Gap Functions and Error Bounds for Set-Valued Vector Quasi Variational Inequality Problems

One of the classical approaches in the analysis of a variational inequality problem is to transform it into an equivalent optimization problem via the notion of gap function. The gap functions are useful tools in deriving the error bounds which provide an estimated distance between a specific point and the exact solution of variational inequality problem. In this paper, we follow a similar approach for set-valued vector quasi variational inequality problems and define the gap functions based on scalarization scheme as well as the one with no scalar parameter. The error bounds results are obtained under fixed point symmetric and locally α-Holder assumptions on the set-valued map describing the domain of solution space of a set-valued vector quasi variational inequality problem.

When the set ( ) reduces to the following strong vector variational inequality ( ) in [1].

Find an *
x K ∈ such that there exists ( ) * * , 1, , x is a single-valued map, and K is a constant map K , then ( ) reduces to the weak Stampacchia vector variational inequality problem ( ) w SVVI studied in [2].
Quasi variational inequality (QVI) problems started with a pioneer work of Bensoussan and Lions in 1973.The terminology quasi variational inequality was coined by Bensoussan et al. [3].A QVI QVI is an extension of a variational inequality (VI) [4] in which the underlying set K depends on the solution vector x.
For further details on QVI and its applications in various domains, the readers can refer to [5] [6] [7] [8] and the references therein.
In 1980, Giannessi [9] introduced and studied vector variational inequality (VVI) in finite-dimensional Euclidean space.Chen and Cheng [10] studied the VVI in infinite-dimensional spaces and applied it to vector optimization problem. Lee et al. [11] [12], Lin et al. [13], Konnov and Yao [14], and Daniilidis and Hadiisawas [15] studied the generalized VVI and obtained some existence results.Very recently, Charitha et al. [2] presented several scalar-valued gap functions for Stampacchia and Minty-type VVIs.A good source of material on VVI is a research monograph [16].Motivated by the extension of VI to VVI, several researchers initiated the study of QVI for vector-valued functions, known as vector quasi variational inequalities (VQVI); see, for instance [11] [12] [13] [14] [15] and the references therein.
In this paper, we first proposed a gap function for ( ) , 1, , ; i SVQVI F i m K =  using a scalarization scheme and then developed another scalar-valued gap function for the same problem but without involving any scalar parameter.Under certain monotonicity conditions and fixed point symmetric assumptions, we developed the error bound results for both kinds of gap functions and their regularized counterparts.Further, we relaxed and replaced the fixed point symmetric condition by a locally α-Holder condition and obtained the same error bound results.Applied Mathematics We now briefly sketch the contents of the paper.In Section 2, we present a scalarization scheme.In Section 3, we develop the classical gap function and the regularized gap function for ( ) with the help of set-valued scalar quasi variational inequality (SSQVI).In Section 4, we introduce another scalar gap function and its regularized version for ( ) both free of any scalar parameter.We also develop the error bounds using fixed point symmetric hypothesis on the underlying map K.In Section 5, we showed that the same error bounds results can be obtained by relaxing the fixed point symmetric property by the α-Holder type hypothesis on K.

Scalarization
In this section, we investigate ( ) via the scalarization approach of Mastroeni [1] and Konnov [17].We introduce SSQVI for ( ) and establish an equivalence between them under certain conditions.
Proof.Note that for each , 1, , are nonempty, convex and compact valued maps. 2) : Then, for each Under assumption (1) and by Proposition 2.1, ( ) * 0 F x is convex and compact which along with assumption (2) and the minmax theorem, yields ( ) ( ) Finally, there exists ( ) completing the requisite result.

Gap Functions by Scalarization
One of the classical approaches in the analysis of VI and QVI and its different variants is to transform the inequality into an equivalent constrained or unconstrained optimization problem by means of the notion of gap function, please see, [5] [18] [19] and references cited therein.The gap functions have potential to play an important role in developing iterative algorithms for solving the inequality, analyzing the convergence properties and obtaining useful stopping rules for iterative algorithms.This prompted us to study and analyze different gap functions for ( )  is said to be a gap function for a ( )

2)
: Proof.Observe that, for ( ) The function 0 F g is not differentiable, in general, an observation that leads to consider the regularized gap function.

Regularized Gap Function by Scalarization
For any 0 θ > , consider the function 0 : ( ) K x is a convex set, then by the minimax theorem ( ) where ( ) Since ( ) , h x ⋅ is a strongly concave function in y so has unique maxima over closed convex set , 1, , are nonempty, convex and compact valued maps.

2)
: Under assumption (1) and by Proposition 2.1, there exists ( ) Take an arbitrary point ( ) , and define a sequence of vectors k y as ( )

Another Scalar Gap Functions for SVQVI
In previous section, we used the scalarization parameter in constructing ( ) 0 , SSQVI F K and then studied the gap function for ( ) It is interesting to ask whether one can develop a gap function for ( ) without taking help of ( ) 0 , SSQVI F K .We make an attempt to construct such a gap function in the discussion to follow.But first a notation.Let ( ) . Then, ( ), 1, , , is the i th component of the vector , , 1, ,

Classical Gap Function
Define a function Theorem 4.1.Consider the following 1) , 1, , are nonempty, convex and compact valued. 2) : Then, g defined in ( 2) is a gap function for ( ) Proof.Since x ∈  , so ( ) Consider * x ∈  .We observe that ( ) if and only if there exists Proof.Let n x ∈  and ( ) . Then there exist ( ) . For any ( ) We now attend to our prime aim that to develop the error bounds for ( ) . We shall be needing the following concepts.
F is said to be monotone if the above inequality holds with 0 µ = .F is said to be strictly monotone if it is monotone and the strict relation in the above inequality holds when x y ≠ .

F x F x ⊆
for any n x ∈  .Note that, if 2 F is strongly monotone with modulus 0 µ > (respectively, monotone and strictly monotone) on n  then,

1
F is also strongly monotone with modulus 0 µ > (respectively, monotone and strictly monotone) on n  .Consequently, recall if u F is strongly monotone with modulus 0 µ > (respectively, monotone and strictly monotone) on n  then, each , 1, , is strongly monotone with modulus 0 µ > (re- spectively, monotone and strictly monotone) on n  .Remark 4.2.Note that if u F is strongly monotone with modulus µ on any set n S ⊆  then each i F is strongly monotone with modulus µ on S [1].However, the converse, in general, may not hold.For instance, consider two maps 1 2 , : . Then, 1 2 , F F are strongly monotone on  with modulus 1 and 3 respectively.But for x K y ∈ The following result provides an error bound in terms of scalar gap function (without scalarize parameter) under strong monotonicity of u F map and fixed pint symmetric K map. 2)K is closed, convex valued and fixed point symmetric map.
3) u F is strongly monotone with modulus 0 µ > on  .Then, for ( ) , , , , , , , For y x = , we have ( ) , , , , , , Therefore, there exists an index x i such that ( ) ( ) Now, from the definition of ( ) g x and by Proposition 2.1, there exists For ( ) ( ), 1, , , by strongly monotonicity of u F and (4), we get ( ) Hence, for any ( ) We observed that the strong monotonicity of u F (that is, assumption (3)) is used only to obtain relation (6).A careful examination reveals that even the following condition can help us to achieve the same error bound for ( ) For any ( ) Hence the error bound given in ( 3) is valid for because under assumption (3) of Theorem 4.2, the set-valued maps , 1, , i F i m =  always satisfy (7).
In particular, if K is a constant map K and each i F is a single-valued map, then (7) states that for any x K ∈ , there exists an index j such that ( ) ( ) For instant, take 1 2 , : ( ) . In this case u F is not strongly monotone that means assumption (3) of Theorem 4.2 fails but the error bound Formula (3) remains valid because 1 2 , F F satisfy (8).
In light of Proposition 4.1, the following is immediate.
2) K is closed, convex valued and fixed point symmetric map.
3) u F is strongly monotone with modulus
2) K is closed, convex valued and fixed point symmetric map.
3) u F is strongly monotone with modulus 0 µ > on  .

Substitution of "Fixed Point Symmetric Assumption"
Aussel [5] obtained the error bounds for a SSQVI by replacing "fixed point symmetric" property on K by the Holder's type hypothesis which motivated us to see if the Holder's type hypothesis on K works for ( ) is a fixed point symmetric map over any set n S ⊆  then K will also be locally α-Holder ( x S ∈ .However, the converse, in general, may not hold.For instance, consider Proposition 3.6 in [5], where the constraints map K is defined, for any  is a continuously differentiable function and Then for some constant γ (see Proposition 3.6 in [5]), the constraint map K is locally γ-Holder at x S ∈ .Note that K is not necessary fixed point symmetric over S .
3) u F is strongly monotone with modulus a singleton then this gap function reduces to the regularized gap function for QVI proposed by Taji [19].Theorem 3.2.Consider the following 1) nonempty, convex, compact valued.
by fixed point symmetric property of K ,

Fg
, the gap function g is not differentiable leading to define the regularized gap function for