Abstract and Applied Analysis
Volume 2014 (2014), Article ID 468065, 11 pages
http://dx.doi.org/10.1155/2014/468065

Research Article

Further Application of -Differentiability to Generalized Complementarity Problems Based on Generalized Fisher-Burmeister Functions

Wei-Zhe Gu1,2 and Mohamed A. Tawhid2,3

¹Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, China
²Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC, Canada V2C 0C8
³Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Moharram Bey, Alexandria 21511, Egypt

Received 16 July 2014; Accepted 28 August 2014; Published 17 November 2014

Academic Editor: Janusz Brzdek

Copyright © 2014 Wei-Zhe Gu and Mohamed A. Tawhid. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP() where and are -differentiable. We describe -differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on the -differentials of and , we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved for , semismooth, and locally Lipschitzian.

1. Introduction

Gowda et al. in [1] introduced the concepts of the -differentiability and -differential for a function . They showed that the Fréchet derivative of a Fréchet differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function [2], the Bouligand subdifferential of a semismooth function [3–5], and the -differential of a -differentiable function [6] are instances of -differentials. In their paper, they noted that -differentials enjoy simple sum, product, and chain rules, -differentiability implies continuity, and any superset of an -differential is an -differential. It is noted in [7] that the -differentiable function needs not be locally Lipschitzian nor directionally differentiable.

There have been many applications of these concepts to optimization, complementarity problems, and variational inequalities, characterizations of and properties when the underlying functions are not necessarily locally Lipschitzian or semismooth (see, e.g., [7–15]).

In this paper, we study a further application of -differentiability to nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP() where and are -differentiable.

We consider a generalized complementarity problem corresponding to -differentiable functions and , denoted by GCP(), which is to find a vector such that

where and .

In the last decades, many researchers have given a lot of attention to this problems in terms of its applications, numerical methods, and formulation; see [16, 17] and the references cited therein. If with some , then GCP is known as the quasi/implicit complementarity problem; see, for example, [17–19]. Also, if , then GCP reduces to the nonlinear complementarity problem NCP. By taking in NCP   with and a vector , then NCP is called a linear complementarity problem LCP.

Our approach is to reformulate GCP as an unconstrained optimization problem through some merit function. We construct a merit function via a GCP function :

For the problem GCP, we define

and we call a GCP function for GCP. A function is said to be a merit function for GCP provided that the global minima of are coincident with the solutions of the original GCP. We consider a GCP function associated with GCP() and its merit function

so that

The organization of the paper is as follows. We state some basic definitions and preliminary results. We describe -differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations and their merit functions. We show that under appropriate show () conditions and column property conditions, local/global minimum of a merit function (or a “stationary point” or “semi-stationary point” of a merit function) based on the generalized Fisher-Burmeister function and its generalizations coincides with the solution of the given generalized complementarity problem. Note that considering GCP functions on the basis of the generalized Fisher-Burmeister function and its generalizations seems to be new.

Moreover, when specializing GCP to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved for , semismooth, and locally Lipschitzian.

2. Preliminaries

Throughout this paper, all vectors in are column vectors. or denotes the inner product between two vectors and in . Vector inequalities are interpreted componentwise. All the operations are performed componentwise. For a set , denotes the convex hull of and denotes the closure of . For a differentiable function , denotes the Jacobian matrix of at . For a matrix , denotes the th row of . denotes the -norm of and denotes the Euclidean norm of . In addition, unless otherwise stated, assume in the sequel is any fixed real number in .

In this section, we first recall some background concepts.

We first recall the definition of -differentiability and examples from [1].

Definition 1. Given a function , where is an open set in and , we say that a nonempty subset , also denoted by , of is an -differential of at if for every sequence converging to , there exist a subsequence and a matrix such that

We say that is -differentiable at if has an -differential at .

A useful equivalent definition of an -differential is as follows: for any sequence with and for all , there exist convergent subsequences and , and such that

Here are some well-known facts about -differentiability; see, for example, [8–10, 15].

Remark 2. (i) Any superset of an -differential is an -differential.
(ii) -differentiability implies continuity, and -differentials enjoy simple sum, product, and chain rules.
(iii) While the Fréchet derivative of a differentiable function, the Clarke generalized Jacobian of a locally Lipschitzian function [2], the Bouligand differential of a semismooth function [4], and the -differential of a -differentiable function [6] are particular instances of -differential, it is shown in [10] by example that an -differentiable function need not be locally Lipschitzian nor directionally differentiable.
(iv) If a function is -differentiable at a point , then there exist a constant and a neighborhood of with

Conversely, if condition (8) holds, then can be taken as an -differential of at .

In [10], the following definition is introduced to generalize the concepts of monotonicity, -property, and their variants for function in [20].

Definition 3. For functions , we say that and are as follows:

(a)relatively monotone if

(b)relatively strictly monotone if

(c)relatively strongly monotone if there exists a constant such that

(d)relatively ()-functions if for any in ,

(e)relatively uniform ()-functions if there exists a constant such that for any ,

3. -Differentials of Some GCP Functions When the Underlying Functions Are -Differentiable

In this section, we compute the -differentials of some GCP functions and their merit functions.

Example 4. Suppose that and are -differentiable at with -differentials, respectively, by and . Consider the following GCP function which is the basis of

where is any fixed real number in the interval and denotes the -norm of ; that is, . The function was noted by Tseng [21]. For further study on this family of NCP functions, see [22]. The th component of this kind of GCP function in (3) is defined as

Now we describe the -differentials of . Let

The -differential of at is given by

where is the set of all quadruples with , , , , and being diagonal matrices satisfying the conditions:

Proof. To see this claim, let with and . By the -differentiability of , there exists a sequence of , , and , such that

Let . With , , , , and satisfying (19), let . We claim that . To see this, we fix an index and show that . Without loss of generality, let . We denote the 1st row of , , and by , , and .
We consider two cases.
Case 1 ( ). In this case, we have

Case 2  ( )
Subcase (1) ( ). In this case,

Subcase (2) ( ). In this case, . Then

Example 5. Consider the following GCP function which is based on proposed family of NCP functions [22] relying on in (14) and some introduced NCP functions in [23]:

where and GCP function in (3) is defined as

where all the operations are performed componentwise.
Let

Then the -differential of at is given by

where is the set of all quadruples with , , , , and being diagonal matrices with

The above calculation relies on the observation that the following is an -differential of the one variable function at any :

Example 6. The following GCP function is based on NCP function in [22]:

We define the GCP function in (3) as

where all the operations in (31) are performed componentwise.
Let

Then the -differential of at is given by

where is the set of all quadruples with , , , , and being diagonal matrices with

Example 7. The following GCP function is based on NCP function in [23]:

We define the GCP function in (3) as

where all the operations in (36) are performed componentwise.
Let

When , .
Then the -differential of at is given by

where is the set of all quadruples with , , , , and being diagonal matrices with

4. Minimizing the Merit Function

In this section, we consider an NCP function corresponding to GCP and let , when the underlying functions and are -differentiable.

It should be recalled that

Let be an GCP function with an -differential given by

The following theorem from [15] describes -differential, at , for the merit function .

Theorem 8. Suppose is -differentiable at with as an -differential. Then is -differentiable at with an -differential given by

We need the following Lemma [10] in our subsequent analysis.

Lemma 9. Suppose that and is one-to-one and onto. Define where . The following hold.

(a) and are relatively (strictly) monotone if and only if is (strictly) monotone.(b)If is Lipschitz-continuous and and are relatively strongly monotone then is strongly monotone.(c) and are relatively ()-functions if and only if is ()-function.(d)If is Lipschitz-continuous and and are relatively uniform -functions, then is uniform -function.

The following result is from [20, 24].

Theorem 10. Under each of the following conditions, is a -function.

(a) is Fréchet differentiable on and, for every , the Jacobian matrix is a -matrix.(b) is locally Lipschitzian on and, for every , the generalized Jacobian consists of -matrices.(c) is semismooth on (in particular, piecewise affine or piecewise smooth) and, for every , the Bouligand subdifferential consists of -matrices.(d) is -differentiable on and, for every , an -differential consists of -matrices.

Remarks. Based on some results in [20, 24], we note the following.

(i)For -conditions, the the converse statements in the above theorem are usually false.(ii)For -conditions in Theorem 10, the converse statements of Item (a) and Item (c) are true, while the converse statements of Item (b) and Item (d) may not hold in general ([20, 24]).

It is easy to see the following Lemma.

Lemma 11. Suppose that and is one-to-one and onto. Define where . Suppose that and are -differentiable at with -differentials, respectively, by and with consisting of nonsingular matrices. Denote . Then is -differentiable at with , where

The following two Lemmas give favorable properties which will be needed in our results.

Lemma 12. We can easily see that , , and in Examples 4–6 satisfy the following properties.

(i) solves GCP .(ii), , when .(iii), when .(iv), when .(v), when , .

Proof. The proof can be easily verified.

Lemma 13. We can easily see that , , and in Example 7 satisfy the following properties:

(i) solves GCP .(ii), when .(iii), when .(iv), when .(v), when , .

Proof. The proof can be easily verified.

Starting with -differentiable functions and , we show that, under appropriate conditions, a vector is a solution of the GCP if and only if zero belongs to .

In the following theorems we will minimize the merit function under -conditions.

Theorem 14. Suppose and are -differentiable at with -differentials, respectively, by and . Suppose is a GCP function of and . Assume that is -differentiable at with an -differential given by

Further suppose that consists of nonsingular matrices and consists of -matrices where . Then

Proof. The proof is similar to Theorem 6 in [15].

Remark 15. Theorem 14 is applicable to GCP functions of Examples 4–7 by the property (ii) in Lemma 12 and the property (ii) in Lemma 13.

A slight modification of the above theorem leads to the following result.

Theorem 16. Suppose and are -differentiable at with -differentials, respectively, by and . Suppose is a GCP function of and . Assume that is -differentiable at with an -differential given by

Further suppose that consists of nonsingular matrices and consists of positive semidefinite matrices where . Then

Proof. Since every positive semidefinite matrix is also a -matrix, the proof follows from Theorem 14.

If is a monotone (strongly monotone) , is positive semidefinite (positive definite) matrix. From Lemma 9, Example 4, and the above theorems, we have the following.

Corollary 17. Suppose and are differentiable at . Assume is continuous and strongly monotone. Moreover, and are relatively monotone (relatively strongly monotone) functions. Suppose is a GCP function of and , which is based on the generalized Fischer-Burmeister function and .
Then is a local minimizer to if and only if solves GCP .

In view of Example 5 and the above results, we have the following.

Corollary 18. Suppose and are semismooth (piecewise smooth or piecewise affine) at with Bouligand subdifferentials, respectively, by and . Assume is continuous, one-to-one, onto and consists of nonsingular matrices. Moreover, and are relatively monotone (relatively strongly monotone) functions. Suppose is a GCP function of and , which is based on the generalized Fischer-Burmeister function and .
Then is a local minimizer to if and only if solves GCP.

We state the result for GCP function which is based on the generalized Fischer-Burmeister function. However, as in Theorem 14, it is possible to state a very general result for any GCP function satisfying the properties in Lemmas 12 and 13. For simplicity, we avoid dealing in such a generality.

Corollary 19. Suppose and are differentiable at . Suppose is a GCP function of and , which is the basis of the generalized Fischer-Burmeister function in Example 4 and . If is nonsingular and the product is -matrix, then is a local minimizer to if and only if solves GCP .

Corollary 20. Suppose and are differentiable at . Assume is continuous, one-to-one, onto and is nonsingular. Moreover, assume and are relatively -functions. Suppose is a GCP function of and , which is based on the generalized square Fischer-Burmeister function in Example 4 and .
Then is a local minimizer to if and only if solves GCP.

Proof. Since is one-to-one and onto and and are relatively -functions, by Lemma 9, the mapping is -function which implies is -matrix; see [20]. The proof follows from Corollary 19.

We recall that a continuous mapping is called a homeomorphism if it is a one-to-one and onto mapping and if its inverse mapping is also continuous.

It is known that a continuous, strongly monotone mapping is a homeomorphism from onto itself and the is positive definite matrix if is (see [20]). So we have the following.

Corollary 21. Suppose and are differentiable at . Assume is continuous and strongly monotone. Moreover, assume and are relatively -functions. Suppose is a GCP function of and , which is based on the generalized Fischer-Burmeister function and . Then is a local minimizer to if and only if solves GCP.

Remark 22. When in Corollary 21, we get Proposition  3.4 in [22].

In a view of Example 2, Theorem 3, Corollary 3 in [24], and the above results, we get the following.

Corollary 23. Suppose and are semismooth (piecewise smooth or piecewise affine) at with Bouligand subdifferentials, respectively, by and . Assume is continuous, one-to-one, onto and consists of nonsingular matrices. Moreover, assume and are relatively -functions. Suppose is a GCP function of and , which is based on the generalized Fischer-Burmeister function and .
Then is a local minimizer to if and only if solves GCP.

Theorem 24. Suppose and are -differentiable at with -differentials, respectively, by and . Suppose is a GCP function of and . Assume that is -differentiable at with an -differential given by

or

Further suppose that consists of nonsingular matrices and consists of -matrices where . Then

Proof. The proof is similar to that of Theorem 9 in [15].

Remark 25. Theorem 24 is applicable to GCP functions of Examples 4–7 by the properties (i) and (iii) (or (i) and (iv)) in Lemma 12 and (i) and (iii) (or (i) and (iv)) in Lemma 13.

Since every positive definite matrix is also a -matrix, now we minimize the merit function under positive semidefinite/definite conditions.

Theorem 26. Suppose and are -differentiable at with -differentials, respectively, by and . Suppose is a GCP function of and . Assume that is -differentiable at with an -differential given by

or

Further suppose that consists of nonsingular matrices and consists of positive definite matrices where . Then

Now we replace the condition by weaker conditions or . In the next two successive theorems, of course, stronger/different conditions on the -differentials of and will be imposed. First, we have the following definition.

Remark 27. As noted in [15], a stationary point of the problem is a point such that where is an -differential of at . By weakening this condition, we may call a point a quasi-stationary point (semistationary point) of the problem if (resp., ). While local/global minimizers of are stationary points, it is not clear how to get or describe semi- and quasi-stationary points.

We will show that under appropriate conditions when is a semistationary point of with , then is a solution of a generalized complementarity problem. That is, starting with -differentiable functions and , we show that under appropriate conditions, a vector is a solution of the GCP if and only if zero belongs to .

Definition 28. Consider a nonempty set in . We say that a matrix is a row representative of if for each index , the th row of is the th row of some matrix . We say that has the row--property (row--property) if every row representative of is a -matrix (-matrix). We say that has the column--property (column--property) if has the row--property (row--property).

We have the result from [9].

Proposition 29. A set has the row--property (row--property) if and only if for each nonzero , there is an index such that and for all .

A simple consequence of this proposition is the following result in [15].

Proposition 30. The following statements hold.

(i)Suppose the set of matrices has the row--property. Then for any collection of nonnegative diagonal matrices, the sum

is a -matrix. In particular, any convex combination of the s is a -matrix.(ii)Suppose the set of matrices has the row--property. Then for any collection of nonnegative diagonal matrices with ,

is a -matrix.

Theorem 31. Suppose and are -differentiable at with -differentials, respectively, by and . Suppose is a GCP function of and . Assume that is -differentiable at with an -differential given by

Further suppose has the column--property where . Then

Proof. Suppose ; then . Conversely, suppose . Then by Carathéodory’s theorem, there exist and scalars , , and for with such that

where . Then (58) can be rewritten as

where and for all . And (59) can reduce to

where . From (60), we have

Then

Now, we claim that . Assume the contrary that ; then for , from Proposition 29, there is an index such that and

Note that , and for ; then we have

for . Then we obtain

(65) contradicts (62). This proves that .

Remark 32. (i) When , the GCP reduces to NCP.
(ii) Theorem 31 is applicable to GCP functions of Examples 4–7 by the properties (ii) in Lemma 12 and (ii) in Lemma 13.

Concluding Remarks. We considered a generalized complementarity problem corresponding to -differentiable functions, with an associated GCP function and a merit function . In this paper, we showed under certain -conditions the global/local minimum or a stationary point of is a solution of GCP. For generalized complementarity problem based on the penalized Fischer-Burmeister function, our results give various results for generalized complementarity problem when the underlying functions are continuously differentiable (locally Lipschitzian, semismooth, and directionally differentiable) functions. For example, we have the following.

(i)When in this paper, then GCP reduces to nonlinear complementarity problem NCP and we get results for nonsmooth NCP based on the generalized Fischer-Burmeister function and its generalizations which seems to be new.(ii)When and are in which case and , our results will be true when the underlying functions are and get the results in [25].(iii)When is and (in which case we can let ), GCP reduces to nonlinear complementarity problem NCP, and the results of this paper will be valid for NCP.(iv)When is locally Lipschitzian with and , our results will be applicable to NCP when the underlying data are locally Lipschitzian.(v)Our results give various results for generalized complementarity problem when -norm is replaced by -norm (or when is an integer greater than ), we can state our results for GCP function based on the Fischer-Burmeister function and we get the following.

(a)When and are in which case and , Theorem 14 reduces to Theorem  3.2 in [26].(b)When is and (in which case we can let ), GCP reduces to NCP and the above result reduces to Prop. 3.4 in [27].(c)When is locally Lipschitzian with and , the above theorem reduces to a result for nonlinear complementarity problem by Fischer [28]. Moreover, when , our result extends/generalizes a result obtained by Geiger and Kanzow [29] for NCP under monotonicity of a function and by Jiang [30] under uniform -property of a directionally differentiable function.

To the best of our knowledge, solving nonsmooth GCP on the basis of the generalized Fischer-Burmeister function and its generalizations seems to be new.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Mohamed A. Tawhid is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). The postdoctoral fellowship of Wei-Zhe Gu is supported by NSERC. The research of Wei-Zhe Gu is supported in part by the National Natural Science foundation of China (Grant no. 11301375; Grant no. 71301118), Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20120032120076), and Tianjin Planing Program of Philosophy and Social Science (Grant no. TJTJ11-004).

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