1. Introduction
In the mathematical optimization theory, the concept of variational inequality problems was initially introduced by Hartman and Stampacchia [
1]. Giannessi [
2,
3] introduced the vector-valued versions of Stampacchia [
4] and Minty [
5] variational inequality problems in the framework of Euclidean space. Due to several real-life applications of vector variational inequality problems, for instance, in traffic equilibrium problems [
6], vector equilibria [
7], and vector optimization problems [
8], vector variational inequality problems have been extensively studied by several researchers in various frameworks (see, for instance, [
9,
10,
11,
12,
13,
14,
15] and the references cited therein). Furthermore, the notion of hemivariational inequality problem was first introduced by Panagiotopoulos [
16], which is based on the properties of Clarke generalized subdifferential for locally Lipschitz functions. These problems have various applications in the field of science and engineering, such as structured analysis, nonconvex optimization, mechanics, and contact problems (see, for instance, [
17,
18]). As a result, hemivariational inequality problems emerged as a significant area of research (see, for instance, [
19,
20,
21] and the references cited therein).
Variational-hemivariational inequality problems are a fusion of variational and hemivariational inequality problems, which incorporate both convex and nonconvex functions. The initial conceptualization and formulation of variational-hemivariational inequality problems are attributed to the work of Motreanu and Rǎdulescu [
22]. Furthermore, these problems have several applications in mathematical modeling and contact mechanics (see, for instance, [
23,
24]). In recent years, the theory of variational-hemivariational inequality problems has gained significant attention from several researchers (see, for instance, [
17,
19,
25] and the references cited therein). Tang and Huang [
26] have studied existence results for variational-hemivariational inequality problems by employing the KKM theorem in reflexive Banach spaces. Moreover, Migórski et al. [
27] have investigated the existence and uniqueness of the solution for variational-hemivariational inequality problems in the setting of reflexive Banach spaces.
It is well known that gap functions play an important role in the study of existence of solutions to variational inequality problems, as one can transform a variational inequality problem into an optimization problem by employing gap functions (see, [
28]). Moreover, gap functions are used to derive the error bounds, which provide an upper estimate of the distance between an arbitrary point of the feasible set and the solution set of an optimization problem (see, [
29,
30]). The notions of gap functions and regularized gap functions for variational inequality problems have been introduced by Auslender [
31] and Fukushima [
32] in the setting of Euclidean space. Moreover, Yamashita and Fukushima [
30] introduced Moreau-Yosida type regularized gap functions for variational inequality problems and further, derived several global error bounds for the solution of the considered problem. Li and He [
33] have formulated gap functions and derived the existence results for generalized vector variational inequality problems under the monotonicity assumption on topological vector spaces. Various gap functions for vector variational inequality problems have been studied by Charitha et al. [
34] in the Euclidean space setting. Furthermore, Hung et al. [
35] have studied gap functions and global error bounds for variational-hemivariational inequality problems in the setting of reflexive Banach spaces. For a detailed exposition regarding gap functions and global error bounds for variational and vector variational inequality problems in various settings, we refer the readers to [
29,
36,
37,
38,
39] and the references cited therein.
Over the last few decades, it has been observed that various real-life optimization problems arising in the field of engineering and science can be effectively formulated in the setting of Riemannian manifolds rather than in the Euclidean space setting, see, for instance, [
40,
41] and the references cited therein. The extension of optimization concepts from the Euclidean space setting to the framework of Riemannian manifolds is associated with several crucial advantages, such as one can transform nonconvex optimization problems in the setting of Euclidean space into convex optimization problems in the framework of Riemannian manifolds. Moreover, constrained optimization problems in the Euclidean space setting can be reformulated as unconstrained optimization problems in the Riemannian manifold framework (see, for instance, [
42,
43] and the references cited therein). Udrişte [
41] introduced the notions of geodesic convex functions on Hadamard manifolds corresponding to the notions of convex functions in the setting of Euclidean space. For further details related to the extension of optimization techniques from linear spaces to Riemannian manifolds, we refer the readers to [
9,
13,
14,
44,
45] and the references cited therein.
In the setting of Hadamard manifolds, Németh [
46] introduced the concept of variational inequality problems and discussed the existence of solutions for the considered problem. Since then, numerous scholars have investigated variational and vector variational inequality problems in the framework of Hadamard manifolds, see, for instance, [
9,
13,
14,
38,
47] and the references cited therein. Chen and Huang [
10] have studied existence results for vector variational inequality problems by employing the KKM lemma in the framework of Hadamard manifolds. Moreover, Jayswal et al. [
44] have studied existence results and gap functions for nonsmooth mixed vector variational inequality problems via Clarke subdifferentials. Existence results for the solution of hemivariational inequality problems have been established by Tang et al. [
21] in the Hadamard manifolds setting. Further, the notions of gap functions and global error bounds for generalized mixed variational inequality problems have been studied by Li et al. [
48]. Ansari et al. [
49] have formulated gap functions and derived global error bounds for nonsmooth variational inequality problems in terms of bifunctions. Moroever, Hung et al. [
20] have derived global error bounds for the solution of considered mixed quasi-hemivariational inequality problems in the setting of Hadamard manifolds.
It is significant to note that several authors have studied existence results for the solutions of vector variational inequality problems and hemivariational inequality problems in various frameworks (see, for instance, [
10,
21,
26,
35,
44]). However, the existence results for the solution of WVVHVIP, which belongs to a broader class of vector variational as well as hemivariational inequality problems, have not been investigated before in the setting of Hadamard manifolds. Moreover, gap functions and global error bounds for variational inequality problems, vector variational inequality problems, and hemivariational inequality problems in the framework of manifolds have been studied by numerous researchers (see, for instance, [
20,
33,
34,
35,
44,
48,
49] and the references cited therein). However, the notions of gap functions and global error bounds for vector variational-hemivariational inequality problems involving nonsmooth locally Lipschitz functions, in particular, WVVHVIP, have not been studied yet in the framework of Hadamard manifolds. Furthermore, it is imperative to note that Hadamard manifolds, in general, are not equipped with a linear structure. Therefore, to cope up with the challenges associated with the nonlinear structure of Hadamard manifolds, several tools from Riemannian geometry will be applied to explore existence results, gap functions, and global error bounds for WVVHVIP via Clarke subdifferentials in the Hadamard manifolds setting.
Inspired by the results established in [
20,
30,
31,
34,
35], in this paper, we consider a class of nonsmooth weak vector variational-hemivariational inequality problems (WVVHVIP) in the framework of Hadamard manifolds. We introduce the notion of the generalized geodesic strong monotonicity of a set-valued vector field in the Hadamard manifold setting. By employing an analogous to KKM lemma, we establish the existence of the solutions for WVVHVIP without using any monotonicity assumptions. Moreover, we establish the uniqueness of the solution to WVVHVIP by using generalized geodesic strong monotonicity assumptions. We formulate several gap functions, in particular, Auslender type, regularized, and Moreau-Yosida type regularized gap functions for WVVHVIP to establish necessary and sufficient conditions for the existence of the solutions to the considered problem, namely, WVVHVIP. Furthermore, we derive the global error bounds for the solution of WVVHVIP by employing the Auslender type as well as regularized gap functions under the assumptions of generalized geodesic strong monotonicity.
The novelty and contributions of this paper are fourfold: Firstly, the existence results for the solution of WVVHVIP generalize the corresponding results derived by Chen and Huang [
10] from nonsmooth weak vector variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problems (WVVHVIP) in the Hadamard manifold setting. Secondly, the existence results for the solution of WVVHVIP generalize the corresponding results derived by Jayswal et al. [
44] from mixed weak vector variational inequality problems to a broader category of vector variational-hemivariational inequality problems. Thirdly, several results related to gap functions and global error bounds for WVVHVIP generalize the corresponding results derived in [
30,
31,
32] from the Euclidean space setting to the framework of Hadamard manifolds, as well as generalize them from variational inequality problems to WVVHVIP, which belongs to a broader class of vector variational and hemivariational inequality problems. Fourthly, the gap functions and error bounds results investigated in this article generalize various corresponding results derived by Charitha et al. [
34] from vector variational inequality problems to WVVHVIP and from the setting of Euclidean space to the framework of Hadamard manifolds. However, existence results, uniqueness results, as well as gap functions and global error bounds for WVVHVIP, have not been investigated before in the setting of Hadamard manifolds. Consequently, the results established in this paper are applicable to study more general classes of vector variational inequality problems and hemivariational inequality problems, as compared to the results existing in the available literature (see, for instance, [
21,
29,
31,
35,
44] and the references cited therein).
The rest of this article is structured as follows. In
Section 2, we recall some basic definitions and mathematical preliminaries related to Hadamard manifolds. By employing an analogous to the KKM lemma, we establish the existence results for the solutions of WVVHVIP in
Section 3. Moreover, a uniqueness result for the solution of WVVHVIP is established under assumptions of generalized geodesic strong monotonicity. Furthermore, we formulate various gap functions and regularized gap functions, in particular, Auslender, regularized, and Moreau-Yosida type regularized gap functions for WVVHVIP in
Section 4. Subsequently, in
Section 5, we derive the global error bounds for the solution of WVVHVIP in terms of the Auslender type and regularized gap functions under generalized geodesic strong monotonicity hypotheses. Finally, in
Section 6, we draw conclusions and suggest all possible future research avenues.
2. Notations and Mathematical Preliminaries
The following definitions and fundamental concepts related to Riemannian and Hadamard manifolds are from [
41,
50,
51].
Let and denote the dimensional Euclidean space and the non-negative orthant of , respectively. The notations and are employed to denote the standard inner product in and interior of , respectively. The notation ∅ is employed to denote an empty set.
Let
be the symbol used to represent an
dimensional connected Riemannian manifold endowed with the Riemannian metric
The tangent space at
is denoted by
which is a vector space of dimension
n over
The dual space of
is represented by
The tangent bundle
is a disjoint union of tangent spaces at all points in the Riemannian manifold
. That is,
The symbols
and
are used to represent the inner product and its associated norm on the tangent space
for every
respectively. Let
be a piecewise differentiable curve joining
u and
v in
. That is,
and
. The length of the curve
is denoted by
, and is defined as:
where
Moreover, the Riemannian distance between
u and
v in
is defined as follows:
A differentiable curve
is said to be a geodesic if its vector field
is parallel to itself, that is,
where ∇ is the unique Levi-Civita connection on
. Furthermore, a geodesic curve joining
is termed as a minimal geodesic if its length is equal to the Riemannian distance between them. The Riemannian manifold
is called geodesically complete at
if every geodesic
emanating from
u is defined for every real number (see, [
50]). Moreover,
is said to be geodesically complete if it is geodesically complete at every point in
From Hopf-Rinow theorem (see, [
50]), every geodesically complete Riemannian manifold is a complete metric space, and there always exists a minimal geodesic between any two points in
A Riemannian manifold is called a Hadamard manifold if it is simply connected, complete, and has non-positive sectional curvature everywhere.
From now onwards, we assume that is an n-dimensional Hadamard manifold.
For any , the exponential map is given by where is the unique geodesic such that and Moreover, the inverse of the exponential map satisfies where is the zero tangent vector in tangent space In addition, for any , the distance between u and v is The parallel transport from u to v along a geodesic is a linear isometry . The symbol is used to represent a parallel transport from u to v along a minimal geodesic joining u and
A function
is known as a locally Lipschitz function at
with rank
K, if there exists a neighborhood
U of
u such that
A function is known as locally Lipschitz on with rank K if it is locally Lipschitz at every with rank K.
The proof of the following lemma will follow on the lines of the proof of Theorem 6 in [
52].
Lemma 1. Let be a finite index set and the functions be lower semicontinuous. Then, the function , defined byis also lower semicontinuous on In the following definition, we recall the notion of a geodesic convex set from Udrişte [
41].
Definition 1. Let and be any two points in Then, is said to be a geodesic convex set, provided that the following inclusion holds: The notion of the geodesic convex hull of a subset of
is recalled in the following definition (see, for instance, [
51,
53]).
Definition 2. Let be a nonempty set. The geodesic convex hull of a set is the smallest geodesic convex subset of containing
Remark 1. The geodesic convex hull of a set can be defined as the intersection of all the geodesic convex sets containing (see, for instance, [51,53]). From now onwards, the geodesic convex hull of a set is represented by unless specified otherwise.
The notion of the geodesic convex combination of a finite number of elements in
is recalled in the following definition (see, for instance, [
49,
53]).
Definition 3. Let be a finite number of elements chosen arbitrarily from Then, the geodesic convex combination of is the geodesic joining to any geodesic convex combination of , denoted by , and is defined as follows:for every with A relationship between geodesic convex hull, geodesic convex set, and geodesic convex combination is provided in the following lemmas (see, [
53]).
Lemma 2. Let Then is said to be a geodesic convex set if and only if it contains all the geodesic convex combinations of its elements.
Lemma 3. Let be any set. Then, geodesic convex hull of consists of all the geodesic convex combinations of elements of
In the following definitions, we recall the notions of generalized directional derivative and generalized subdifferential of a locally Lipschitz function from Bento et al. [
54].
Definition 4. Let be a locally Lipschitz function on .
- (i)
For any the generalized directional derivative of at u in some direction is denoted by , and is defined as follows:where is the differential of exponential map at - (ii)
The generalized subdifferential of function at is denoted by , and is given by:
In the following lemmas, we discuss some basic properties of the generalized directional derivative and generalized subdifferential for a locally Lipschitz function from Hosseini and Pouryayevali [
55].
Lemma 4. Let be a locally Lipschitz function on with rank K. Then, the following conditions hold:
- (i)
The function is finite, positively homogeneous and subadditive on tangent space for every Moreover, satisfies the following condition: - (ii)
is upper semicontinuous as a function of and, as a function of ν alone, is Lipschitz of rank K on .
- (iii)
, for every and
Lemma 5. Let be a locally Lipschitz function on with rank K and be an arbitrary element. Then:
- (i)
The Clarke subdifferential is a nonempty, convex, compact subset of and , for every
- (ii)
Let and be arbitrary sequences in and respectively, such that for each i, and converges to u. Further, we assume that ζ is a cluster point of the sequence Then, we have
The following definition of a geodesic convex function is from Udrişte [
41].
Definition 5. Let be a geodesic convex set and be any real-valued function. Then, is said to be a geodesic convex function on if for any , the following condition holds: Remark 2. It is worth noting that α-strongly geodesic convex () and strictly geodesic convex functions are geodesic convex functions on a geodesic convex set (see, [41,56]). Now, we furnish an example of a real-valued geodesic convex function defined on the Poincaré half-plane.
Example 1. Consider the following Poincaré half-plane:Furthermore, is a Hadamard manifold of dimension 2 and has constant negative sectional curvature (see, [41]). The tangent space at any point is . The Riemannian metric induces an inner product on , for all . That is,where . For any arbitrary element and , the exponential map is given by:
If If whereThe inverse of exponential map is given by:whereLet and . Then the unique minimal geodesic joining u and w is given by:This implies that is a geodesic convex set. Now, we define a function as follows:Let and ThenTherefore, is a geodesic convex function on . Let us define a set-valued vector field
, such that
for every
The domain of
, denoted by
is defined as follows:
The graph of set-valued vector field
, denoted by
is defined as follows:
In the following definition, we introduce the generalized version of geodesic strong monotonicity of a set-valued vector field in the setting of Hadamard manifolds. For further details related to monotone vector fields, we refer the readers to [
57,
58].
Definition 6. Let be any set-valued vector field and Then, is said to be geodesic strongly monotone with respect to σ, if there exists a positive constant such that for every and for every the following condition holds: Remark 3. - 1.
If then the geodesic strong monotonicity of with respect to σ reduces to the strong monotonicity of the vector field presented by Li et al. [57]. - 2.
If then geodesic strong monotonicity of set-valued vector field with respect to σ reduces to the corresponding definition of strong monotonicity presented by Barani [59]. - 3.
If is a finite-dimensional real Hilbert space and then , , . In this case, geodesic strong monotonicity of with respect to σ reduces to the corresponding definition of strong monotonicity presented by Tang and Huang [29].
Now, we provide an example of a geodesic strong monotone vector field with respect to in the framework of real symmetric positive definite matrices.
Example 2. Let and denote the sets of all real symmetric positive definite matrices and real symmetric matrices of order respectively. For any matrix and denote the trace and determinant of matrix U, respectively. Equivalently, can be defined as follows:Moreover, is a Riemannian manifold endowed with the following Riemannian metric (see, [42])From [42], it follows that is a Hadamard manifold with tangent space Therefore, which is a nonempty set. The exponential map for is defined as follows:The inverse of the exponential map is defined as follows:where Log denotes the usual logarithmic function on Let be a real-valued function. Then, the Riemannian gradient of function is given as follows (see, for instance, [42]):where denotes the Euclidean gradient of at Let I be an identity matrix of order , and let be defined as follows:Let Then, one can verify that for every and for every there exists a positive constant such that the following inequality holds:Therefore, is a geodesic strongly monotone vector field with respect to The following lemma from Li et al. [
60] will be used in the sequel.
Lemma 6. Let and z be arbitrary elements of such that Then, one has Remark 4. In view of Lemma 6, we have (see, for instance, [48]) The following lemma will be employed in the sequel (see, for instance, [
48]).
Lemma 7. Let the set-valued vector field and the function be given. Moreover, if we assume that h and are upper semicontinuous and the values of are compact, then the function defined byis upper semicontinuous. Now, we recall the definition of the KKM map and an analogous to the KKM lemma from Zhou and Huang [
61]. For further details, we refer to [
62].
Definition 7. Let be a closed geodesic convex set and be any finite set. Further, assume that is a set-valued map. Then, is known as a KKM map if the following inclusion holds: Lemma 8. Let be a closed geodesic convex set and be a KKM map. Further, we assume that the following conditions hold:
- (i)
is a closed set for every .
- (ii)
There exists , such that is a compact set.
3. Existence and Uniqueness Results for WVVHVIP
In this section, by employing an analogous to the KKM lemma, we derive the existence of the solutions for WVVHVIP via Clarke subdifferentials. Moreover, the uniqueness of the solution of WVVHVIP is established under generalized geodesic strong monotonicity assumptions.
In the rest of the paper, we assume that the following conditions hold:
- (B1)
Let be a nonempty, closed, and geodesic convex subset of . Moreover, we assume that .
- (B2)
Let be an index set and be locally Lipschitz functions on for every .
- (B3)
Let for every be a continuous function.
Now, we consider a nonsmooth weak vector variational-hemivariational inequality problem in the framework of Hadamard manifolds as follows:
Nonsmooth Weak Vector Variational-Hemivariational Inequality Problem (WVVHVIP): Find
, such that there exist
, satisfying:
for all
Remark 5. - 1.
It is worthwhile to note that if for every is a constant function, then Therefore, if for every , is a constant function, and then WVVHVIP reduces to the mixed weak vector variational inequality problem of the form: Find and such that as considered by Jayswal et al. [44]. - 2.
If for every , is a constant function, then WVVHVIP reduces to weak Stampacchia vector variational inequality problem (WSVVIP) as discussed in [9,10]. - 3.
If , for every and where is a single-valued vector field, then WVVHVIP reduces to a hemivariational inequality problem (HVIP()) of the form: Find such that as considered by Tang et al. [21]. - 4.
If for every , is a single-valued vector field foe every , and is a constant function, then In this case, WVVHVIP reduces to a variational inequality problem of the form: Find such that as introduced by Németh [46]. - 5.
If is a single-valued vector field for every , and , then WVVHVIP reduces to the weak Stampacchia vector variational inequality problem (SVVI)w of the form: Find such that as considered by Charitha et al. [34]. - 6.
If , , and is a constant function, then Moreover, if is a single-valued vector field for every , then WVVHVIP reduces to the variational inequality problem of the form: Find such that as considered by Yamashita and Fukushima [30].
From now onwards, we assume that is a nonempty, compact, and geodesic convex subset of , unless specified otherwise.
In the following theorem, we establish the existence of the solutions to WVVHVIP without relying on the monotonicity assumption on
Theorem 1. Let for every be a geodesic convex function on for every . Moreover, we suppose that for every and for every the following setis a geodesic convex set. Then, WVVHVIP has a solution in Proof.
Let
be an arbitrary element. A set-valued map
is defined as follows:
Notably,
is a nonempty set for every
.
To prove the existence of a solution for WVVHVIP, we divide the proof into two parts:
- (i)
In this part, we prove that
is a KKM map. On the contrary, we suppose that there exists a finite set
such that for some
we have
This implies that for every
and for every
, the following inclusion relation holds:
Hence, for every
,
, and for every
, we have
Let us consider the following set, which is defined as follows:
Notably,
is a nonempty subset of
. From the given hypotheses,
being an intersection of geodesic convex sets is a geodesic convex set. Therefore,
which implies that
for every
which is a contradiction. Therefore,
is a KKM map.
- (ii)
In this part, we show that is a closed set-valued map for every
Let
and
such that
as
Then there exist
satisfying:
Employing continuity of
upper semicontinuity of
and Lemma 5, there exist
such that
Therefore,
Moreover,
and
is a compact set implies that
is a bounded set for every
. In view of (ii),
is a closed set for every
. Therefore,
is a KKM map such that
is a compact set for every
. From Lemma 8, there exists
such that
Therefore, there exist
such that
for every
. Hence,
is a solution of WVVHVIP. This completes the proof. □
Remark 6. Theorem 1 generalizes Theorem 3.5 derived by Jayswal et al. [44] from mixed weak vector variational inequality problem to weak vector variational-hemivariational inequality problem, namely WVVHVIP. Moreover, Theorem 1 is applicable to a broader class of weak vector variational inequality problems and hemivariational inequality problems as the proof of Theorem 1 has not utilized the monotonicity assumption as employed in prior works (see, for instance, [35,44]). In the following example, we illustrate the significance of Theorem 1.
Example 3. Let us consider the Hadamard manifold as considered in Example 2.
Let I be an identity matrix of order and be a nonempty, compact, and geodesic convex subset of . Moreover, let be defined as follows:where The Clarke subdifferentials of and are given by:Define as follows:It can be verified that, are geodesic convex functions on for every The functions are given by:The Clarke generalized directional derivatives of and are given by:Consider the following nonsmooth weak vector variational-hemivariational inequality problem (WVVHVIP1): Find such that there exist satisfying:for any and It can be verified that for every and for every the following setis a geodesic convex set. Moreover, is a geodesic convex function for every Therefore, all the hypotheses of Theorem 1 are satisfied, which concludes that there exists , such that is a solution of WVVHVIP1. Furthermore, one can verify that is a solution of WVVHVIP1. In the following theorem, we establish the uniqueness of the solution to WVVHVIP under generalized geodesic strong monotonicity assumptions.
Theorem 2. Let for every , be a geodesic convex function on for every . Further, we assume that the following conditions hold:
- (i)
The set-valued vector field is geodesic strongly monotone with respect to σ having a positive constant
- (ii)
There exists a constant such that for every and
- (iii)
There exists a constant such that for all and
Furthermore, we assume that Then WVVHVIP has a unique solution.
Proof.
In view of Theorem 1, there exists at least one solution of WVVHVIP. On the contrary, we suppose that
and
in
are two distinct solutions of WVVHVIP. Therefore, there exist
,
, and
such that the following inequalities hold:
On adding (
7) and (
8) we get
Therefore, from the given hypotheses (i)–(iii), we obtain the following inequality
In view of the given hypotheses, we have
Then from (
9), it follows that
. This completes the proof. □
Example 4. Let be the same manifold as considered in Example 1.
Let be a nonempty, compact and geodesic convex subset of . Further, let be defined as follows:The Clarke subdifferentials of and at are given as follows:Let us define as follows:Let be two real-valued functions defined as follows:Then, the generalized directional derivatives of and at in the direction for any are given as:Now, we consider the following nonsmooth weak vector variational-hemivariational inequality problem for : (WVVHVIP2): Find , such that there exist for every satisfying:for every It can be verified that is geodesic strongly monotone with respect to σ with a positive constant Moreover, all the hypotheses (ii)–(iii) of Theorem 2 are satisfied with and Therefore, there exists a unique solution of WVVHVIP2.
4. Gap Functions for WVVHVIP
In this section, we formulate various gap functions, in particular, Auslender, regularized, and Moreau-Yosida type regularized gap functions for WVVHVIP. These gap functions are employed to establish the necessary and sufficient conditions for the existence of the solutions to WVVHVIP.
In the following definition, we provide the definition of a gap function for WVVHVIP.
Definition 8. A function is called a gap function for WVVHVIP, if it satisfies the following conditions:
- (i)
- (ii)
is a solution of WVVHVIP if and only if
Now, we consider function
, which is defined as follows:
where
, and
For
and
, we let
In the following theorem, we prove that the function
defined in (
10) is a gap function for WVVHVIP.
Theorem 3. Let for every and , be a geodesic convex function on . Then the function is a gap function for WVVHVIP.
Proof.
On the contrary, we assume that there exists
such that
In view of the fact that
is a compact set and
is a continuous function for every
, there exists
, satisfying:
It follows that for every
, we have
which is a contradiction for
Therefore, we have
Let
for some
. Since
is a compact set and
is continuous, there exists
for every
such that the following condition holds:
It follows that for every
, we have
Therefore, for any
there exists some
such that the following inequality holds:
Applying the parallel transport from
w to
in (
12), we have
Therefore, we have
for every
This implies that
is a solution of WVVHVIP.
For the converse part, we assume that
is a solution of WVVHVIP. This implies that there exist
, satisfying:
. It follows that for every
there exists some
such that
Applying parallel transport from
to
w in (
13), we infer that
Hence, for every
, we get the following inequality
On taking supremum over
in (
14), we have
From (
11) and (
15), it follows that
This completes the proof. □
Remark 7. - 1.
It is worthwhile to note that if for every is a constant function, then In addition to this, if and , then Theorem 3 reduces to Theorem 3.3 deduced by Jayswal et al. [44]. - 2.
Let be a constant function . Moreover, if is a single-valued vector field then Theorem 3 reduces to Theorem 4.1 established by Charitha et al. [34]. - 3.
If , , , and is a constant function, then Moreover, if is a single-valued vector field for all , then Theorem 3 reduces to Lemma 2.1 derived by Yamashita and Fukushima [30]. - 4.
It is worth noting that Hadamard manifolds, in general, represent a nonlinear space. For instance, for any However, in the framework of Euclidean space Therefore, the techniques that have been successfully employed in the context of linear spaces cannot be applied to the optimization problems defined on Hadamard manifolds. This limitation underscores the significant challenges associated with the development of optimization techniques in the framework of Hadamard manifolds.
Now, we provide a non-trivial example to illustrate the significance of Theorem 3.
Example 5. Let us consider the sets and functions as defined in Example 3. Now, we define a function for as follows:Let such that . Then, the function Π for is given by:For such that the function is given by:Let such that Then, we haveNotably, if and only if Therefore, Π is a gap function for WVVHVIP1. Now, we introduce a regularized gap function for WVVHVIP. Let
be any arbitrary parameter and a function
be defined as follows:
where
, and
For any
and
, we let
In the following theorem, we establish that the function
defined in (
16) is a gap function for WVVHVIP.
Theorem 4. Let be geodesic convex functions on for every . Then, for any , the function is a gap function for WVVHVIP.
Proof.
On the contrary, we suppose that there exists
such that
Since
is continuous and
is a compact set for every
, this implies that there exist
such that
It follows that for every
, we have
which is a contradiction for
Therefore, we have
Let
for some
. In view of the continuity of the function
and the compactness of the set
, there exist
such that
That is, for every
, we have
In view of the fact that
for every
we get from (
19)
Now consider an arbitrary but fixed . For let . Since is a geodesic convex set, it is evident that .
In view of (
20) and the fact that
, we have
By employing geodesic convexity assumptions on
, positive homogeneous property of
, and
in (
21), it follows that
In light of the fact that parallel transport preserves the inner product, we yield the following from (
22):
Letting
in (
23), we have
This implies that there exists
depending on
such that
By following the similar steps for any
, we have
Therefore, is a solution of WVVHVIP.
For the converse part, we assume that there exists a solution
of WVVHVIP in
. This implies that there exist
such that the following condition holds:
for every
. Equivalently, for every
, there exists some
such that we have
Applying parallel transport from
to
w in (
25) we get
From (
26) it follows that
From (
17) and (
27) we have
This completes the proof. □
Remark 8. - 1.
If is a single-valued vector field , and is a constant function, then Theorem 4 reduces to Theorem 4.2 deduced by Charitha et al. [34]. - 2.
The results derived in Theorem 4 generalize the corresponding results established in Theorem 3.1 deduced by Fukushima [32] from the Euclidean space to a more general space, namely Hadamard manifolds, as well as generalize them from variational inequality problem to a broader category of vector variational-hemivariational inequality problems, in particular, WVVHVIP.
In the following example, we illustrate the significance of Theorem 4.
Example 6. Consider the following Riemannian manifold (see, [47]):The Riemannian manifold is endowed with the following Riemannian metric (see, [41]):Further, is a Hadamard manifold and the tangent space at every is the set of real numbers. That is, (see, [41]). For any and , the exponential map is defined by:Moreover, the inverse of the exponential map is given by:Let be a nonempty, compact, and geodesic convex subset of . Further, let be defined as follows:The Clarke subdifferentials of and are given by:Let the functions be defined as follows:It can be verified that the functions are geodesic convex functions on . The functions are defined as follows:The generalized directional derivative of functions are given by:Let and Now, we define a function as follows:It can be verified that serves as a gap function for the following variational-hemivariational inequality problem (WVVHVIP2): Find such that there exist , satisfying:. In the following lemma, we prove the lower semicontinuity of the gap function
defined in (
16).
Lemma 9. Let for every the function be a geodesic convex function on for every . Then, is a lower semicontinuous function on .
Proof.
Let
. Then we define
where
The functions
,
are continuous and the function
is lower semicontinuous. Therefore, in view of Lemma 1, the function
defined in (
28) is a lower semicontinuous function in the argument
for every
Then, we have
is a lower semicontinuous function. This implies that—
is upper semicontinuous. In view of the fact that
is upper semicontinuous with compact values, we have from Lemma 7
is a lower semicontinuous function on
. □
Now, we define the Moreau-Yosida regularization of in the setting of Hadamard manifolds.
Let
be two fixed parameters and
be Moreau-Yosida regularization of
, which is defined as follows:
In the following theorem, we prove that the function
defined in (
30) is a gap function for WVVHVIP.
Theorem 5. Let for every and , be a geodesic convex function on . Then, for any is a gap function for WVVHVIP.
Proof. From Theorem 4,
Therefore, from (
30), we have
Let
be an arbitrary solution of WVVHVIP. This implies that
. Therefore,
From (
31) and (
32), it follows that
Conversely, let
be an element such that
That is,
Hence, it follows that there exists a minimizing sequence
such that
Therefore, there exists a sequence
such that
and
as
. Since,
is lower semicontinuous such that
, it follows that
Therefore, which in turn implies from Theorem 4 that is a solution of WVVHVIP. This completes the proof. □
Remark 9. Theorem 5 generalizes Theorem 2.4 deduced by Yamashita and Fukushima [30] from the Euclidean space setting to the framework of Hadamard manifolds, as well as from variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problem, namely, WVVHVIP. 5. Global Error Bounds for WVVHVIP
In this section, by employing the Auslender type, regularized, and Moreau-Yosida type regularized gap functions, we derive the global error bounds for the solution of WVVHVIP under generalized geodesic monotonicity assumptions and properties of Clarke subdifferential.
Theorem 6. Let for every and , be a geodesic convex function on . Further, we assume that the following conditions hold:
- (i)
The set-valued map is geodesic strongly monotone with respect to σ having a positive constant
- (ii)
There exists a constant such that for every and
- (iii)
There exists a constant such that for all and
Further, we assume that . Then, there exists a unique solution of WVVHVIP such that the following condition holds: Proof.
From the given hypotheses and Theorem 2, there exists a unique solution, namely
of WVVHVIP in
. This implies that there exist
satisfying:
for every
Equivalently, for every
there exists some
such that
From the definition of
in (
10), there exists
, satisfying:
From the given hypotheses (i)–(iii) and the fact that
is a solution of WVVHVIP, it follows that for every
there exists
such that the following inequalities hold:
From (
36)–(
38), the following inequality holds:
Hence, we complete the proof. □
Theorem 7. Let be a fixed parameter such that all the hypotheses in Theorem 6 are satisfied with Then, there exists a unique solution of WVVHVIP such that the following inequality holds: Proof.
From the given hypotheses and Theorem 2, there exists a unique solution, namely
of WVVHVIP in
. This implies that for every
there exist
satisfying:
Equivalently, for every
there exists some
such that
From the definition of
in (
16), there exist
, satisfying:
From the given hypotheses (i)–(iii) and the fact that
is a solution of WVVHVIP, it follows that for every
there exists
such that the following inequalities hold:
From (
40), (
41), and (
42), the following inequality holds:
Hence, we complete the proof. □
Remark 10. Theorem 6 generalizes Lemma 4.1 derived by Yamashita and Fukushima [30] from Euclidean space to an even more general space, namely Hadamard manifolds and from variational inequality problem to a more general problem, in particular, WVVHVIP. In the following example, we illustrate the significance of Theorem 6.
Example 7. Let be the manifold as considered in Example 1. Moreover, we consider as defined in Example 4.
Now, we define a function corresponding to as follows:On simplifying (43), it follows thatMoreover, if and only if Therefore, for is a regularized gap function for WVVHVIP2. Furthermore, and is the unique solution of WVVHVIP2. In view of Example 4, all the hypotheses of Theorem 6 are satisfied. Therefore, the following inequality holds:Equivalently, Theorem 8. Let and be fixed parameters, such that . Furthermore, we assume that all the hypotheses of Theorem 6 are satisfied. Then, there exists a unique solution of WVVHVIP such that the following inequality holds: Proof.
From the given hypotheses and Theorem 2, there exists a unique solution of WVVHVIP in , namely .
Let
w be an arbitrary element of
It follows from Theorem 7 that
From (
44) and (
45), the following conclusion holds:
Hence, we complete the proof. □
Remark 11. If then , for all Further, if , is a constant function, is a single-valued vector field and for all then Theorem 8 reduces to Theorem 4.3 deduced by Yamashita and Fukushima [30]. Now, we provide a non-trivial example to demonstrate the significance of Theorem 8.
Example 8. Consider as defined in Example 7. Then, the function corresponding to and is defined as follows:Moreover, if and only if Therefore, for is a Moreau-Yosida regularized gap function for WVVHVIP2. From Examples 4 and 7, all the hypotheses of Theorem 6 are satisfied, which leads to the following inequality:Equivalently, for every , the following conditions hold: 6. Conclusions and Future Research Directions
In this paper, we have investigated a class of nonsmooth weak vector variational-hemivariational inequality problems, namely, WVVHVIP in the setting of Hadamard manifolds. By employing an analogous to the KKM lemma, we have established the existence of solutions for WVVHVIP without using any monotonicity assumptions. Moreover, we have established the uniqueness of the solution to WVVHVIP under generalized geodesic strong monotonicity assumptions. We have formulated several gap functions for WVVHVIP, in particular, Auslender, regularized, and Moreau-Yosida type regularized gap functions. Subsequently, we have established necessary and sufficient conditions for the existence of the solutions to the considered vector variational-hemivariational inequality problem. Furthermore, the global error bounds for the solution of WVVHVIP have been derived in terms of the Auslender type, regularized, and Moreau-Yosida type regularized gap functions under generalized geodesic strong monotonicity assumptions of Clarke subdifferentials. Several non-trivial examples have been provided in the Hadamard manifold setting to demonstrate the significance of the established results.
The results derived in this paper extend and generalize several well-known results existing in the literature. In particular, the results derived in this article generalize the corresponding results established by Chen and Huang [
10] from nonsmooth weak vector variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problems in the framework of Hadamard manifolds. Various existence results for the solution of WVVHVIP generalize the corresponding results derived by Jayswal et al. [
44] from nonsmooth mixed weak vector variational inequality problems to nonsmooth weak vector variational-hemivariational inequality problems in the Hadamard manifold setting. Furthermore, the results established in the present article generalize the corresponding results derived in [
30,
31,
32] from the Euclidean space setting to the framework of Hadamard manifolds and from variational inequality problems to WVVHVIP. In addition, several results related to gap functions and global error bounds for WVVHVIP generalize the corresponding results derived by Charitha et al. [
34] from vector variational inequality problems to nonsmooth weak vector-variational hemivariational inequality problems and generalize them from the framework of Euclidean space to the setting of Hadamard manifolds.
The results derived in the present article can be applied to various real-life problems in the domains of mechanics and engineering, such as optimal control and frictional contact problems (see, [
23,
24]). Moreover, it is significant to observe that all the results derived in this paper use the linear property of the inner product in tangent spaces. However, several variational inequalities involve bifunctions (see, for instance, [
20,
49]), which may not necessarily be linear, then the results derived in the present article may no longer be applicable for nonsmooth vector variational-hemivariational inequalities involving bifunctions. We intend to address this limitation in our future research work.
The results established in the present article suggest various potential avenues for future research work. For instance, investigating gap functions and regularized gap functions for mixed vector quasi-variational-hemivariational inequality problems would be an interesting research problem. Furthermore, in view of the work of Oyewole [
63], we would like to develop subgradient extragradient-type algorithms for WVVHVIP in our future course of study.