Some aspects of nonsmooth variational inequalities on Hadamard manifolds

This is the first paper dealing with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using gap functions in the setting of Hadamard manifolds. We also provide some characterizations of these two sufficiency properties. We conclude the paper with a discussion of the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds.


Introduction
The variational inequality problem, introduced by Hartman and Stampacchia [20], plays an important role in current mathematical technology. It has been extended and generalized to study a wide class of problems arising in optimization, nonlinear programming, physics, economics, transportation equilibrium problems, and engineering sciences. For further details, see [1-4, 7, 8, 21-23, 25, 32, 36] and the references therein.
The weak sharp minimum property for the convex optimization problem was introduced and studied by Ferris [13,17]. It has significant applications in sensitivity analysis, error bounds, and finite convergence of algorithms for solving convex optimization problems, see [12,17,18,27]. In 1992, Ferris and Mangasarian [19] studied the minimum principle sufficiency property for convex programming and proved that this property is equivalent to the weak sharp minimum property of convex programming. Marcotte and Zhu [24] extended the minimum principle sufficiency property for variational inequalities and characterized the weak sharp solutions for variational inequalities using pseudomonotone + and continuous mapping on a compact polyhedral set. Extending the results of Marcotte and Zhu [24], Wu and Wu [35] established the maximum principle sufficiency property for variational inequalities and also characterized the weak sharp solutions of variational inequalities. Recently, Wu [34] characterized the minimum principle sufficiency property when the mapping in a variational inequality is constant and pseudomonotone + .
The variational inequality problem can be used to solve any differentiable optimization problem over a convex feasible region. However, in many practical problems, the function involved in the optimization problem need not be differentiable but has some kind of directional derivative. In this case, the relevant optimization problem can be considered by using a nonsmooth variational inequality with a bifunction, see [10]. In 2016, Alshahrani et al. [6] defined the minimum and maximum principle sufficiency properties for nonsmooth variational inequalities (NVI) by using a gap function that is similar to that of Wu and Wu [35] and provided some characterizations for these two properties. They also discussed error bounds for nonsmooth variational inequalities. For further related results, see [5,11,33] and the reference therein.
On the other hand, in the last two decades, several classical problems have been extended from a linear space setting to Riemannian/Hadamard manifolds setting (nonlinear space) because some nonconvex and constrained optimization problems in Euclidean space become convex and unconstrained ones in Riemannian/Hadamard manifolds, see [31]. In 2003, Németh [25] extended the notion of variational inequalities to Hadamard manifolds and related it to geodesic convex optimization problems. He also proved the existence and uniqueness results for variational inequalities on Hadamard manifolds. Then, Colao et al. [15] developed the equilibrium theory in Hadamard manifolds and also proved some existence results. In 2017, we [9] derived the generalized convexity of a real-valued function defined on a Riemannian manifold in terms of a bifunction h. We also defined the generalized monotonicity of the bifunction h. We also proved the relationship between the generalized convexity of a real-valued function and the generalized monotonicity of h. In 2020, we [8] extended the notion of the nonsmooth variational inequality problem (NVIP) and the Minty nonsmooth variational inequality problem (MNVIP) in the setting of Hadamard manifolds and proved some existence results for the nonsmooth variational inequality problem. We gave some relations among (NVIP), (MNVIP), and optimization problems in the setting of Hadamard manifolds. The gap functions for (NVIP) and (MN-VIP) were also studied in the setting of Hadamard manifolds.
Motivated by the above results, in the present paper, we extend the work of Alshahrani et al. [6] in the setting of Hadamard manifolds. Hence, we defined the minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using a gap function in the Hadamard manifolds setting and provide several characterizations for these two properties. We also study the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds.
Let M be a finite-dimensional differentiable manifold. For each x ∈ M, let T x M be the tangent space at the point x to M, which is a real vector space of the same dimension as M. The collection of all tangent spaces on M is called a tangent bundle and it is denoted by TM. A C ∞ mapping A : M → TM, which assigns a tangent vector A(x) at x for each x ∈ M, is called a vector field on M. We denote by ·, · x the scalar product on T x M with the associated norm · x , where the subscript x can be omitted if there is no confusion.
A scalar product ·, · x is called the Riemannian metric on T x M. A C ∞ tensor field g of type (0, 2) on M is called the Riemannian metric on M if for each x ∈ M, the tensor g(x) is a Riemannian metric on T x M, and the pair (M, g) is called the Riemannian manifold.
For any x, y ∈ M, let γ : [0, 1] → M be a piecewise-smooth curve joining x to y. The arc length of γ is defined by whereγ (t) denotes the tangent vector at γ (t). For any x, y ∈ M, the Riemannian distance from x to y is defined by where the infimum is taken over all piecewise-smooth curves γ : [0, 1] → M joining x to y.
Let ∇ be the Levi-Civita connection on M. A curve γ : [0, 1] → M joining x to y is said to be a geodesic if A geodesic γ : [0, 1] → M joining x to y is said to be minimal if its arc length equals its Riemannian distance between x and y.
A Riemannian manifold M is said to be complete if for any x ∈ M, all the geodesics emanating from x are defined for all t ∈ R.
A simply connected complete Riemannian manifold M of nonpositive sectional curvature is called a Hadamard manifold [16].
Let M be a Hadamard manifold. The exponential mapping [31] In particular, the exponential mapping and its inverse are continuous on a Hadamard manifold.
Let M be a Hadamard manifold. We denote by P γ (b),γ (a) the parallel transport from T γ (a) M to T γ (b) M along the geodesic γ with respect to ∇ and it is defined by where A is the unique vector field such that ∇γ (t) v = 0 for all t and A(γ (a)) = v.

Note that
(i) P γ (a),γ (a) = Id, the identity transformation of T γ (a) M.
Definition 1 A subset K of a Hadamard manifold M is said to be geodesic convex if for any pair of distinct points x, y ∈ K , the geodesic γ joining x to y belongs to K , that is, if for any γ : [0, 1] → M such that γ (0) = x and γ (1) = y, then γ (t) = exp x (t exp -1 x y) ∈ K for all t ∈ [0, 1].

Formulation of the problems
Throughout the paper, unless otherwise specified, we assume that M is a Hadamard manifold, K is a nonempty geodesic convex subset of M and h : Recently, we [8] considered the following nonsmooth variational inequality problems: Minty-type nonsmooth variational inequality problem (in short, MTNVIP): We denote by S * and S * the solution set of NVIP (1) and MTNVIP (2), respectively, and assume that they are nonempty.
Consider the following optimization problem (in short, OP): where f : K → R is a real-valued function. Assume that the solution setS = {x ∈ K : f (x) ≤ f (y) for all y ∈ K} of OP (3) is nonempty.
Recently, we [8] established the following equivalence result between the solution set of NVIP (1) and MTNVIP (2) under the pseudomonotonicity and geodesic upper sign continuity assumptions of the bifunction h.

Lemma 1 ([8]) Let h : K × TM → R ∪ {±∞} be a pseudomonotone and geodesic upper sign continuous bifunction such that h is positively homogeneous in the second argument, that is, for all α > 0 and v ∈ T x M, h(x; αv) = αh(x; v). Then,x ∈ K is a solution of NVIP (1) if and only if it is a solution of MTNVIP (2).
We consider the following condition, which was first considered by Wu and Wu [35], to develop the weak sharpness of variational inequalities in Hilbert spaces and later it was considered by Alshahrani et al. [6].
Note that U(x) is nonempty for all x ∈ S * and also V(y) is nonempty for all y ∈ K , since S * is assumed to be nonempty. If h is pseudomonotone, then any exp -1 x y from U(x) also belongs to V(y) for all x, y ∈ K .
The following result shows the equivalence between S * and S * without pseudomonotonicity and geodesic upper sign continuity of h, but under the assumption of Condition A.
x y) ≥ 0 for all y ∈ K , and therefore, exp -1 x y ∈ U(x) for all y ∈ K . By hypothesis, U(x) = V(y) for all y ∈ K and thus, exp -1 x y ∈ V(y) for all y ∈ K . Hence, h y; -P y,x exp -1 x y ≤ 0, for all y ∈ K.
The primal gap function ϕ(x) associated with NVIP (1) is defined by and we set In a similar way, the dual gap function (x) associated with MTNVIP (2) is defined by and we set Note that both the functions ϕ and are nonnegative on K and vanish on S * and S * , respectively. Therefore, they are also gap functions for NVIP (1) and MTNVIP (2), respectively.

Characterizations of solution sets
Definition 5 ([9]) A function f : K → R is said to be geodesic radially upper semicontinuous (respectively, geodesic radially lower semicontinuous) on K if for every pair of distinct points x, y ∈ K , the function f is upper semicontinuous (respectively, lower semicontinuous) along the geodesic segment γ xy (t) for all t ∈ [0, 1], that is, t → f (γ xy (t)) is upper semicontinuous (respectively, lower semicontinuous) on [0, 1]. The function f is said to be geodesic radially continuous on K if it is both geodesic radially upper semicontinuous as well as geodesic radially lower semicontinuous on K . ( Definition 7 ([9]) A function f : K → R is said to be geodesic h-pseudoconvex if for any pair of distinct points x, y ∈ K , we have The function f is called geodesic h-pseudoconcave if in Definition (7), the inequality < is replaced by >.
Further, assume that one of the following conditions holds: (a) f is geodesic radially continuous; .
Again, by Theorem 1 and (5), we obtain Therefore, w ∈S and henceS is geodesic convex.
Forx ∈S, consider the sets For anyx ∈S, consider the set Proof Let x ∈S. By Theorem 2, w = exp¯x s exp -1 x x ∈S for all s ∈ (0, 1). By Theorem 3, we have 0 = h w; exp -1 wx = sh w; P w,x exp -1 xx = h w; P w,x exp -1 xx , since s > 0.
As x ∈S, by Theorem 3 and the oddness of h in the second argument, we have By combining (6) and (7), we obtainS ⊆S 3 . Conversely, assume that x ∈S 3 , and taking t = 1 in particular, we obtain Therefore, by Theorem 3, we have x ∈S, and hence,S =S 3 .
For anyx ∈S, consider the sets Theorem 5 Let f : K → R be geodesic h-pseudolinear and h : K × TM → R ∪ {±∞} be positively homogeneous in the second argument such that condition (4) holds. Assume that one of the following conditions holds: (a) f is geodesic radially continuous; (b) h is odd in the second argument. Ifx ∈S, thenS =S 4 =S 5 .
Proof By Theorem 3, we haveS ⊆S 4 . For the converse, assume that x ∈S 4 , that is, However,x is a solution of OP (3), and we have f (x) ≤ f (x). Therefore, f (x) = f (x), which yields x ∈S, and hence,S =S 4 . In a similar way, we obtainS =S 5 .
For anyx ∈S, consider the set where w = exp¯x t exp -1 x x for all t ∈ [0, 1]. Thus, x ∈S 6 , and hence,S ⊆S 6 . Therefore,S = S 6 .
For anyx ∈S, consider the sets

Relations among S * , S * , (x) and (x)
In the present section, we study the relationships among the solution set of NVIP (1) and MTNVIP (2) and sets (x) and (x). The following proposition follows directly from the definitions.
Proposition 4 Letx,ȳ ∈ K . Then, the following statements are equivalent: Proof (a) ⇒ (b): Letx ∈ S * andȳ ∈ S * . Then, for all x, y ∈ K , we have h x; exp -1 x x ≥ 0 and h y; exp -1 yȳ ≤ 0. Thus, from the definitions of S * and S * , we obtain the required result.
Therefore, (a) is obtained.
For the conclusion part, by using the Lemma 1 and the parts (a) and (b), we obtain the required result.

Condition B
For any x, y, z ∈ K , there exists α, β > 0 such that is called the Dikin metric (see [28]). In particular, for n = 1, we have M = (R ++ , X -2 ). In particular, we take h(x; ·) = Ax, · for all x ∈ M, where A : M → TM is a vector field and let Ax = 1 for all x ∈ M. Then, Ax, exp -1 x y = exp -1 x y. Now, for any two points x, y ∈ M, we have For any x, y, z ∈ M, we can obtain This implies that Therefore, the condition B holds good with α = 1 and β = x z .
(b) If either h(x; exp -1 xȳ ) = 0 or h(ȳ; exp -1 yx ) = 0, then it follows from part (a). Let us suppose that h(x; exp -1 xȳ ) = 0 andx ∈ S * . Then, h(x; exp -1 x z) ≥ 0 for all z ∈ K . Therefore, by Condition B, we have α, β > 0 such that Since h is subodd in the second argument, we have for all z ∈ K , Therefore,ȳ ∈ S * . In a similar manner, the converse part can be proved.
Definition 9 A bifunction h : K × TM → R ∪ {±∞} is said to be pseudomonotone * if it is pseudomonotone and for some k < 0 and for all x, y ∈ K , Remark 4 Since forx ∈ S * andȳ ∈ (x), we have h(ȳ; exp -1 yx ) = 0, assertions (d) and (e) in Proposition 8 are equivalent to h(x; exp -1 xȳ ) = 0. Therefore, if we add the assumption pseudomonotone * of h in the Proposition 8, then (a) to (e) in Proposition 8 are equivalent to each other.

Minimum principle sufficiency property for nonsmooth variational inequalities
The NVIP (1) has the minimum principle sufficiency property if (x) = S * , for allx ∈ S * .

Theorem 9
Suppose that the Condition B holds. Then, the following statements hold: Proof (a) Follows directly by using the part (a) of Theorem 8.
Since h is subodd in the second argument, we have By the suboddness of h in the second argument, we obtain Therefore by Proposition 6(a), we obtain h(x; exp -1 xȳ ) = 0. Now, on using (12), we have ϕ(x) = 0, which means thatx ∈ S * . Hence, by Proposition 7(b),ȳ ∈ S * .
Remark 5 By Proposition 5 and Theorem 9, the NVIP (1) has the minimum sufficiency property if S * ⊆ S * and Condition A holds for allx ∈ S * and for allȳ ∈ (x) and Condition B holds.

Maximum principle sufficiency property for nonsmooth variational inequalities
The NVIP (1) has the maximum principle sufficiency property if (x) = S * , for allx ∈ S * .
Remark 6 In the case S * ⊆ S * , the expression S * ∪ S * in Theorem 11 can be reduced to S * and the equality h(ȳ; exp -1 yx ) = 0 in (d) can also be omitted.
Under geodesic upper sign continuity and pseudomonotonicity * of h, all the statements of Theorem 11 hold, as shown below. Proof If h is geodesic upper sign continuous and pseudomonotone, then S * = S * . Since (d) can be easily verified forx ∈ S * andȳ ∈ (x), then by Theorem 11, (a)-(e) hold.

Theorem 14
Assume that h is odd in the second argument and strongly pseudomonotone with constant α > 0. Then, where d(x, S * ) = inf y∈S * d(x, y) is the Riemannian distance between the point x and the solution set S * .
As h is odd in the second argument and -exp -1 y x = P y,x exp -1 x y, we have -h y; P y,x exp -1 x y = h y; exp -1 yx ≤ -αd 2 (y,x), for all y ∈ K, and hence, h y; P y,x exp -1 x y ≥ αd 2 (y,x), for all y ∈ K.
Since Therefore, if h is odd in the second argument and strongly pseudomonotone with constant α > 0, then by Theorem 14, the NVIP (1) has weak sharp solutions.

Conclusion
In this paper, we define the notions minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using gap functions in the setting of Hadamard manifolds. We also characterize these two sufficiency properties in the setting of Hadamard manifolds. We conclude our paper by introducing the idea of the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds. The main objective of the paper is to include the existing results in nonlinear space, namely Hadamard manifolds. This is the first paper dealing with these two notions for nonsmooth variational inequalities. Therefore, in the future one can extend the concept of this paper in many other directions.