Geodesic r-preinvex functions on Riemannian manifolds
© Khan et al.; licensee Springer. 2014
Received: 27 November 2013
Accepted: 24 March 2014
Published: 9 April 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. ).
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.
for all .
Remark 3.1 
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 
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 
for all .
Definition 3.3 
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.
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.
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.
for all .
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.
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 .
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 .
for all , where . The set of all proximal subgradient of is denoted by .
then is a global minimum of (P).
for all .
As S is a geodesic invex set with respect to η, there exists a unique geodesic γ such that , , , for all .
for all . Therefore, for each , , which is a contradiction to (5). Hence the result. □
for all .
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 .
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 .
If we set .
Theorem 6.1 (Mean value inequality)
is true for all .
which shows that f is geodesic r-preinvex function on S. □
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