OPTIMALITY CONDITIONS AND DUALITY FOR MULTIOBJECTIVE SEMI-INFINITE PROGRAMMING PROBLEMS ON HADAMARD MANIFOLDS USING GENERALIZED GEODESIC CONVEXITY

. This paper deals with multiobjective semi-infinite programming problems on Hadamard manifolds. We establish the sufficient optimality criteria of the considered problem under generalized geodesic convexity assumptions. Moreover, we formulate the Mond-Weir and Wolfe type dual problems and derive the weak, strong and strict converse duality theorems relating the primal and dual problems under generalized geodesic convexity assumptions. Suitable examples have also been given to illustrate the significance of these results. The results presented in this paper extend and generalize the corresponding results in the literature.


Introduction
In theory of optimization, semi-infinite programming is the class of mathematical programming problems that deals with finitely many decision variables and in which the feasible set is defined by infinitely many constraints.The concepts and mathematical theory of semi-infinite programming were conceived by Haar [24].The term 'semi-infinite programming' was later coined by Charnes et al. [11] in 1962.Semi-infinite programming has a very wide range of applications in various practical problems of mathematical physics, game theory, engineering design, etc., see [12,13,16,22,23,27,29,45,50,54] and the references cited therein.
Although semi-infinite programming over finite or infinite dimensional Banach spaces has been extensively studied, it is observed that a lot of programming problems that arise in various real life applications require the problem to be formulated on Riemannian manifolds.One of the very first attempts in this direction is due to Ekeland [19], who discussed applications of variational principles on Riemannian manifolds.The generalizations of optimization methods from Euclidean space to Riemannian manifolds have important advantages.For example, constrained optimization problems can be viewed as unconstrained problems from the Riemannian geometry perspective.Moreover, nonconvex optimization problems can be converted to convex optimization problems through introduction of a suitable Riemannian metric (see for instance, [17,41,43]).Some results on convex optimization problems were extended to Riemannian manifolds by Rapcsák [46] and Udrişte [59] by introducing a generalization of convexity notion, namely, geodesic convexity.Further, Udrişte [59] introduced the notions of geodesic pseudoconvex and quasiconvex functions in Riemannian manifold setting.Constrained optimization problems and weak sharp minimizers on Hadamard manifold were discussed by Li et al. [31].Recently, many authors have generalized various other notions and concepts of optimization of R  to Riemannian manifolds; see for instance, [1, 4, 6-10, 44, 53, 57, 58] and the references cited therein.
Optimality and duality conditions play a very crucial role in optimization theory.Duality theory is important to understand the nature of the original (primal problem) from the perspective of a dual problem.Many authors have developed many interesting results on optimality and duality in R  , see for instance, [20,21,32,38,54,63] and the references cited therein.Two of the most important types of dual of a primal problem are Mond-Weir dual problem [60] and Wolfe dual problem [61], which have been referred to in this paper.
The optimality conditions for nonlinear programming problems on Riemannian manifolds were studied by Yang et al. in [62].Bergmann and Herzog [8] developed the intrinsic formulation of Karush-Kuhn-Tucker conditions and constraint qualification on smooth manifolds.The necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds have been discussed by Ruiz-Garzón in [47].Characterizations of solution sets of convex optimization problems in Riemannian manifolds were investigated by Barani and Hosseini in [6].Further, Chen [15] studied the Karush-Kuhn-Tucker type optimality criteria for interval valued objective function on Hadamard manifolds.Optimality and duality for multiobjective semi-infinite programming on Hadamard manifolds was investigated by Tung and Tam in [56].
Motivated by the works of [15,56] and the references cited therein, we consider a class of multiobjective semi-infinite programming problems on Hadamard manifold.We establish Karush-Kuhn-Tucker type sufficient optimality criteria for the considered problem under generalized geodesic convexity assumptions.Moreover, we formulate the Mond-Weir and Wolfe type dual problems and establish weak, strong and strict converse duality theorems relating the primal and the dual problems under generalized geodesic convexity assumptions.The results presented in this paper extend and generalize some known results in the literature to a more general space, namely, Hadamard manifold, as well as to more general classes of generalized geodesic convex functions.In particular, the results of this paper generalize the corresponding results of Tung and Tam [56] to a more general class of generalized geodesic convex function.Moreover, the results of the paper generalize some other well known results in R  , see for instance, [3,[34][35][36].
The paper is organized as follows.In Section 2, we recall the basic notions of Riemannian and Hadamard manifolds, that will be used in the sequel.Moreover, we consider a multiobjective semi-infinite programming problem on a Hadamard manifold.In Section 3, we establish sufficient optimality criteria of an efficient solution of the considered problem under generalized geodesic convexity assumptions.In Section 4, we formulate the Mond-Weir and Wolfe type dual problems for the considered primal problem.Moreover, we derive weak, strong and strict converse duality theorems relating the primal and the dual problems under generalized geodesic convexity assumptions.
The notation  ⊀  (respectively,  ) indicates the negation of  ≺  (respectively,  ⪯ ).If E is a -dimensional linear subspace of R  , then E inherits the inner product from R  , denoted by ⟨•, •⟩ E = ⟨•, •⟩.Moreover, the topology in R  is induced to E. One can further obtain a natural isometry between R  and E (see for instance, [9]).For a subset  ⊂ E, the closure and convex hull of  in E is denoted by cl() and co(), respectively.The positive conic hull of , denoted by pos(), is the convex cone containing the origin generated by  ⊂ E, and is defined as where, N denotes the set of all natural numbers.The negative polar cone of , denoted by  − , is defined by For any two Euclidean spaces Now, we recall some fundamental concepts and definitions of Riemannian and Hadamard manifolds (see for instance, [2,9,25,28,30]).
Let us consider that H be a topological space.Then H is said to be topological -manifold or a topological manifold of dimension  if H is Hausdorff, second-countable, and each point of H is contained in some neighborhood that is homeomorphic to an open subset of R  .Any pair (, ), where  is an open set in H , and  is a homeomorphism from  to some open set in R  is called a chart or a co-ordinate chart on H .For any two charts (, ) and (, ), such that  ∩  is non empty, the composite map  ∘  −1 : ( ∩  ) → ( ∩  ) is called the transition map from  to .Two charts (, ) and (, ) are said to be smoothly compatible if either  ∩  is empty or the transition map  ∘  −1 is infinitely continuously differentiable.A collection of charts such that the corresponding open sets cover H is called an atlas.An atlas  for H is said to be smooth if any two charts in  are smoothly compatible with each other.A smooth atlas  on H is said to be maximal if it is not properly contained in any larger smooth atlas.A maximal smooth atlas on H is called a smooth structure.A smooth manifold is a pair (H , ) where H is a topological manifold and  is a smooth structure on H .For an element  ∈ H , a curve  : (−, ) → H is said to be of class C 1 about the point  if (0) = , and  ∘  is of class C 1 for any chart (, ) about the point .Let  1 ,  2 : (−, ) → H be any two C 1 curves about .Then  1 and  2 are said to be equivalent if and only if there exists some chart (, ) about the point , such that A tangent vector to H at the point  is any equivalence class of The following definitions of geodesic convex sets and geodesic convex functions on a Riemannian manifold are from Udrişte [59] (Page 57) and Rapcsák [46] When the preceding inequality is strict, for  ̸ =  and  ∈ (0, 1), the function  is said to be geodesic strictly convex at  ∈ .
In particular, if H is a Hadamard manifold, then  is geodesic convex at  if and only if the following holds When the preceding inequality is strict, for  ̸ =  and  ∈ (0, 1), the function  is said to be geodesic strictly convex at  ∈ .
The following definitions of geodesic pseudoconvex and quasiconvex functions on Hadamard manifolds are taken from Definition 2.1 of Barani [5].
Definition 2.6.Let  ⊆ H be a geodesic convex set.Then (i) A map  :  → R is said to be geodesic pseudoconvex at  ∈ , if for any arbitrary point  ∈ , we have A map  :  → R is said to be geodesic strictly pseudoconvex at  ∈ , if for any arbitrary point  ∈ ,  ̸ = , we have Equivalently, a map  :  → R is said to be geodesic pseudoconvex at  ∈ , if for any arbitrary point  ∈ , we have A map  :  → R is said to be geodesic strictly pseudoconvex at  ∈ , if for any arbitrary point  ∈ ,  ̸ = , we have (ii) A map  :  → R is said to be geodesic quasiconvex at  ∈ , if for any arbitrary point  ∈ , we have Equivalently, a map  :  → R is said to be geodesic quasiconvex at  ∈ , if for any arbitrary point  ∈ , we have Remark 2.7.
(2) In view of Definition 10.1 in Udrişte [59] and Definition 13.2.1 in Rapcsák [46], if  :  → R in the above definitions is a geodesic convex function, then it is also a geodesic pseudoconvex and a geodesic quasiconvex function.
For more details on generalized geodesic convex functions on Hadamard manifolds, we refer the reader to [14,37,51,52] and the references cited therein.
The following theorem is from Shahi and Mishra [49] (see Thm. 3.2a).We present a proof of the theorem here for the sake of convenience of the readers.Proof.Let  denote the set of all geodesics connecting ,  ∈ .Since  is geodesic convex, then ∀,  ∈ , ∀γ  ∈ , we have Since  is affine, then ∀,  ∈ , ∀γ  ∈ , we have From (2.1) and (2.2), we get that, Then, it follows that lim ]︂ ≥ 0. (2.5) Therefore, we get the following lim an indeterminate form as since,  is positive.Thus, we get Therefore,   is a geodesic pseudoconvex function on .This completes the proof.
In this paper, we consider the following multiobjective semi-infinite programming problem on Hadamard manifold: (MSIP) Minimize  () = ( 1 (), . . .,   ()), where,   :  → R, ( ∈  = {1, 2, . . ., }),   :  → R ( ∈ ), are smooth functions defined on an open geodesic convex set  ⊂ H .The index set  is arbitrary.The feasible set of the problem (MSIP), denoted by  , is defined by The index set of all active inequality constraints at a feasible point  ∈  , denoted by (), is defined as For any feasible point  ∈  , we denote the set of all active contraint multipliers at  ∈  by A (), that is Now, we recall the notions of efficient solution and weakly efficient solution of (MSIP) (see for instance, [33,56]).grad   (x) and the set pos ⋃︀ ∈(x) grad   (x) is closed.

Optimality conditons
In this section, we derive a sufficient optimality criteria for (MSIP) using generalized geodesic convex functions.
To begin with, we state the Karush-Kuhn-Tucker type necessary optimality criteria for (MSIP) from Tung and Tam [56].Then the following statements are true.
(i) If   is geodesic pseudoconvex at x ∈  for all  ∈ , and   is geodesic quasiconvex at x ∈  for all  ∈ , then x is a weakly efficient solution of (MSIP).(ii) If   is geodesic strictly pseudoconvex at x ∈  for all  ∈ , and   is geodesic quasiconvex at x ∈  for all  ∈ , then x is an efficient solution of (MSIP).
Proof.Since  ∈ A (x), there exists a finite subset  of (x), such that From condition (3.1), we have Let  ∈  be an arbitrary feasible point.Then, As   () is geodesic quasiconvex at x on  for all  ∈ , we have From inequality (3.3) and equation (3.2), we have (3.4) (i) On the contrary, let us assume that x is not a weakly efficient solution of (MSIP).Then, there exists some x ∈  , such that   (x) <   (x), ∀ ∈ .
As   () is geodesic pseudoconvex at x ∈  for all  ∈ , we have which is a contradiction to (3.4).This proves that x is a weakly efficient solution of (MSIP).
(ii) On the contrary, let us assume x is not an efficient solution of (MSIP).Then, there exists some x ∈  , such that , for at least one  ∈ .
The above inequalities imply that x ̸ = x.Since,   () is geodesic strictly pseudoconvex at x ∈  for all  ∈ , we have which is a contradiction to (3.4).This proves that x is an efficient solution of (MSIP).
The following example illustrates the significance of Theorem 3.2.
Example 3.3.Let us consider the Poincaré half-plane defined as follows Then, H is a Riemannian manifold equipped with the inner product (see for instance, Example 5, Page 2 in [59]), as follows , and ⟨•, •⟩ is the standard inner product on R 2 .Since the sectional curvature of H is −1, it is also a Hadamard manifold.The second Christoffel symbols are as follows: Let us consider the following open geodesic convex set on the Hadamard manifold H as follows:

}︂ •
We consider the following semi-infinite programming problem on the Hadamard Manifold H .
Here   ,   :  → R,  = 1, 2 are real valued functions and  ∈ .The feasible set  of the problem is
Now, we see that, the (hyperbolic) Hessian, or the second-order covariant derivative, of 2 is a positive semidefinite matrix as all its eigen values are non negative.Thus, )︂ is a positive semidefinite matrix.That is, ln 2 1+ 1 2 2 is also a convex function.Thus,  1 () and  2 () are both ratios of geodesic convex functions and positive affine functions.Then, from Theorem 2.8, it follows that  1 ,  2 are geodesic pseudoconvex functions.Also, we have Since the matrix     () is positive semidefinite,   is quasiconvex for all  ∈ .This shows that the conditions of Theorem 3.2i hold.It can be verified that x is a weakly efficient solution of the problem ( 1 ).

Duality
In this section, we formulate the Mond-Weir [60] and Wolfe [61] type dual problems for (MSIP) and establish weak, strong and strict converse duality theorems relating the primal problem (MSIP) and the dual problems under generalized geodesic convexity assumptions.

Mond-Weir duality
Let us consider that  ∈  ⊂ H , where  is an open geodesic convex set in H ,  ∈ R  + ∖ {0} and  ∈ R || + .The Mond-Weir dual problem of (MSIP), denoted by (MSID MW ), is formulated as follows: The feasible set of MSID MW , denoted by F MW , is given by The following definitions of efficient solution and weakly efficient solution of the Mond-Weir dual problem (MSID MW ) are from Tung and Tam [56].
Let us consider the following geodesic convex set on the Hadamard manifold H as follows:
The following theorem establishes strong duality relating (MSIP) and (MSID MW ).
Moreover, the following statements are true.
The following example illustrates strong duality theorem relating (MSIP) and (MSID MW ).
Example 4.6.Let us consider the Poincaré half-plane defined as follows Let us consider the following geodesic convex set on the Hadamard manifold H as follows: Let us consider the multiobjective semi-infinite problem ( 1 ) as defined in Example 3.3.We denote the feasible set of ( 1 ) by  .The Mond-Weir dual problem related to ( 1 ), denoted by (  1 ), may be formulated as follows: The feasible set of (  1 ), denoted by   , is given by Let us consider the point x = (0, 1 2 ) ∈  .Then, it can be verified that Also, we have the following )︂ , grad  2 (x) = (0, 0), grad   (x) = (0, −).
Then, it follows from Example 3.3 that (ACQ) is satisfied at the feasible point x = (0, 1 2 ).We can check that x is an efficient solution of (P 1 ).Thus, we see that all the assumptions for strong duality of Mond-Weir dual problem (Thm.4.5) are satisfied.
Hence, there exist The following theorem establishes the strict converse duality relating (MSIP) and (MSID MW ).
Theorem 4.7 (Strict converse duality).Let  * be a weakly efficient solution of (MSIP) such that Abadie constraint qualification (ACQ) is satisfied at  * .Let (x, ᾱ, λ) be a weakly efficient solution of (MSID MW ).If   is geodesic strictly pseudoconvex at x for all  ∈  and ∑︀ ∈     is geodesic quasiconvex at x, then  * = x.
Proof.On the contrary, let us assume that  * ̸ = x.Since  * is a weakly efficient solution of (MSIP) and (ACQ) is satisfied at  * , we can infer from Theorem 4.5 that there exist Further, it also follows from Theorem 4.5 that ( * ,  * ,  * ) is an efficient solution of (MSID MW ).Since  * ∈  and (x, ᾱ, λ) ∈    , then from Theorem 4.4ii, it follows that which is a contradiction.This completes the proof.Now, we give an example to illustrate the results obtained for Mond-Weir duality.
Example 4.8.Let us consider the set H ⊂ R 2 as follows Then H is a Riemannian manifold (see for instance, [7,46,56], and Example 4.4 of [42]).H is equipped with the metric as defined below ⟨, where ⟨•, •⟩ is the standard inner product on R 2 and Since the sectional curvature of H is 0, which is non positive, H is also a Hadamard manifold.Also, H is a geodesic convex set.The second Christoffel symbols are as follows: The Riemannian distance between  = ( 1 ,  2 ) ,  = ( 1 ,  2 ) ∈ H is given by The exponential map exp  :   H → H for any  ∈   H is given by The inverse of the exponential map exp −1  : H →   H for any  ∈ H is given by )︂ .
We consider the following semi-infinite programming problem on H Here, ,   : H → R 2 .The feasible set  for the problem is The Mond-Weir dual problem related to ( 2 ) may be formulated as The feasible set of (P MW ) is given by Let us consider the feasible point x = (, ) ∈  MW .Since we have (x) = .Let  be any arbitrary element in the contingent cone T (, x).Then, there exist   ↓ 0 and Letting  to infinity, we can conclude that  1 ≥ 0,  2 ≥ 0.
Substituting x = (, ) for  = ( 1 ,  2 ) in the above equations, we get Hence, we obtain the following is closed.This implies that Abadie constraint qualification (ACQ) holds at x.We can check that x is an efficient solution of (P 2 ).Thus, we see that all the assumptions for strong duality of Mond-Weir dual problem (Thm.4.5) are satisfied.
Now, we see that, the (hyperbolic) Hessian, or the second-order covariant derivative, of  is given by is a positive semidefinite matrix as all its eigen values are non negative.Thus,  is geodesic convex at (0, 1  2 ).Then, ℎ is a ratio of a geodesic convex function and a positive affine function.Then, from Theorem 2.8, it follows that ℎ is a geodesic pseudoconvex function.It can also be easily verified that  (x) ⊀ L (x, , ).Thus, the weak duality theorem (Thm.4.11) is verified.
Moreover, the following statements are true.
This proves that (x, ᾱ, λ) is a weakly efficient solution of (MSID W ).
This proves that (x, ᾱ, λ) is an efficient solution of (MSID W ).
Since  1 ,  2 > 0, hence, the (hyperbolic) Hessian, or the second-order covariant derivative of ℎ() is positive semidefinite.Remark 4.16.In view of Definition 10.1 in Udrişte [59] and Definition 13.2.1 in Rapcsák [46], every geodesic convex function is geodesic pseudoconvex and geodesic quasiconvex.Thus, the results presented in this paper generalize the corresponding results of optimality and duality from Tung and Tam [56].

Conclusion
In this paper, we have considered a class of multiobjective semi-infinite programming problems on Hadamard manifold (MSIP) and established the Karush-Kuhn-Tucker type sufficient optimality criteria for (MSIP) under generalized geodesic convexity assumptions.The sufficient optimality condition derived in this paper extend the sufficient optimality result derived by Tung and Tam [56] from geodesic convexity assumptions to geodesic pseudoconvexity and geodesic quasiconvexity assumptions.Moreover, related to (MSIP), we have formulated the Mond-Weir type dual problem (MSIP MW ) and Wolfe type dual problem (MSIP MW ) and derived the weak, strong and strict converse duality theorems.The weak and strong duality results derived in this paper extend the corresponding results of Tung and Tam [56] from geodesic convexity assumptions to geodesic pseudoconvexity and geodesic quasiconvexity assumptions.In particular, the results of the paper generalize some other well known results in R  , see for instance, [3,[34][35][36].Several non-trivial examples have been given to illustrate the significance of these results.Our work in this paper leaves various avenues for future research.For example, it would be interesting to extend the results in this paper for non-smooth multiobjective semi-infinite problems on Hadamard manifolds.Further, we intend to investigate multiobjective semi-infinite problems on Hadamard manifolds with uncertain data in objective functions.

Theorem 2 . 8 .
Let us assume that the function () = () () , where  and  are smooth functions on an open geodesic convex set  of a Riemannian manifold H .If  be a geodesic convex function and  be a positive affine function, then  be a geodesic pseudoconvex function.

Then
From the second-order covariant derivative, it follows that   ℎ() = ∇ 2 ℎ() − ∇ℎ()Γ [48]ves through  on H modulo the equivalence relation defined above.The set of all tangent vectors at the point  in H is termed as the tangent space to H at  and is denoted by the symbol   H .A Riemannian metric on a smooth manifold H is defined as a 2-tensor field G , such that G is symmetric and positive definite.An inner product is induced on every tangent space   H by a Riemannian metric and this is denoted by G (, ) = ⟨, ⟩  for all ,  ∈   H .A smooth manifold together with a given Riemannian metric is called a Riemannian manifold.The exponential map exp  :   H → H is defined by exp  () =  , (1) for any  ∈   H , where  , is the geodesic starting at  with a velocity .A Riemannian manifold H is said to be geodesic complete if for every  ∈ H , the exponential map exp  () is defined for all  ∈   H .A complete, simply connected Riemannian manifold with nonpositive sectional curvature everywhere is called a Hadamard manifold.Henceforth, we shall use to the symbol H to denote a Hadamard manifold, unless otherwise specified.The following theorem, known as Hadamard-Cartan theorem, is from Sakai[48](Thm.4.1, Page 221).Let H be a Hadamard manifold.Then for every  ∈   H , the exponential map exp  :   H → H is a diffeomorphism with the inverse map exp−1 : H →   H satisfying exp −1  () = 0  .Moreover, for any  ∈ H there exists a unique minimal geodesic  , : [0, 1] → H satisfying  , () = exp  (exp −1  ()).The contingent cone for a subset of a Hadamard manifold is defined as follows.Definition 2.2.Let  ⊆ H and  ∈ cl().The contingent cone of  at , denoted by T (, ) is defined by (Def.6.1.2,Page 64), respectively.Let us consider that H be a Riemannian manifold.Then, (i) A subset  of H is called a geodesic convex set in H , if for every pair of distinct points ,  ∈  and for any geodesic  , : [0, 1] → H joining  to , we have  , () ∈ , ∀ ∈ [0, 1].(ii) Let  be a geodesic convex subset of H and  :  → R be a function on .Then, the function  is said to be geodesic convex at  ∈ , if for any point  ∈  and for any geodesic  , : [0, 1] → H joining  to , we have Let  ⊂ H ⊂ R  be an open geodesic convex set, and  :  → R be a twice continuously differentiable function.Then,  is geodesic convex on  if and only if the geodesic Hessian (or second covariant () is the matrix of first partial derivatives,    (()) denotes the Hessian matrix of the function  by  at (), and Γ() is the matrix of second Christoffel symbols with respect to the Riemannian metric of H , is positive semidefinite at all the points of each geodesic convex coordinate neighbourhood () of .Theorem 2.5.Let  ⊂ H = R  be an open geodesic convex set, and  :  → R be a twice continuously differentiable function.Then,  is geodesic convex on  if and only if the following matrix   () = ∇ 2  () + ∇ ()Γ,where Γ is the matrix of second Christoffel symbols with respect to the Riemannian metric of R  , ∇ () and ∇ 2  () are the (Euclidean) gradient and (Euclidean) Hessian of the function  at  in the usual sense, is positive semidefinite at all the points of each geodesic convex coordinate neighbourhood (), with  : Definition 4.1.Let (ũ, α, λ) ∈    .Then, (ũ, α, λ) is said to be an efficient solution of (MSID MW ), if there does not exist any other (, , ) ∈    satisfying Let (ũ, α, λ) ∈    .Then, (ũ, α, λ) is said to be a weakly efficient solution of (MSID MW ), if there does not exist any other (, , ) ∈    satisfying Example 4.3.Let us consider the Poincaré half-plane defined as follows satisfying   is a weakly efficient solution of the Mond-Weir dual problem.
Thus, ℎ() is geodesic convex function.Since, L (, ) is the ratio of a geodesic convex function (ℎ()) and a positive affine function, this implies that