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.
Let K : ℝ n ⇉ ℝ n be a set-valued map such that K ( x ) , for any x ∈ ℝ n , is a closed convex set in ℝ n . Let F i : ℝ n ⇉ ℝ n , i = 1, ⋯ , m be set-valued maps such that F i ( x ) , i = 1, ⋯ , m is convex and compact for all x ∈ ℝ n . Denote by
ℝ + n = { y = ( y 1 , ⋯ , y n ) ∈ ℝ n | y i ≥ 0, i = 1, ⋯ , n }
i n t ℝ + n = { y = ( y 1 , ⋯ , y n ) ∈ ℝ n | y i > 0, i = 1, ⋯ , n }
The set-valued vector quasi variational inequality (SVQVI) problem associated with F i , i = 1 , ⋯ , m and K, denoted by S V Q V I ( F i , i = 1, ⋯ , m ; K ) , consists of finding an x * ∈ K ( x * ) such that there exists
f i x * ∈ F i ( x * ) , i = 1, ⋯ , m
and
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * ) ,
where 〈 ⋅ , ⋅ 〉 denotes the inner product in ℝ n .
Throughout this work, we denote the solution set of S V Q V I ( F i , i = 1, ⋯ , m ; K ) by s o l ( S V Q V I ( F i , i = 1, ⋯ , m ; K ) ) .
When the set K ( x ) is a constant set K ¯ on ℝ n then S V Q V I ( F i , i = 1, ⋯ , m ; K ) reduces to the following strong vector variational inequality S V V I ( F i , i = 1, ⋯ , m ; K ¯ ) in [
Find an x * ∈ K ¯ such that there exists f i x * ∈ F i ( x * ) , i = 1, ⋯ , m and
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ¯ .
Note that if each F i , i = 1, ⋯ , m is a single-valued map, and K is a constant map K ¯ , then S V Q V I ( F i , i = 1, ⋯ , m ; K ) reduces to the weak Stampacchia vector variational inequality problem ( S V V I ) w studied in [
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. [
In 1980, Giannessi [
In this paper, we first proposed a gap function for S V Q V I ( F i , i = 1, ⋯ , 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.
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 S V Q V I ( F i , i = 1, ⋯ , m ; K ) 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 S V Q V I ( F i , i = 1, ⋯ , m ; K ) , 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.
In this section, we investigate S V Q V I ( F i , i = 1, ⋯ , m ; K ) via the scalarization approach of Mastroeni [
Define functions F 0 , F , F u : ℝ n ⇉ ℝ n by following
F 0 ( x ) = c o n v { F i ( x ) } i = 1 , ⋯ , m = { z ∈ ℝ n | z = ∑ i = 1 m λ i f i , f i ∈ F i ( x ) , λ i ∈ ℝ + , ∑ i = 1 m λ i = 1 }
F ( x ) = ∏ i = 1 m F i ( x )
F u ( x ) = ∪ i = 1 m F i ( x )
Lemma 2.1. Let F i ( x ) , i = 1 , ⋯ , m be nonempty subsets of ℝ n . Then
F u ( x ) ⊆ F 0 ( x ) = c o n v { F u ( x ) }
where c o n v means convex hull.
Proof. Note that for each i = 1 , ⋯ , m , F i ( x ) ⊆ c o n v { F i ( x ) } i = 1 , ⋯ , m , hence
F u ( x ) ⊆ F 0 ( x )
Moreover, F 0 ( x ) is convex, thus,
c o n v { ∪ i = 1 m F i ( x ) } = c o n v { F u ( x ) } ⊆ F 0 ( x )
Conversely, let x ∈ F 0 ( x ) . Then, there exist x i ∈ F i ( x ) ⊆ ∪ i = 1 m F i ( x ) , i = 1 , ⋯ , m and λ i ≥ 0 , i = 1 , ⋯ , m with ∑ i = 1 m λ i = 1 , such that x = ∑ i = 1 m λ i x i , implying x ∈ c o n v { ∪ i = 1 m F i ( x ) } . Hence the requisite result follows.
Proposition 2.1. [
The SSQVI associated with set-valued maps F 0 and K, denoted by S S Q V I ( F 0 , K ) , consists of finding an x * ∈ K ( x * ) such that there exists f 0 x * ∈ F 0 ( x * ) and
〈 f 0 x * , y − x * 〉 ≥ 0, ∀ y ∈ K ( x * )
Throughout this paper, the solution set of S S Q V I ( F 0 , K ) is represented by s o l ( S S Q V I ( F 0 , K ) ) .
Theorem 2.1. Consider the following
1) F i , i = 1 , ⋯ , m are nonempty, convex and compact valued maps.
2) K : ℝ n ⇉ ℝ n is closed, convex valued map.
Then, for each x ∈ ℝ n , s o l ( S S Q V I ( F 0 , K ) ) = s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) .
Proof. Let x * ∈ s o l ( S S Q V I ( F 0 , K ) ) . Then there exist f 0 x * ∈ F 0 ( x * ) such that
〈 f 0 x * , y − x * 〉 ≥ 0, ∀ y ∈ K ( x * )
By definition of F 0 , there exists λ ∈ ℝ + m , with ∑ i = 1 m λ i = 1 and f i x * ∈ F i ( x * ) , i = 1, ⋯ , m , such that
〈 ∑ i = 1 m λ i f i x * , y − x * 〉 ≥ 0 , ∀ y ∈ K ( x * )
which implies that, for every y ∈ K ( x * ) , there exists an index i y , such that
〈 f i y x * , y − x * 〉 ≥ 0
It follows that
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
so, s o l ( S S Q V I ( F 0 , K ) ) ⊂ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) .
Conversely, let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Hence, x * ∈ K ( x * ) , and there exists f i x * ∈ F i ( x * ) , i = 1, ⋯ , m , such that
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
thus, for each y ∈ K ( x * ) , there exists an index i y such that
〈 f i y x * , y − x * 〉 ≥ 0
Observe that f i y x * ∈ F 0 ( x * ) , hence for each y ∈ K ( x * ) , there exist f 0 * = f i y x * ∈ F 0 ( x * ) such that
〈 f 0 * , x * − y 〉 ≤ 0
Consequently,
sup y ∈ K ( x * ) min f 0 * ∈ F 0 ( x * ) 〈 f 0 * , x * − y 〉 ≤ 0
Under assumption (1) and by Proposition 2.1, F 0 ( x * ) is convex and compact which along with assumption (2) and the minmax theorem, yields
min f 0 * ∈ F 0 ( x * ) sup y ∈ K ( x * ) 〈 f 0 * , x * − y 〉 ≤ 0
Finally, there exists f 0 x * ∈ F 0 ( x * ) such that
〈 f 0 x * , y − x * 〉 ≥ 0, ∀ y ∈ K ( x * )
completing the requisite result.
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, [
Definition 3.1. A function g : ℝ n → ℝ is said to be a gap function for a S V Q V I ( F i , i = 1 , ⋯ , m ; K ) on any set K ⊆ ℝ n if it satisfies the following properties:
1) g ( x ) ≥ 0, ∀ x ∈ K ,
2) g ( x * ) = 0, x * ∈ K ⇔ x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) .
Consider the function g F 0 : ℝ n → ℝ defined by
g F 0 ( x ) = inf f x ∈ F 0 ( x ) sup y ∈ K ( x ) 〈 f x , x − y 〉 (1)
Theorem 3.1. Consider the following
1) F i , i = 1 , ⋯ , m are nonempty, convex and compact valued maps.
2) K : ℝ n ⇉ ℝ n is closed, convex valued map.
Then, g F 0 defined in (1) is a gap function for S S Q V I ( F 0 , K ) on K = { x ∈ ℝ n | x ∈ K ( x ) } .
Proof. Observe that, for x ∈ K , x ∈ K ( x ) which implies g F 0 ( x ) ≥ 0 .
Next for x * ∈ K , g F 0 ( x * ) = 0 if and only if
inf f x * ∈ F 0 ( x * ) sup y ∈ K ( x * ) 〈 f x * , x * − y 〉 = 0
By Proposition 2.1, since F 0 ( x * ) is compact set on K and x * ∈ K , there exists f 0 x * ∈ F 0 ( x * ) such that
sup y ∈ K ( x * ) 〈 f 0 x * , x * − y 〉 = 0
therefore, we have
〈 f 0 x * , y − x * 〉 ≥ 0, ∀ y ∈ K ( x * ) .
By invoking Theorem 2.1, x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) .
The function g F 0 is not differentiable, in general, an observation that leads to consider the regularized gap function.
For any θ > 0 , consider the function g θ F 0 : ℝ n → ℝ defined by
g θ F 0 ( x ) = inf f x ∈ F 0 ( x ) sup y ∈ K ( x ) ( 〈 f x , x − y 〉 − 1 2 θ ‖ x − y ‖ 2 )
If, for x ∈ ℝ n , each F i ( x ) , i = 1, ⋯ , m is a compact set and K ( x ) is a convex set, then by the minimax theorem
g θ F 0 ( x ) = sup y ∈ K ( x ) inf f x ∈ F 0 ( x ) ( 〈 f x , x − y 〉 − 1 2 θ ‖ x − y ‖ 2 ) = sup y ∈ K ( x ) h ( x , y )
where h ( x , y ) = inf f x ∈ F 0 ( x ) ( 〈 f x , x − y 〉 − 1 2 θ ‖ x − y ‖ 2 ) .
Since h ( x , ⋅ ) is a strongly concave function in y so has unique maxima over closed convex set K ( x ) , then follow from [
Note that if F 0 ( x ) is a singleton then this gap function reduces to the regularized gap function for QVI proposed by Taji [
Theorem 3.2. Consider the following
1) F i , i = 1 , ⋯ , m are nonempty, convex and compact valued maps.
2) K : ℝ n ⇉ ℝ n is closed, convex valued map.
Then, g θ F 0 is a gap function for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) over K .
Proof. Clearly, for x ∈ K , g θ F 0 ( x ) ≥ 0 .
Let g θ F 0 ( x * ) = 0 and x * ∈ K . Then,
inf f x * ∈ F 0 ( x * ) sup y ∈ K ( x * ) ( 〈 f x * , x * − y 〉 − 1 2 θ ‖ x * − y ‖ 2 ) = 0
Under assumption (1) and by Proposition 2.1, there exists f 0 x * ∈ F 0 ( x * ) such that
s u p y ∈ K ( x * ) ( 〈 f 0 x * , x * − y 〉 − 1 2 θ ‖ x * − y ‖ 2 ) = 0,
which implies
〈 f 0 x * , x * − y 〉 − 1 2 θ ‖ x * − y ‖ 2 ≤ 0, ∀ y ∈ K ( x * )
Take an arbitrary point z ∈ K ( x * ) , and define a sequence of vectors y k as
y k = x * + 1 k ( z − x * ) , k ∈ ℕ
K ( x * ) being convex, so y k ∈ K ( x * ) , k ∈ ℕ , therefore
〈 f 0 x * , x * − y k 〉 ≤ 1 2 θ ‖ x * − y k ‖ 2
which when k → ∞ yields
〈 f 0 x * , z − x * 〉 ≥ 0
Hence x * ∈ s o l ( S S Q V I ( F 0 , K ) ) , which implies that x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) also.
Conversely, let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Then, by Theorem 2.1, x * ∈ s o l ( S S Q V I ( F 0 , K ) ) . Hence x * ∈ K ( x * ) and there exists f 0 x * ∈ F 0 ( x * ) such that
sup y ∈ K ( x * ) 〈 f 0 x * , x * − y 〉 ≤ 0
therefore
g θ F 0 ( x * ) = inf f x * ∈ F 0 ( x * ) sup y ∈ K ( x * ) ( 〈 f x * , x * − y 〉 − 1 2 θ ‖ x * − y ‖ 2 ) ≤ 0
But g θ F 0 ( x * ) ≥ 0 , which gives g θ F 0 ( x * ) = 0 .
In previous section, we used the scalarization parameter λ in constructing S S Q V I ( F 0 , K ) and then studied the gap function for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) . It is interesting to ask whether one can develop a gap function for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) without taking help of S S Q V I ( F 0 , K ) . We make an attempt to construct such a gap function in the discussion to follow. But first a notation.
Let x , y ∈ K ( x ) and let f x = ( f 1 x , ⋯ , f m x ) ∈ F ( x ) . Then, f i x ∈ F i ( x ) , i = 1, ⋯ , m and denote
〈 f x , y − x 〉 = ( 〈 f 1 x , y − x 〉 , 〈 f 2 x , y − x 〉 , … , 〈 f m x , y − x 〉 ) ,
i.e., 〈 f i x , y − x 〉 is the ith component of the vector 〈 f x , y − x 〉 , i = 1, ⋯ , m .
Define a function g : ℝ n → ℝ such that
g ( x ) = inf f x ∈ F ( x ) sup y ∈ K ( x ) min 1 ≤ i ≤ m 〈 f i x , x − y 〉 (2)
Theorem 4.1. Consider the following
1) F i , i = 1 , ⋯ , m are nonempty, convex and compact valued.
2) K : ℝ n ⇉ ℝ n is closed, convex valued map.
Then, g defined in (2) is a gap function for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) on K = { x ∈ ℝ n | x ∈ K ( x ) } .
Proof. Since x ∈ K , so x ∈ K ( x ) which implies g ( x ) ≥ 0 .
Consider x * ∈ K . We observe that g ( x * ) = 0 if and only if there exists f x * ∈ F ( x * ) such that
sup y ∈ K ( x * ) min 1 ≤ i ≤ m 〈 f i x * , x * − y 〉 = 0
that is,
min 1 ≤ i ≤ m 〈 f i x * , y − x * 〉 ≥ 0, ∀ y ∈ K ( x * )
Equivalently,
〈 f x * , y − x * 〉 ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
Hence, x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) .
Proposition 4.1. For each x ∈ ℝ n , g F 0 ( x ) ≥ g ( x ) .
Proof. Let x ∈ ℝ n and f 0 x ∈ F 0 ( x ) . Then there exist f i x ∈ F i ( x ) or equivalently, f x = ( f 1 x , ⋯ , f m x ) ∈ F ( x ) and λ i ≥ 0 , i = 1 , ⋯ , m with ∑ i = 1 m λ i = 1 such that f 0 x = ∑ i = 1 m λ i f i . For any y ∈ K ( x ) ,
min 1 ≤ i ≤ m 〈 f i x , x − y 〉 ≤ 〈 ∑ i = 1 m λ i f i x , x − y 〉 = 〈 f 0 x , x − y 〉
It follows that
g ( x ) = inf f x ∈ F ( x ) sup y ∈ K ( x ) min 1 ≤ i ≤ m 〈 f i x , x − y 〉 ≤ inf f 0 x ∈ F 0 ( x ) sup y ∈ K ( x ) 〈 f 0 x , x − y 〉 = g F 0 ( x )
We now attend to our prime aim that to develop the error bounds for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) . We shall be needing the following concepts.
Definition 4.1. [
〈 f x − f y , x − y 〉 ≥ μ ‖ x − y ‖ 2 , ∀ f x ∈ F ( x ) , f y ∈ F ( y )
F is said to be monotone if the above inequality holds with μ = 0 . F is said to be strictly monotone if it is monotone and the strict relation in the above inequality holds when x ≠ y .
Remark 4.1. Let F 1 , F 2 : ℝ n ⇉ ℝ n be two set-valued maps with F 1 ( x ) ⊆ F 2 ( x ) for any x ∈ ℝ n . Note that, if F 2 is strongly monotone with modulus μ > 0 (respectively, monotone and strictly monotone) on ℝ n then, F 1 is also strongly monotone with modulus μ > 0 (respectively, monotone and strictly monotone) on ℝ n . Consequently, recall if F u is strongly monotone with modulus μ > 0 (respectively, monotone and strictly monotone) on ℝ n then, each F i , i = 1 , ⋯ , m is strongly monotone with modulus μ > 0 (respectively, monotone and strictly monotone) on ℝ n .
Remark 4.2. Note that if F u is strongly monotone with modulus μ on any set S ⊆ ℝ n then each F i is strongly monotone with modulus μ on S [
Definition 4.2. [
if y ∈ K ( x ) then x ∈ K ( y )
The following result provides an error bound in terms of scalar gap function (without scalarize parameter) under strong monotonicity of F u map and fixed pint symmetric K map.
Theorem 4.2. Let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Suppose the following hold
1) F i , i = 1 , ⋯ , m are nonempty, convex, compact valued.
2) K is closed, convex valued and fixed point symmetric map.
3) F u is strongly monotone with modulus μ > 0 on K .
Then, for x ∈ K ( x * ) , we have
‖ x − x * ‖ ≤ g ( x ) μ (3)
Proof. Since x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) , there exists f i x * ∈ F i ( x * ) , i = 1 , ⋯ , m such that
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
For y = x , we have
( 〈 f 1 x * , x − x * 〉 , 〈 f 2 x * , x − x * 〉 , ⋯ , 〈 f m x * , x − x * 〉 ) ∉ − i n t ℝ + m
Therefore, there exists an index i x such that f i x x * ∈ F i x ( x * ) ⊆ F u ( x * ) and
〈 f i x x * , x − x * 〉 ≥ 0 (4)
Now, from the definition of g ( x ) and by Proposition 2.1, there exists f x ∈ F ( x ) , f x = ( f 1 x , ⋯ , f m x ) such that
g ( x ) = sup y ∈ K ( x ) min 1 ≤ i ≤ m 〈 f i x , x − y 〉
which gives
g ( x ) ≥ min 1 ≤ i ≤ m 〈 f i x , x − y 〉 , ∀ y ∈ K ( x )
Since x ∈ K ( x * ) , by fixed point symmetric property of K , x * ∈ K ( x ) , thus taking y = x * in above inequality, we have
g ( x ) ≥ min 1 ≤ i ≤ m 〈 f i x , x − x * 〉 (5)
For f i x ∈ F i ( x ) ⊆ F u ( x ) , i = 1 , ⋯ , m , by strongly monotonicity of F u and (4), we get
g ( x ) ≥ min 1 ≤ i ≤ m 〈 f i x − f i x x * , x − x * 〉 + 〈 f i x x * , x − x * 〉 ≥ μ ‖ x − x * ‖ 2 (6)
Hence, for any x ∈ K ( x * ) ,
‖ x − x * ‖ ≤ g ( x ) μ
Remark 4.3. We observed that the strong monotonicity of F u (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 S V Q V I ( F i , i = 1 , ⋯ , m ; K ) :
For any x ∈ K ( x * ) and for any f x = ( f 1 x , ⋯ , f m x ) ∈ F ( x ) , there exists an index j ∈ { 1, ⋯ , m } , and f j x * ∈ F j ( x * ) satisfying (4) and
min 1 ≤ i ≤ m 〈 f i x − f j x * , x − x * 〉 ≥ μ ‖ x − x * ‖ 2 (7)
Hence the error bound given in (3) is valid for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) because under assumption (3) of Theorem 4.2, the set-valued maps F i , i = 1 , ⋯ , m always satisfy (7).
In particular, if K is a constant map K ¯ and each F i is a single-valued map, then (7) states that for any x ∈ K ¯ , there exists an index j such that
min 1 ≤ i ≤ m 〈 F i ( x ) − F j ( x * ) , x − x * 〉 ≥ μ ‖ x − x * ‖ 2 (8)
For instant, take F 1 , F 2 : ℝ ⇉ ℝ given as F 1 ( x ) = { 2 x } and F 2 ( x ) = { x − 1 2 } and K ¯ = [ − 1 , + 1 ] . For this, s o l ( S V V I ( F 1 , F 2 ; K ) ) = [ 0 , 1 2 ] . In this case F u is
not strongly monotone that means assumption (3) of Theorem 4.2 fails but the error bound Formula (3) remains valid because F 1 , F 2 satisfy (8).
In light of Proposition 4.1, the following is immediate.
Corollary 4.2.1. Let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Suppose the following hold
1) F i , i = 1 , ⋯ , m are nonempty, convex, compact valued.
2) K is closed, convex valued and fixed point symmetric map.
3) F u is strongly monotone with modulus μ > 0 on ℝ n .
Then, for x ∈ K ( x * ) ,
‖ x − x * ‖ ≤ g F 0 ( x ) μ
Similar to g F 0 , the gap function g is not differentiable leading to define the regularized gap function for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) .
For θ > 0 , define a function g θ : ℝ n → ℝ as
g θ ( x ) = inf f x ∈ F ( x ) sup y ∈ K ( x ) min 1 ≤ i ≤ m ( 〈 f i x , x − y 〉 − 1 2 θ ‖ x − y ‖ 2 ) (9)
For each x, define the function
φ ( x , y ) = min 1 ≤ i ≤ m ( 〈 f i x , x − y 〉 − 1 2 θ ‖ x − y ‖ 2 )
Here, φ ( x ,. ) is a strongly concave function of y. When K ( x ) is a closed convex set for any x ∈ ℝ n then, φ ( x ,. ) attains maximum at a unique point in K ( x ) . If F ( x ) is a compact set in ℝ m then, it follow from [
Theorem 4.3. Consider the following
1) F i , i = 1 , ⋯ , m are nonempty, convex and compact valued.
2) K : ℝ n ⇉ ℝ n is closed, convex valued map.
Then, g θ defined in (9) is a gap function for S V Q V I ( F i , i = 1 , ⋯ , m ; K ) over the set K .
Proof. Since x ∈ K , so x ∈ K ( x ) which implies g θ ( x ) ≥ 0 .
Let x * ∈ K . We observe that g θ ( x * ) = 0 if there exists f x * ∈ F ( x * ) such that
sup y ∈ K ( x * ) min 1 ≤ i ≤ m { 〈 f i x * , x * − y 〉 − 1 2 θ ‖ x − y ‖ 2 } = 0
By similar arguments given in Theorem 3.2, we can work out that
min 1 ≤ i ≤ m 〈 f i x * , x * − y 〉 ≤ 0, ∀ y ∈ K ( x * )
which is equivalent to
〈 f x * , y − x * 〉 ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
that is, x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) .
For the converse part, let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Then x * ∈ K ( x * ) and there exists f i x * ∈ F i ( x * ) , i = 1, ⋯ , m such that
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
Hence for any arbitrary but fixed z ∈ K ( x * ) , there exists an index i z , depending on z , and there exists f i z x * ∈ F i z ( x * ) , such that
〈 f i z x * , x * − z 〉 ≤ 0
In other words,
min 1 ≤ i ≤ m { 〈 f i x * , x * − y 〉 − 1 2 θ ‖ x * − y ‖ 2 } ≤ 0 , ∀ y ∈ K ( x * )
which implies
g θ ( x * ) = inf f x * ∈ F ( x * ) sup y ∈ K ( x * ) min 1 ≤ i ≤ m ( 〈 f i x * , x * − y 〉 − 1 2 θ ‖ x * − y ‖ 2 ) ≤ 0
We conclude that g θ ( x * ) = 0 , and hence the result follows.
Theorem 4.4. Let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Suppose the following hold
1) F i , i = 1 , ⋯ , m are nonempty, convex, compact valued.
2) K is closed, convex valued and fixed point symmetric map.
3) F u is strongly monotone with modulus μ > 0 on K .
Then, for θ > 1 2 μ and for any x ∈ K ( x * ) ,
‖ x − x * ‖ ≤ g θ ( x ) ( μ − 1 2 θ )
Proof. Since x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) , there exists f i x * ∈ F i ( x * ) , i = 1, ⋯ , m such that
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
Taking y = x ∈ K ( x * ) , we have
( 〈 f 1 x * , x − x * 〉 , 〈 f 2 x * , x − x * 〉 , ⋯ , 〈 f m x * , x − x * 〉 ) ∉ − i n t ℝ + m
There exists an index i x such that f i x x * ∈ F i x ( x * ) ⊆ F u ( x * ) , and
〈 f i x x * , x − x * 〉 ≥ 0 (10)
Proceeding along the lines of Theorem 4.2, we can easily obtain, for x ∈ K ( x * ) ,
g θ ( x ) ≥ min 1 ≤ i ≤ m 〈 f i x − f i x x * , x − x * 〉 + 〈 f i x x * , x − x * 〉 − 1 2 θ ‖ x − x * ‖ 2 ≥ ( μ − 1 2 θ ) ‖ x − x * ‖ 2
where the last inequality follows from strongly monotonicity of F u and (10), yielding the requisite result.
Aussel [
Definition 5.1. [
K ( x * ) ∩ B ( x * , δ ) ⊂ K ( x ) + B ( 0, L ‖ x − x * ‖ α )
where B represents a ball in ℝ n .
Remark 5.1. If K : ℝ n ⇉ ℝ n is a fixed point symmetric map over any set S ⊆ ℝ n then K will also be locally α-Holder ( α ≥ 1, L ≥ 1 ) at any point x ∈ S . However, the converse, in general, may not hold. For instance, consider Proposition 3.6 in [
K ( x ) = { y ∈ ℝ n | ϕ ( y ) ≤ ψ ( x ) }
where ϕ : ℝ n → ℝ is a continuously differentiable function and ψ : ℝ n → ℝ is an α-Holder continuous on ℝ n . Let x ∈ S be such that ∇ ϕ ( x ) ≠ 0 . Then for some constant γ (see Proposition 3.6 in [
Recall the map F u : ℝ n ⇉ ℝ n . For if F u is a compact valued map then define
M = s u p { ‖ f ‖ : f ∈ F u ( x ) , ∀ x ∈ B ¯ ( x * ,1 ) }
where B ¯ ( x * ,1 ) indicates the closed unit ball in ℝ n centered at x * .
Theorem 5.1. Let x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) . Suppose the following hold
1) F i , i = 1 , ⋯ , m are nonempty, convex, compact valued.
2) K is closed, convex valued and locally α-Holder with α > 2 at x * and δ ∈ ( 0,1 ) .
3) F u is strongly monotone with modulus μ > M L δ α − 2 > 0 .
Then, for any θ > ( L + 1 ) 2 2 ( μ − M L δ α − 2 ) , and for any
x ∈ B ( x * , δ ) ∩ K ( x * ) , ‖ x − x * ‖ ≤ g θ ( x ) ρ δ
where ρ δ = ( μ − M L δ α − 2 − ( L + 1 ) 2 2 θ ) .
Proof. Since x * ∈ s o l ( S V Q V I ( F i , i = 1 , ⋯ , m ; K ) ) , there exists f i x * ∈ F i ( x * ) , i = 1, ⋯ , m such that
( 〈 f 1 x * , y − x * 〉 , 〈 f 2 x * , y − x * 〉 , ⋯ , 〈 f m x * , y − x * 〉 ) ∉ − i n t ℝ + m , ∀ y ∈ K ( x * )
Taking y = x in above relation
( 〈 f 1 x * , x − x * 〉 , 〈 f 2 x * , x − x * 〉 , ⋯ , 〈 f m x * , x − x * 〉 ) ∉ − i n t ℝ + m
Hence, there exists an index i x and f i x x * ∈ F i x ( x * ) such that
〈 f i x x * , x − x * 〉 ≥ 0 (11)
Also,
g θ ( x ) = inf f x ∈ F ( x ) sup y ∈ K ( x ) min 1 ≤ i ≤ m ( 〈 f i x , x − y 〉 − 1 2 θ ‖ y − x ‖ 2 )
Using Proposition 2.1, there exists f x ∈ F ( x ) such that
g θ ( x ) ≥ min 1 ≤ i ≤ m ( 〈 f i x , x − y 〉 − 1 2 θ ‖ y − x ‖ 2 ) , ∀ y ∈ K ( x )
For any y ∈ K ( x ) , there exists an index i y and f i y x ∈ F i y ( x ) such that
g θ ( x ) ≥ 〈 f i y x , x − y 〉 − 1 2 θ ‖ y − x ‖ 2
Consequently,
g θ ( x ) ≥ 〈 f i y x , x − y 〉 − 1 2 θ ‖ y − x ‖ 2 = 〈 f i y x , x − x * 〉 + 〈 f i y x , x * − y 〉 − 1 2 θ ‖ y − x ‖ 2 = 〈 f i y x − f i x x * , x − x * 〉 + 〈 f i x x * , x − x * 〉 + 〈 f i y x , x * − y 〉 − 1 2 θ ‖ y − x ‖ 2 ≥ μ ‖ x − x * ‖ 2 + 〈 f i y x , x * − y 〉 − 1 2 θ ( ‖ y − x * ‖ + ‖ x * − x ‖ ) 2 , (12)
where the last inequality is due to assumption (3), (11) and triangular inequality of ‖ . ‖ .
Since K is locally α-Holder at x * , for all x ∈ B ( x * , δ ) ∩ K ( x * ) , we have
‖ x * − y ‖ ≤ L ‖ x * − x ‖ α , ∀ y ∈ K ( x )
Taking into account that ‖ x − x * ‖ < 1 , inequality (12), we have, for x ∈ B ( x * , δ ) ∩ K ( x * ) ,
g θ ( x ) ≥ μ ‖ x − x * ‖ 2 − M ‖ x * − y ‖ − 1 2 θ ( ‖ y − x * ‖ + ‖ x − x * ‖ ) 2 ≥ μ ‖ x − x * ‖ 2 − M L ‖ x − x * ‖ α − 1 2 θ ( L ‖ x − x * ‖ α + ‖ x − x * ‖ ) 2 = μ ‖ x − x * ‖ 2 − M L ‖ x − x * ‖ α − 1 2 θ ( L 2 ‖ x − x * ‖ 2 α + 2 L ‖ x − x * ‖ α + 1 + ‖ x − x * ‖ 2 ) ≥ ( μ − M L ‖ x − x * ‖ α − 2 − 1 2 θ − L θ − L 2 2 θ ) ‖ x − x * ‖ 2 ≥ ( μ − M L δ α − 2 − ( L + 1 ) 2 2 θ ) ‖ x − x * ‖ 2 = ρ δ ‖ x − x * ‖ 2 ,
where ρ δ = ( μ − M L δ α − 2 − ( L + 1 ) 2 2 θ ) .
Then, for all x ∈ B ( x * , δ ) ∩ K ( x * ) , if x ≠ x * we have g θ ( x ) > 0 because ρ δ > 0 , thus proving that x * is the unique solution of S V Q V I ( F i , i = 1 , ⋯ , m ; K ) over B ( x * , δ ) ∩ K ( x * ) .
Gupta, R. and Mehra, A. (2017) Gap Functions and Error Bounds for Set-Valued Vector Quasi Variational Inequality Problems. Applied Mathematics, 8, 1903-1917. https://doi.org/10.4236/am.2017.812135