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

and Applied Analysis 3 (c) relatively strongly monotone if there exists a constant μ > 0 such that ⟨f (x) − f (y) , g (x) − g (y)⟩ ≥ μ 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 2 ∀x, y ∈ Rn, (11) (d) relatively P0(P)-functions if for any x ̸ = y inR, max i:xi ̸ =yi [f (x) − f (y)] i [g (x) − g (y)] i ≥ (>) 0, (12) (e) relatively uniform (P)-functions if there exists a constant η > 0 such that for any x, y ∈ R,


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][4][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.
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.
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][18][19].Also, if () = , then GCP(, ) reduces to the nonlinear complementarity problem NCP().By taking in NCP() Abstract and Applied Analysis () =  +  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  :  2 → :  (, ) = 0 ⇐⇒  = 0,  ≥ 0,  ≥ 0. ( For the problem GCP(, ), we define Φ () = [ ( 1 () ,  1 ()) , . . .,  (  () ,   ()) , . . .,  (  () ,   ())] and we call Φ() a GCP function for GCP(, ).A function Ψ :   → [0, ∞) is said to be a merit function for GCP(, ) provided that the global minima of Ψ are coincident with the solutions of the original GCP(, ).We consider a GCP function Φ :   →   associated with GCP(, ) and its merit function solves GCP (, ) ⇐⇒ Φ () = 0 ⇐⇒ Ψ () = 0. ( 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 P 0 (P) conditions and column P 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  1 , semismooth, and locally Lipschitzian.

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  ⊆   , co  denotes the convex hull of  and co  denotes the closure of co .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 (1, ∞).
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  : Ω ⊆ R  → R  , where Ω is an open set in R  and  * ∈ Ω, we say that a nonempty subset ( * ), also denoted by   ( * ), of R × is an differential of  at  * if for every sequence   ∈ Ω converging to  * , there exist a subsequence    and a matrix  ∈ ( * ) such that We say that  is -differentiable at  * if  has an differential at  * .
(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  : Ω ⊆   →   is -differentiable at a point , then there exist a constant  > 0 and a neighborhood (, ) of  with Conversely, if condition (8) holds, then () := R × can be taken as an -differential of  at .
In [10], the following definition is introduced to generalize the concepts of monotonicity, P 0 -property, and their variants for function in [20].
We consider two cases.

Minimizing the Merit Function
In this section, we consider an NCP function Φ corresponding to GCP and let Ψ := (1/2)‖Φ‖ 2 , when the underlying functions  and  are -differentiable.
Theorem 10.Under each of the following conditions,  :   →   is a P 0 (P)-function.
(a)  is Fréchet differentiable on   and, for every  ∈   , the Jacobian matrix ∇() is a P 0 (P)-matrix.
(b)  is locally Lipschitzian on   and, for every  ∈   , the generalized Jacobian () consists of P 0 (P)-matrices.
Remarks.Based on some results in [20,24], we note the following.
(i) For P-conditions, the the converse statements in the above theorem are usually false.
(ii) For P 0 -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.
The following two Lemmas give favorable properties which will be needed in our results.
Lemma 12.We can easily see that Φ, , and  in Examples 4-6 satisfy the following properties.
Proof.The proof can be easily verified.
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  Ψ ().
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  : R  → R  and  : R  → R  are -differentiable at  with -differentials, respectively, by   () and   ().Suppose Φ is a GCP function of  and .Assume that Ψ := (1/2)‖Φ‖ 2 is -differentiable at  with an -differential given by Further suppose that   () consists of nonsingular matrices and () consists of positive semidefinite matrices where Proof.Since every positive semidefinite matrix is also a P 0matrix, the proof follows from Theorem 14.
If  is a monotone (strongly monotone)  1 , ∇() 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 Φ is a GCP function of  and , which is based on the generalized Fischer-Burmeister function and Then  is a local minimizer to Ψ if and only if  solves GCP (, ).Proof.Since  is one-to-one and onto and  and  are relatively P 0 -functions, by Lemma 9, the mapping  ∘  −1 is P 0 -function which implies ∇()∇() −1 is P 0 -matrix; see [20].The proof follows from Corollary 19.

In view of
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  1 (see [20]).So we have the following.
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 P-matrix, now we minimize the merit function under positive semidefinite/definite conditions.Theorem 26.Suppose  : R  → R  and  : R  → R  are -differentiable at  with -differentials, respectively, by   () and   ().Suppose Φ is a GCP function of  and .Assume that Ψ := (1/2)‖Φ‖ 2 is -differentiable at  with an -differential given by Further suppose that   () consists of nonsingular matrices and () consists of positive definite matrices where () := { −1 :  ∈   (),  ∈   ()}.Then Now we replace the condition 0 ∈  Ψ () by weaker conditions 0 ∈ co  Ψ () or 0 ∈ co  Ψ ().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 min () is a point  * such that 0 ∈ co   ( * ) where   ( * ) is an -differential of  at  * .By weakening this condition, we may call a point  * a quasi-stationary point (semistationary point) of the problem min () if 0 ∈   ( * ) (resp., 0 ∈ co   ( * )).While local/global minimizers of min () 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 min Ψ with Ψ := (1/2)‖Φ‖ 2 , 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 co  Ψ ( * ).
Definition 28.Consider a nonempty set C in R × .We say that a matrix  is a row representative of C if for each index  = 1, 2, . . ., , the th row of  is the th row of some matrix  ∈ C. We say that C has the row-P 0 -property (row-Pproperty) if every row representative of C is a P 0 -matrix (Pmatrix).We say that C has the column-P 0 -property (column-P-property) if C  = {  :  ∈ C} has the row-P 0 -property (row-P-property).
We have the result from [9].Proposition 29.A set C has the row-P 0 -property (row-Pproperty) if and only if for each nonzero  ∈ R  , there is an index  such that   ̸ = 0 and   ()  ≥ 0(> 0) for all  ∈ C.
A simple consequence of this proposition is the following result in [15].Proposition 30.The following statements hold.
(i) Suppose the set of matrices { 1 ,  2 , . . .,   } has the row-P 0 -property.Then for any collection { 1 ,  2 , . . .,   } of nonnegative diagonal matrices, the sum is a P 0 -matrix.In particular, any convex combination of the   s is a P 0 -matrix.
(ii) Suppose the set of matrices { 1 ,
(a)  and  are relatively (strictly) monotone if and only if ℎ is (strictly) monotone.(b) If  is Lipschitz-continuous and  and  are relatively strongly monotone then ℎ is strongly monotone.(c)  and  are relatively P 0 (P)-functions if and only if ℎ is P 0 (P)-function.(d) If  is Lipschitz-continuous and  and  are relatively uniform P-functions, then ℎ is uniform P-function.

of Some GCP Functions When the Underlying Functions Are 𝐻-Differentiable
Suppose that  and  are -differentiable at  with -differentials, respectively, by   () and   ().

Lemma 11 .
Suppose that ,  : R  → R  and  is one-toone and onto.Define ℎ : R  → R  where ℎ :=  ∘  −1 .Suppose that  and  are -differentiable at  with differentials, respectively, by   () and   () with   () consisting of nonsingular matrices.Denote  := .Then ℎ is -differentiable at  with  ℎ (), where Example 5 and the above results, we have the following.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 Φ is a GCP function of  and , which is based on the generalized Fischer-Burmeister function and Ψ := (1/2)‖Φ‖ 2 .Then  is a local minimizer to Ψ if and only if  solves GCP(, ).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 Φ satisfying the properties in Lemmas 12 and 13.For simplicity, we avoid dealing in such a generality.Suppose  :   →   and  :   →   are differentiable at . Suppose Φ is a GCP function of  and , which is the basis of the generalized Fischer-Burmeister function in Example 4 and Ψ := (1/2)‖Φ‖ 2 .If ∇() is nonsingular and the product ∇()∇() −1 is P 0 -matrix, then  is a local minimizer to Ψ if and only if  solves GCP (, ).Suppose  :   →   and  :   →   are differentiable at . Assume  is continuous, one-to-one, onto and ∇() is nonsingular.Moreover, assume  and  are relatively P 0 -functions.Suppose Φ is a GCP function of  and , which is based on the generalized square Fischer-Burmeister function in Example 4 and Ψ := (1/2)‖Φ‖ 2 .Then  is a local minimizer to Ψ if and only if  solves GCP(, ).

Corollary 21 .
Suppose  :   →   and  :   →   are differentiable at . Assume  is continuous and strongly monotone.Moreover, assume  and  are relatively P 0 -functions.Suppose Φ is a GCP function of  and , which is based on the generalized Fischer-Burmeister function and Ψ := (1/2)‖Φ‖ 2 .Then  is a local minimizer to Ψ if and only if  solves GCP(, ).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 P 0 -functions.Suppose Φ is a GCP function of  and , which is based on the generalized Fischer-Burmeister function and Ψ := (1/2)‖Φ‖ 2 .Suppose  : R  → R  and  : R  → R  are -differentiable at  with -differentials, respectively, by   () and   ().Suppose Φ is a GCP function of  and .Assume that Ψ := (1/2)‖Φ‖ 2 is -differentiable at  with an -differential given by Then  is a local minimizer to Ψ if and only if  solves GCP(, ).