]≠0,

for all x∈Sr and all t∈[0, 1].

In this case, applying the homotopy property of 0-epi mappings, we havethat xC[βαy(βAα )]T (x)APK [[β2y(βAα )]T (x)]G0 has a solution inBr ; that is, there exists x∏∈Br such that

x∏C[βαy(βAα )]T (x∏ )APK [[β2y(βAα )]T (x∏ )]G0. (14)

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From (14), we have

[(βAα )yβ ]x∏CαT (x∏ )APK [βT (x∏ )]G0,

and by an algebraic calculus we obtain that

x∏Aα f (x∏ )GPK [x∏Aβ f (x∏ )]. (15)

Using (15), as in the proof of Theorem 5.2, we obtain that problemNCP( f, K ) is feasible.

(II) For every rH0, there exists xr∈Sr and tr∈[0, 1] such that

xrCtr[βαy(βAα )]T (xr )APK [[β2y(βAα )]T (xr )]G0. (16)

We observe that it is impossible to have trG0. If trG1, we can show againthat, in this case, problem NCP( f, K ) is feasible.

If in (16) we have tr∈]0, 1[, we deduce that

PK [T (xr )]G[(βAα )yβ2tr ]xrC(αyβ )T (xr ). (17)

Using (17) and the same argument as in the proof of Theorem 5.2, we canshow that, for the pair (α , β ), either problem NCP( f, K ) is feasible or thereexists an (α , β )-exceptional family of elements in the sense of Definition 6.1for f with respect to K. h

Remark 6.1. Modulo some details, it is possible to extend Theorem5.2 from completely continuous fields of the form f (x)G(1yβ )xAT (x) tok-set fields of the form f (x)G(1yβ )xAT (x), where T is a k-set contractionswith an appropriate k.

7. Conclusions and Open Problems

Theorems 5.2 and 6.1 have as consequence the necessity to study theclass of functions without (α , β )-exceptional family of elements with respectto a closed convex cone. Results obtained recently in this sense can be foundin Refs. 5–8 and 9–14. The class of functions without exceptional familiesof elements with respect to a closed convex cone supports the idea thatthe same study is now necessary for the class of functions without (α , β )-exceptional families of elements.

Open Problems.

(P1) It is interesting to know if Theorems 5.2 and 6.1 can be extendedto a cone K with the property that K⊂K*.

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(P2) Can the method presented in this paper be adapted to the studyof strict feasibility?

(P3) In his thesis (Ref. 12), Zhao introduced the concept of d-orien-tation family of elements as a variant of the notion of exceptionalfamily of elements. It is interesting to know if the concept of d-orientation family of element can be used to study the feasibilityof problem NCP( f, K ).

References

1. COTTLE, R. W., PANG, J. S., and STONE, R. E., The Linear ComplementarityProblem, Academic Press, New York, NY, 1992.

2. ISAC, G., Complementarity Problems, Lecture Notes in Mathematics, SpringerVerlag, New York, NY, Vol. 1528, 1992.

3. BERSHCHANSKII, Y. M., and MEEROV, M. V., The Complementarity Problem:Theory and Methods of Solution, Automation and Remote Control, Vol. 44,pp. 687–710, 1983.

4. FERRIS, M. C., and PANG, J. S., Engineering and Economic Applications ofComplementarity Problems, SIAM Review, Vol. 39, pp. 669–713, 1997.

5. ISAC, G., BULAVSKI, V., and KALASHNIKOV, V., Exceptional Families, Topo-logical Degree, and Complementarity Problems, Journal of Global Optimization,Vol. 10, pp. 207–225, 1997.

6. BULAVSKI, V. A., ISAC, G., and KALASHNIKOV, V. V., Application of Topologi-cal Degree Theory to Complementarity Problems, Multilevel Optimization: Algo-rithms and Applications, Edited by A. Migdalas et al., Kluwer AcademicPublishers, Dordrecht, Holland, pp. 333–358, 1998.

7. CARBONE, A., and ISAC, G., The Generalized Order Complementarity Problem:Applications to Economics and an Existence Result, Nonlinear Studies, Vol. 5,pp. 129–151, 1998.

8. ISAC, G., Exceptional Families of Elements for k-Set Fields in Hilbert Spacesand Complementarity Theory, Proceedings of ICOTA–1998, Perth, Australia,pp. 1135–1143.

9. ISAC, G., and CARBONE, A., Exceptional Families of Elements for ContinuousFunctions: Some Applications to Complementarity Theory, Journal of GlobalOptimization (to appear).

10. ISAC, G., and OBUCHOWSKA, W. T., Functions without Exceptional Familiesof Elements and Complementarity Problems, Journal of Optimization TheoryApplications, Vol. 99, pp. 147–163, 1998.

11. ZHAO, Y. B., Exceptional Family and Finite-Dimensional Variational Inequalityover Polyhedral Convex Sets, Applied Mathematics and Computation, Vol. 87,pp. 111–126, 1997.

ps899$p253 09-03-:0 12:45:47

JOTA: VOL. 104, NO. 3, MARCH 2000588

12. ZHAO, Y. B., Existence Theory and Algorithms for Finite-DimensionalVariational Inequality and Complementarity Problems, PhD Dissertation, Insti-tute of Applied Mathematics, Chinese Academy of Sciences, Beijing, China,1998.

13. ZHAO, Y. B., and HAN, J. Y., Exceptional Family of Elements for a VariationalInequality Problem and Its Applications, Journal of Global Optimization, Vol.14, pp. 313–330, 1999.

14. ZHAO, Y. B., HAN, J. Y., and QI, H. D., Exceptional Families and ExistenceTheorems for Variational Inequality Problems, Journal of Optimization Theoryand Applications, Vol. 101, pp. 475–495, 1999.

15. ISAC, G., KOSTREVA, M. M., and WIECEK, M. M., Multiple-Objective Approxi-mation of Feasible but Unsolvable Linear Complementarity Problems, Journal ofOptimization Theory and Applications, Vol. 86, pp. 389–405, 1995.

16. MEGIDDO, N., A Monotone Complementarity Problem with Feasible Solutionsbut No Complementary Solutions, Mathematics Programming Study, Vol. 12,pp. 131–132, 1977.

17. MORE, J. J., Classes of Functions and Feasibility Conditions in NonlinearComplementarity Problems, Mathematical Programming, Vol. 6, pp. 327–338,1974.

18. FURI, M., MARTELLI, M., and VIGNOLI, A., On the Solvability of NonlinearOperator Equations in Normed Spaces, Annali di Matematica Pura ed Applicata,Vol. 124, pp. 321–343, 1980.

19. TARAFDAR, E. U., and THOMPSON, H. B., On the Solvability of Nonlinear Non-compact Operator Equations, Journal of the Australian Mathematical Society,Vol. 43A, pp. 103–106, 1987.

20. SMITH, T. E., A Solution Condition for Complementarity Problems with Appli-cation to Spatial Price Equilibrium, Applications of Mathematical Computation,Vol. 15, pp. 61–69, 1984.

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