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Geodesic r-preinvex functions on Riemannian manifolds
Journal of Inequalities and Applicationsvolume 2014, Article number: 144 (2014)
In this article, we introduce a new class of functions called r-invexity and geodesic r-preinvexity functions on a Riemannian manifolds. Further, we establish the relationships between r-invexity and geodesic r-preinvexity on Riemannian manifolds. It is observed that a local minimum point for a scalar optimization problem is also a global minimum point under geodesic r-preinvexity on Riemannian manifolds. In the end, a mean value inequality is extended to a Cartan-Hadamard manifold. The results presented in this paper extend and generalize the results that have appeared in the literature.
Convexity is one of the most frequently used hypotheses in optimization theory. It is well known that a local minimum is also a global minimum for a convex function. A significant generalization of convex functions is that of an invex function introduced by Hanson . Hanson’s initial results inspired a great deal of subsequent work, which has greatly expanded the role and applications of invexity in non-linear optimization and other branches of pure and applied sciences.
Ben-Israel and Mond  introduced a new generalization of convex sets and convex functions, Craven  called them invex sets and preinvex functions, respectively. Jeyakumar  studied the properties of preinvex functions and their role in optimization and mathematical programming. Jeyakumar and Mond  introduced a new class of functions, namely V-invex functions, and established sufficient optimality criteria and duality results in the multiobjective programming problems. Antczak  introduced the concept of r-invexity and r-preinvexity in mathematical programming. Making a step forward Antczak  introduced the concept of -invexity for differentiable multiobjective programming problems, which is a generalization of V-invex functions  and r-invex functions .
On the other hand, in the last few years, several important concepts of non-linear analysis and optimization problems have been extended from Euclidean space to a Riemannian manifolds. In general, a manifold is not a linear space, but naturally concepts and techniques from linear spaces to Riemannian manifold can be extended. Rapcsak  and Udriste  considered a generalization of convexity, called geodesic convexity, and extended many results of convex analysis and optimization theory to Riemannian manifolds. The notion of invex functions on Riemannian manifolds was introduced by Pini  and Mititelu , and they investigated its generalization. Barani and Pouryayevali  introduced the geodesic invex set, geodesic η-invex function, and geodesic η-preinvex functions on a Riemannian manifold and found some interesting results. Further, Agarwal et al.  generalized the notion of geodesic η-preinvex functions to geodesic α-preinvex functions. Recently, Zhou and Huang  introduced the concept of roughly B-invex set and functions on Riemannian manifolds.
Motivated by work of Barani and Pouryayevali  and Antczak [6, 7], we introduce the concept of geodesic r-preinvex functions and r-invex functions on Riemannian manifolds, which is a generalization of preinvexity as defined in [6, 12]. Some relations between r-invex and geodesic r-preinvex functions are investigated. The existence conditions for global minima of these functions under proximal subdifferential of lower semicontinuity are also explored. In the end, a mean value inequality is also derived.
In this section we recall some basic definitions and some basic results of Riemannian manifolds, for further study these materials are available in (cf. ).
Let M be a -manifold modeled on a Hilbert space H, either finite or infinite dimensional, endowed with a Riemannian metric on a tangent space . The corresponding norm is denoted by and the length of a piecewise curve is defined by
For any point , we define
then d is a distance which induces the original topology on M. We know that on every Riemannian manifold there exists exactly one covariant derivative called a Levi-Civita connection, denoted by , for any vector fields ; we also recall that a geodesic is a -smooth path γ whose tangent is parallel along the path γ, that is, γ satisfies the equation . Any path γ joining p and q in M such that is a geodesic and is called a minimal geodesic. The existence theorem for ordinary differential equation implies that for every , there exist an open interval containing 0 and exactly one geodesic with . This implies that there is an open neighborhood of the submanifold M of TM such that for every is there is defined and the restriction of exp to a fiber in is denoted by for every . We use parallel transport of vectors along the geodesic. Recall that for a given curve , a number , and a vector , there exists exactly one parallel vector field along such that . Moreover, the mapping defined by is a linear isometry between the tangent spaces and , for each . We denote this mapping by and we call it the parallel translation from to along the curve γ.
If f is a differentiable map from the manifold M to manifold N, then , denotes the differential of f at x. We also recall that a simply connected complete Riemannian manifold of non-positive sectional curvature is called a Cartan-Hadamard manifold.
3 Geodesic r-invex functions
In this section, we define geodesic r-invex functions and r-preinvex functions. Barani and Pouryayevali  define the invex sets as follows.
Definition 3.1 Let M be a Riemannian manifold and such that for every , . A non-empty subset S of M is said to be a geodesic invex set with respect to η if for every , there exists a unique geodesic such that
for all .
Remark 3.1 
If we consider M to be a Cartan-Hadamard manifold (either infinite or finite dimensional), then on M there exists a natural map η playing the role of in the . Indeed we define the function η as
for all . Here is the unique minimal geodesic joining y to x (see [, p.253]) as follows:
for all . Therefore, every geodesic convex set is a geodesic convex set with respect to η defined in above equation. The converse is not true in general.
Example 3.1 
Let M be a Cartan-Hadamard manifold and , . Let for some , where is an open ball with center x and radius r. We define
then S is not a geodesic convex set because every geodesic curve passing through and does not completely lie in S. Now we define the function such that
For every , consider defined by
for all .
Hence , . Barani and Pouryayevali  showed that S is a geodesic invex set with respect to η.
Let S be a geodesic convex subset of a finite dimensional Cartan-Hadamard manifold M and , then there exists a unique point such that for each , . The point is called the projection of x onto S (see [, p.262]).
Definition 3.2 
Let M be an n-dimensional Riemannian manifold and S be an open subset of M which is geodesic invex set with respect to . Let f be a real valued function such that . Then f is said to be an η-invex function with respect to η if
for all .
Definition 3.3 
Let M be a Riemannian manifold and be a geodesic η-invex set with respect to . The function is said to be geodesic η-preinvex if for any
for all , where is the unique geodesic defined in Definition 3.1. If the above inequality is strict, then f is called a strictly geodesic preinvex function.
Now we define an r-invex function and a geodesic r-preinvex function on M.
Definition 3.4 Let M be a Riemannian manifold and be a geodesic invex set with respect to . Let f be a real differentiable function S. Then f is said to be r-invex with respect to η if
Definition 3.5 Let M be a Riemannian manifold and be a geodesic invex set with respect to . The function is said to be geodesic r-preinvex if for any , we have
If the above inequality is strict, then f is called a strictly geodesic r-preinvex function.
We give the following non-trivial example for a geodesic r-preinvex function that is yet not geodesic η-preinvex.
Example 3.2 Let and defined by with , and . If then f is a geodesic r-preinvex function but not a geodesic η-preinvex function at , , since at .
Proposition 3.1 If is a geodesic r-preinvex function with respect to and , then for any real number , the level set is a geodesic invex set.
Proof For any and , we have , . Since f is geodesic r-preinvex function, then we have
Therefore, for all , and the result is proved. □
4 Geodesic r-preinvexity and differentiability
In this section, we discuss property and condition (say condition (C)) introduced by Barani and Pouryayevali  on the function , which will be used in the subsequent analysis.
Pini  define the following property.
Definition 4.1 Let M be a Riemannian manifold and be a curve on M such that and . Then is said to possess the property (P) with respect to if
for all .
Pini  also proved the following conditions as follows:
for all , which taken together are called condition (C).
Theorem 4.1 Let M be a Riemannian manifold and S be an open subset of M which is a geodesic invex set with respect to . Let be a differentiable and geodesic r-preinvex function on S. Then f is an r-invex function on S.
Proof Since S is a geodesic invex set with respect to η, then for all , there exists a unique geodesic , , for all . By the differentiability of f at , we have
But f is geodesic r-preinvex for , and we have
Dividing by t and taking the limit , we get
Hence, f is an r-invex function on S. □
Theorem 4.2 Let M be a Riemannian manifold and S be an open subset of M, which is a geodesic invex set with respect to . Let be a differentiable function, η satisfies the condition (C), then f is geodesic r-preinvex on S if f is r-invex on S.
Proof We know that for a geodesic invex set with respect to η for every , there exists a unique geodesic such that , , , for all .
Fix and set , then by geodesic r-invexity of f on S, we have
On multiplying (1) by t and (2) by , respectively, and then adding we get
By the condition (C), we have
This together with (3) implies
Hence, f is geodesic r-preinvex on S. □
5 Geodesic r-preinvexity and semi-continuity
In this section, we discuss geodesic r-preinvexity on Riemannian manifold under proximal subdifferential of a lower semi-continuous function. First, we recall the definition of a proximal subdifferentiable of a function defined on a Riemannian manifold in .
Definition 5.1 Let M be a Riemannian manifold and be a lower semi-continuous function. A point is said to be proximal subgradient of f at , if there exist a positive number δ and σ such that
for all , where . The set of all proximal subgradient of is denoted by .
Theorem 5.1 Let M be a Riemannian manifold and S be an open subset of M, which is geodesic invex with respect to . Let be geodesic r-preinvex, if is a local minimum of the problem
then is a global minimum of (P).
Proof Let be a local minimum; then there exists a neighborhood such that
for all .
If is not a global minimum of f, then there exists a point such that
As S is a geodesic invex set with respect to η, there exists a unique geodesic γ such that , , , for all .
If we choose such that , then . From the geodesic r-preinvexity of f, we have
Equivalently, we have
for all . Therefore, for each , , which is a contradiction to (5). Hence the result. □
Theorem 5.2 Let M be a Cartan-Hadamard manifold and S be an open subset of M, which is geodesic r-preinvex with respect to with for all . Assume that is a lower semi-continuous geodesic r-preinvex function and , . Then there exists a positive number δ such that
for all .
Proof From the definition of , there are positive numbers δ and σ such that
for all .
Now, fix . Since S is a geodesic invex set with respect to η, there exists a unique geodesic such that , , , for all .
Since M is a Cartan-Hadamard manifold, then for each (see [, p.253]). If we choose , then for all .
From the geodesic r-preinvexity of f, we get
Using (6) for each , we get
Since M is a Cartan-Hadamard manifold, for each , we have
Thus we have
Thus from (7) and (8), we have
By further calculation we arrive at
taking the limit
This proves the theorem completely. □
6 Mean value inequality
Definition 6.1 
Let S be a non-empty subset of a Riemannian manifold M, which is a geodesic η-invex set with respect to , and let x and u be two arbitrary points of S. Let be the unique geodesic such that , , , for all .
A set is said to be a closed η-path joining the points u and , if
An open η-path joining the point u and v is a set of the form
If we set .
Theorem 6.1 (Mean value inequality)
Let M be a Cartan-Hadamard manifold and S be an open subset of M, which is a geodesic invex set with respect to such that for all , . Let for every , and . Then a necessary and sufficient condition for a function to be geodesic r-preinvex is that the inequality
is true for all .
Proof Let be a geodesic preinvex function, and . If or then (9) is true trivially. If , then , for some . From the geodesic η-invexity of S, we have and
Since f is geodesic preinvex on S, it follows that
Using the value of t we get
For sufficiency suppose that the mean value inequality (9) is true. Let and , for some . Then , and we have , from (9)
which shows that f is geodesic r-preinvex function on S. □
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The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.