A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds

Abstract

In this manuscript, we define the $(E,F)$-invex set, $(E,F)$-invex functions, and $(E,F)$-preinvex functions on Euclidean space, i.e., simply vector space. We extend these concepts on the Riemannian manifold. We also detail the fundamental properties of $(E,F)$-preinvex functions and provide some examples that illustrate the concepts well. We have established a relation between $(E,F)$-invex and $(E,F)$-preinvex functions on Riemannian manifolds. We introduce the conditions $\mathcal{A}$ and define the $(E,F)$-proximal sub-gradient. $(E,F)$-preinvex functions are also used to demonstrate their applicability in optimization problems. In the last, we establish the points of extrema of a non-smooth $(E,F)$-preinvex functions on $(E,F)$-invex subset of the Riemannian manifolds by using the $(E,F)$-proximal sub-gradient.

1. Introduction

One of the most crucial aspects of mathematical programming is the search for convexity and generalized convexity because these concepts play a vital role in optimization theory. Several notable generalizations of convexity have been made over the recent few decades (see [1,2,3,4,5,6,7,8]). Youness provided the concept of E-convex sets and E-convex functions. He also explored the E-convex programming problems in his paper [9]. Yang and Chen found some results to be incorrect in the paper written by [9]. Then, they modified those results (see [10,11]). Jian [12] came up with the concept of $(E,F)$-convex sets and functions with exciting applications regarding generalizations.

Recently, Wedad Saleh explored the concept of $(E,F)$-convex sets and functions on the Riemannian manifold and developed some fruitful results in his paper [13].

One can find the clear idea and development of the geodesic theory of convexity in the books authored by Udriste [14] and Rapcsák [15]. And, for more details on E-convex sets and functions, we refer [13,16], from where the idea of $(E,F)$-convex sets and functions was mainly generated. Convexity and its generalizations on the Riemannian manifolds have received more attention from researchers during the recent few decades, and a vast amount of literature has been developed on the subject (see [17,18,19,20,21,22,23,24,25,26]).

Motivated by the above works and results, particularly in [12,13], we provide the definitions of $(E,F)$-invex sets, $(E,F)$-invex functions and $(E,F)$-preinvex functions. We explore this concept on Riemannian manifolds, which is the generalization of $(E,F)$-convexity sets and functions. We establish a relation between $(E,F)$-invex and $(E,F)$-preinvex functions on Riemannian manifolds. We introduce the conditions $\mathcal{A}$ and define the $(E,F)$-proximal sub-gradient. To explore and demonstrate its applicability on problems of optimization, $(E,F)$-preinvexity is applied. In the last section of the present paper, we have obtained extremum points for a non-smooth $(E,F)$-preinvex function defined on the $(E,F)$-invex subset of the Riemannian manifold, which became possible due to use of the $(E,F)$-proximal sub-gradient.

2. Preliminaries

Definition 1

([13]). Let $\mathcal{A}$ be a non-empty subset of ${\mathcal{R}}^n$, and let $E,F:\mathcal{A}\to {\mathcal{R}}^n$ be two maps. If $r_1,s_1\in \mathcal{A}$ and $\mu \in [0,1]$, $F\left(s_1\right)+\mu (E\left(r_1\right)-F\left(s_1\right))\in \mathcal{A}$, then $\mathcal{A}$ is known as an $(E,F)$-convex set.

In this section, we recall some well-known definitions on Riemannian manifolds.

Let $\mathcal{O}$ be a smooth Riemannian manifold ($C^{\infty }$), which means that ${⟨,⟩}_s$, the metric on the tangent space $T_s\mathcal{O}\times T_s\mathcal{O}$ of manifold $\mathcal{O}$, induces a norm ${\parallel ,\parallel }_s$. If z and w are two points on $\mathcal{O}$ and $\beta :[c,d]\to \mathcal{O}$ is a piecewise smooth curve connecting $\beta \left(c\right)=z$ to $\beta \left(d\right)=w$, its length $L\left(\beta \right)$ is given by

$$L\left(\beta \right)={\int }_c^d{\parallel {\beta }^{\prime }\left(t\right)\parallel }_{{\beta }^{\prime }\left(t\right)}dt.$$

Now, we define

$$d(z,w)=inf\{L\left(\gamma \right):\beta is a piecewise C^1 curve connecting z and w\}$$

for any $z,w\in \mathcal{O}$. The original topology on $\mathcal{O}$ is then induced by a distance d. A Levi-Civita connection, denoted by ${\nabla }_{A}B$, for any vector fields $A,B\in T\mathcal{O}$ also known as a covariant derivative, is known to exist uniquely on every Riemannian manifold. We also recall that a geodesic is a $C^{\infty }$ smooth path β whose tangent is parallel along the path β, i.e., β satisfies the equation ${\nabla }_{{\displaystyle \frac{d\beta \left(t\right)}{dt}}}{\displaystyle \frac{d\beta \left(t\right)}{dt}}=0$. Any path β connecting z and w in $\mathcal{O}$, such that $L\left(\beta \right)=d(z,w)$, is a geodesic, and it is called a minimizing geodesic. Recall that for a given curve $\beta :J\to \mathcal{O}$, a point $t_o\in J$, and a vector $u_o\in T_{\beta \left(t_o\right)}\mathcal{O}$, there exists exactly one parallel vector field $U\left(t\right)$ along $\beta \left(t\right)$ s.t. $U\left(t_o\right)=u_o$. Moreover, the mapping defined by $u_o⟼U\left(t\right)$ is a linear isometry between the tangent spaces $T_{\beta \left(t_o\right)}$ and $T_{\beta \left(t\right)}$, for each $t\in J$. We denote this mapping by $P_{t_o,\beta }^t$ and we call it the parallel translation from $T_{\beta \left(t_o\right)}\mathcal{O}$ to $T_{\beta \left(t\right)}\mathcal{O}$ along the curve β. A simply connected complete Riemannian manifold with non-positive sectional curvature is called a Hadamard manifold.

3. $(\mathit{E},\mathit{F})$-Invex Sets and $(\mathit{E},\mathit{F})$-Preinvex Functions

Definition 2.

Let $E,F:\mathcal{A}\to {\mathcal{R}}^n$ be functions, defined on non-empty subset $\mathcal{A}$ of ${\mathcal{R}}^n$, and let $G:{\mathcal{R}}^n\times {\mathcal{R}}^n\to {\mathcal{R}}^n$ be another function. If $\forall r_1,s_1\in \mathcal{A}$ and $\mu \in [0,1]$, $F\left(s_1\right)+\mu G(E\left(r_1\right),F\left(s_1\right))\in \mathcal{A}$, then $\mathcal{A}$ is known as an $(E,F)$-invex set.

Example 1.

Let $\mathcal{A}=[0,4]\cup [6,8]$ be a non-empty subset of $\mathcal{R}$, and let $E,F:\mathcal{A}\to \mathcal{R}$ be two maps defined as

$$F\left(s_1\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}\mathrm{if}1\le s_1\le 3\hfill \\ 1,\mathrm{if}{s}_1<1\hfill \\ 0,\mathrm{if}{s}_1>3\hfill \end{array}\right.,E\left(r_1\right)=\left\{\begin{array}{c}3,\mathrm{if}1\le r_1\le 3\hfill \\ 2,\mathrm{if}{r}_1<1\hfill \\ 9,\mathrm{if}{r}_1>3,\hfill \end{array}\right.$$

and let be $G:\mathcal{R}\times \mathcal{R}\to \mathcal{R}$ a bi-function defined as $G(E\left(r_1\right),F\left(s_1\right))=F\left(s_1\right)$.

Now, for $r_1,s_1>3$ and $\mu =1$, we have $F\left(s_1\right)+\mu (E\left(r_1\right)-F\left(s_1\right))=9\notin \mathcal{A}$.

Then, it is obvious that the set $\mathcal{A}$ is not a $(E,F)$-convex set with respect to the maps E and F. Now,

$$F\left(s_1\right)+\mu G(E\left(r_1\right),F\left(s_1\right))=F\left(s_1\right)+\mu F\left(s_1\right)=(1+\mu )F\left(s_1\right).$$

(1)

It is clear that $\forall r_1,s_1$ and $\mu \in [0,1]$. Thus, we have $F\left(s_1\right)+\mu G(E\left(r_1\right),F\left(s_1\right))\in \mathcal{A}$. Hence, the set $\mathcal{A}$ is an $(E,F)$-invex set.

Remark 1.

If we take $E=F$ in the Definition (2), then the set $\mathcal{A}$ becomes an E-invex set.

Remark 2.

If we take $G(E\left(r_1\right),F\left(s_1\right))=E\left(r_1\right)-F\left(s_1\right)$ in the Definition (2), then the set $\mathcal{A}$ becomes an $(E,F)$-convex set.

Remark 3.

If we take the $E=F$ identity map in the Definition (2), then the set $\mathcal{A}$ becomes an invex set.

Remark 4.

If we take E and F to be the identity maps on $\mathcal{A}$ and $G(E\left(r_1\right),F\left(s_1\right))=E\left(r_1\right)-F\left(s_1\right)$ in the Definition (2), then the set $\mathcal{A}$ becomes a convex set.

Definition 3.

A real-valued function $\mathcal{H}:\mathcal{A}\to \mathcal{R}$ defined on the $(E,F)$-invex set $\mathcal{A}$ is known as an $(E,F)$-preinvex function if

$$\begin{array}{c}\hfill \mathcal{H}(F\left(s_1\right)+\mu G(E\left(r_1\right),F\left(s_1\right)))\le \mu \mathcal{H}\left(E\left(r_1\right)\right)+(1-\mu )\mathcal{H}\left(F\left(s_1\right)\right)\end{array}$$

(2)

holds for $\forall r_1,s_1\in \mathcal{A}$ and $\forall \mu \in [0,1]$.

Theorem 1.

Let $\mathcal{A}$ be a non-empty $(E,F)$-invex subset of ${\mathcal{R}}^n$. Let functions ${\mathcal{H}}_i,i=1,2,..,n$ be $(E,F)$-preinvex functions on the $(E,F)$-invex set $\mathcal{A}$ and ${\alpha }_i\ge 0,i=1,2,...,n$. Then, ${\sum }_{i=1}^n{\alpha }_i{\mathcal{H}}_i$ is also an $(E,F)$-preinvex function on the $(E,F)$-invex set $\mathcal{A}$.

Definition 4.

Let $\mathcal{H}$ be a real-valued differentiable function on the open interval $\mathcal{A}$ of ${\mathcal{R}}^n$. The function $\mathcal{H}$ is known as an $(E,F)$-invex on A if there we have the maps $E,F:\mathcal{A}\to {\mathcal{R}}^n$ and a bi-function $G:{\mathcal{R}}^n\times {\mathcal{R}}^n\to {\mathcal{R}}^n$ such that the inequality

$$\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right)\ge G{(E\left(r_1\right),F\left(s_1\right))}^T\nabla \mathcal{H}\left(F\left(s_1\right)\right)$$

holds for each $r_1,s_1\in \mathcal{A}$ and $\mu \in [0,1]$.

Now, we will explore all the above definitions in Riemannian manifolds.

4. Geodesic $(\mathit{E},\mathit{F})$-Invex Sets and $(\mathit{E},\mathit{F})$-Preinvex Functions on Riemannian Manifolds

Definition 5.

Let $\mathcal{B}$ be a non-empty subset of the Riemannian manifold $\mathcal{O}$, and let $E,F:\mathcal{B}\to \mathcal{O},G:\mathcal{O}\times \mathcal{O}\to T\mathcal{O}$ be functions such that for every $E\left(r_1\right),F\left(s_1\right)\in \mathcal{O},G(E\left(r_1\right),F\left(s_1\right))\in T_{F\left(s_1\right)}\mathcal{O}$. Then, the subset $\mathcal{B}$ of $\mathcal{O}$ is said to be geodesic $(E,F)$-invex if, for each $E\left(r_1\right),F\left(s_1\right)\in \mathcal{B}$, there exists exactly one geodesic ${\beta }_{E\left(r_1\right),F\left(s_1\right)}:[0,1]\to \mathcal{O}$ such that

$${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

Now, we define the $(E,F)$-invex function on an open geodesic $(E,F)$-invex subset of a Riemannian manifold.

Definition 6.

Let $\mathcal{B}$ be a non-empty open subset of the Riemannian manifold $\mathcal{O}$, with Riemannian metric g and $\mathcal{B}$ being a geodesic invex set. A function $\mathcal{H}:\mathcal{B}\to \mathcal{R}$ is said to be $(E,F)$-invex on $\mathcal{B}$ if the following inequality holds:

$$\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right)\ge ⟨\nabla {\mathcal{H}}_{F\left(s_1\right)},G(E\left(r_1\right),F\left(s_1\right))⟩\forall E\left(r_1\right),F\left(s_1\right)\in \mathcal{B}.$$

Definition 7.

Let $\mathcal{B}$ be a non-empty geodesic $(E,F)$-invex subset of the Riemannian manifold $\mathcal{O}$. A function $\mathcal{H}:\mathcal{B}\to \mathcal{R}$ is said to be $(E,F)$-preinvex if, for each $E\left(r_1\right),F\left(s_1\right)\in \mathcal{B}$ and $\forall s\in [0,1]$,

$$\begin{array}{c}\hfill \mathcal{H}\left({\beta }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(s\right)\le s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right)\end{array}$$

(3)

holds, where ${\beta }_{E\left(r_1\right),F\left(s_1\right)}$ is the unique geodesic in Definition 5. If the above inequality (3) is strict for $E\left(r_1\right)\ne F\left(s_1\right)$ and $s\in (0,1)$, then $\mathcal{H}$ is said to be a strict $(E,F)$-preinvex function.

Example 2.

Let $\mathcal{O}=[-8,8]$ be a non-empty subset of $\mathcal{R}$, and let $E,F:\mathcal{O}\to \mathcal{R}$ be two maps defined as

$$E\left(r_1\right)=3r_{1}andF\left(s_1\right)=s_1,$$

and let $G:\mathcal{R}\times \mathcal{R}\to \mathcal{R}$, a point-to-point bi-function, be defined as

$$G(E\left(r_1\right),F\left(s_1\right))={\displaystyle \frac{E\left(r_1\right)}{3}}-F\left(s_1\right).$$

Now, consider the geodesic β as

$$\begin{array}{ccc}\hfill {\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)& =& F\left(s_1\right)+s\left[{\displaystyle \frac{E\left(r_1\right)}{3}}-F\left(s_1\right)\right]\hfill \\ & =& s_1+s(r_1-s_1).\hfill \end{array}$$

It can be easily seen that

$${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{O},\forall s\in [0,1].$$

So, $\mathcal{O}$ is a geodesic $(E,F)$-invex set. Now, we define a function $\mathcal{H}:\mathcal{O}\to \mathcal{R}$ as $\mathcal{H}\left(r_1\right)=r_1+r_1^2$. Then, $\mathcal{H}$ is an $(E,F)$-preinvex function on the set $\mathcal{O}$.

Proposition 1.

Let $\mathcal{B}$ be a non-empty geodesic $(E,F)$-invex subset of the Riemannian manifold $\mathcal{O}$. Consider the function $\mathcal{H}:\mathcal{B}\to \mathcal{R}$, an $(E,F)$-preinvex. Thus, we have

(a) Every lower level set of $\mathcal{H}$ defined by

$$P(\mathcal{H},t):=\{E\left(r_1\right)\in \mathcal{B}|\mathcal{H}\left(E\left(r_1\right)\right)\le t\}$$

is a geodesic $(E,F)$-invex set.

(b) An optimization problem is presented as follows:

$$\begin{array}{cc}\hfill & min\mathcal{H}\left(w\right)\hfill \\ \hfill & subject\mathrm{to}w\in \mathcal{B}\hfill \end{array}$$

(4)

where the set L of the solution of the problem (4) is a geodesic $(E,F)$-invex set. Also, if $\mathcal{H}$ is strictly an $(E,F)$-preinvex function, then $\mathcal{O}$ contains (at most) one point.

Proof. 

$\left(a\right)$ Take $E\left(r_1\right),F\left(s_1\right)\in P(\mathcal{H},t)$. Then, we have $\mathcal{H}\left(E\left(r_1\right)\right)\le tand\mathcal{H}\left(F\left(s_1\right)\right)\le t$. As $\mathcal{B}$ is a geodesic $(E,F)$-invex set, there exists exactly one geodesic ${\beta }_{E\left(r_1\right),F\left(s_1\right)}:[0,1]\to \mathcal{O}$ such that

$${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

Since $\mathcal{H}$ is $(E,F)$-preinvex, then, by definition, we have

$$\begin{array}{ccc}\hfill \mathcal{H}\left({\beta }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(s\right)& \le & s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right)\hfill \\ & \le & st+(1-s)t\hfill \\ & =& t.\hfill \end{array}$$

Hence, ${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in P(\mathcal{H},t)$ for any $s\in [0,1]$.

(b) If we take $\beta :={inf}_{E\left(r_1\right)\in \mathcal{B}}\mathcal{H}\left(E\left(r_1\right)\right)$, then it is clear that $L={\cap }_{t>\gamma }P(\mathcal{H},t)$. Therefore, L is an intersection of a geodesic $(E,F)$-invex set. Hence, it is a geodesic $(E,F)$-invex set as well.

Now, if $\mathcal{H}$ is strictly an $(E,F)$-preinvex function, then, for the geodesic $(E,F)$-invexity of L, there exists exactly one geodesic ${\gamma }_{E\left(r_1\right),F\left(s_1\right)}:[0,1]\to \mathcal{O}$ such that

$${\gamma }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\gamma }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\gamma }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

By the strictly $(E,F)$-preinvexity of $\mathcal{H}$, we have

$$\begin{array}{ccc}\hfill \beta =\mathcal{H}\left({\gamma }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(s\right)& <& s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right)\hfill \\ & =& s\beta +(1-s)\beta \hfill \\ & =& \beta \hfill \end{array}$$

which is not possible. □

Theorem 2.

Let $\mathcal{B}$ be a non-empty open geodesic $(E,F)$-invex subset of the Riemannian manifold $\mathcal{O}$. Consider that $\mathcal{H}:\mathcal{B}\to \mathcal{O}$ is a differentiable and $(E,F)$-preinvex function. Then, $\mathcal{H}$ is an $(E,F)$-invex function.

Proof. 

Since $\mathcal{B}$ is a geodesic $(E,F)$-invex set of the Riemannian manifold, there exists exactly one geodesic ${\beta }_{E\left(r_1\right),F\left(s_1\right)}:[0,1]\to \mathcal{O}$ for every $E\left(r_1\right),F\left(s_1\right)\in \mathcal{B}$, such that

$${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

Now, as $\mathcal{H}$ is the $(E,F)$-preinvex for $s\in (0,1)$, we obtain

$$\mathcal{H}\left({\beta }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(s\right)\le s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right).$$

This may also be written as

$$\mathcal{H}\left({\beta }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(s\right)-\mathcal{H}\left(F\left(s_1\right)\right)\le s[\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right)],$$

we divide both sides by s, and we obtain

$${\displaystyle \frac{\mathcal{H}\left({\beta }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(s\right)-\mathcal{H}\left(F\left(s_1\right)\right)}{s}}\le [\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right)].$$

Taking the limit as $s\to 0$, we have

$$⟨\nabla \mathcal{H}\left({\beta }_{E\left(r_1\right),F\left(s_1\right)}\right)\left(0\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)⟩\le \mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right).$$

Therefore,

$$⟨\nabla {\mathcal{H}}_{F\left(s_1\right)},G(E\left(r_1\right),F\left(s_1\right))⟩\le \mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right).$$

Hence, $\mathcal{H}$ is an $(E,F)$-invex function. □

Definition 8.

Let the functions $E,F:\mathcal{O}\to \mathcal{O}$ be defined on a Riemannian manifold $\mathcal{O}$. Then, we say that the function $G:\mathcal{O}\times \mathcal{O}\to T\mathcal{O}$ satisfies the condition $\mathcal{A}$, if, for every $E\left(r_1\right),F\left(s_1\right)\in \mathcal{O}$ and for the geodesic $\beta :[0,1]\to \mathcal{O}$ satisfying ${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right))$, we have

$\left({\mathcal{A}}_1\right)$$P_{s,\beta }^o\left[G(F\left(s_1\right),\beta \left(s\right))\right]=-sG(E\left(r_1\right),F\left(s_1\right))$

$\left({\mathcal{A}}_2\right)$$P_{s,\beta }^o\left[G(E\left(r_1\right),\beta \left(s\right))\right]=(1-s)G(E\left(r_1\right),F\left(s_1\right))$

for all $s\in [0,1]$.

Theorem 3.

Let $\mathcal{B}$ be a non-empty open geodesic $(E,F)$-invex subset of the Riemannian manifold $\mathcal{O}$. Suppose that the function $\mathcal{H}:\mathcal{B}\to \mathcal{R}$ is differentiable. If $\mathcal{H}$ is an $(E,F)$-invex on $\mathcal{B}$ and $\mathcal{H}$ satisfies the condition $\mathcal{A}$, then $\mathcal{H}$ is an $(E,F)$-preinvex function on $\mathcal{B}$.

Proof. 

Since $\mathcal{B}$ is a geodesic $(E,F)$-invex set of the Riemannian manifold, then, for every $E\left(r_1\right),F\left(s_1\right)\in \mathcal{B}$, there exists exactly one geodesic ${\beta }_{E\left(r_1\right),F\left(s_1\right)}:[0,1]\to \mathcal{O}$ such that

$${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

Now, fix $s\in [0,1]$ and set $F\left(\overline{s_1}\right):={\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)$. Then, we have

$$\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(\overline{s_1}\right)\right)\ge ⟨\nabla {\mathcal{H}}_{F\left(\overline{s_1}\right)},G(E\left(r_1\right),F\left(\overline{s_1}\right))⟩$$

(5)

$$\mathcal{H}\left(E\left(s_1\right)\right)-\mathcal{H}\left(F\left(\overline{s_1}\right)\right)\ge ⟨\nabla {\mathcal{H}}_{F\left(\overline{s_1}\right)},G(E\left(s_1\right),F\left(\overline{s_1}\right))⟩.$$

(6)

Multiplying Inequality (5) by s and Inequality (6) by $1-s$, respectively, and then carrying out addition, will lead to

$$\begin{array}{ccc}\hfill s\mathcal{H}\left(E\left(r_1\right)\right)& +& \\ \hfill (1-s)\mathcal{H}\left(F\left(s_1\right)\right)-\mathcal{H}\left(F\left(\overline{s_1}\right)\right)& \ge & ⟨\nabla {\mathcal{H}}_{F\left(\overline{s_1}\right)},sG(E\left(r_1\right),F\left(\overline{s_1}\right)⟩+(1-s)G(E\left(r_1\right),\left(F\left(\overline{s_1}\right)\right).\hfill \end{array}$$

From Condition $\mathcal{A}$, we obtain

$$\begin{array}{ccc}\hfill sG(E\left(r_1\right),F\left(\overline{s_1}\right))+(1-s)G(E\left(r_1\right),\left(F\left(\overline{s_1}\right)\right)& =& s(1-s)P_{s,\beta }^o[G(E\left(r_1\right),\left(F\left(s_1\right)\right)]\hfill \\ & & +(1-s)(-s)[G(E\left(r_1\right),\left(F\left(s_1\right)\right)]\hfill \\ & =& 0.\hfill \end{array}$$

Therefore, we obtain

$$s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right)\ge \mathcal{H}\left(F\left(\overline{s_1}\right)\right).$$

Hence, $\mathcal{H}$ is an $(E,F)$-preinvex function on the $(E,F)$-invex set $\mathcal{O}$. □

Theorem 4.

Let $\mathcal{B}$ be a non-empty open geodesic $(E,F)$-invex subset of the Riemannian manifold $\mathcal{O}$. Suppose that the function $\mathcal{H}:\mathcal{B}\to \mathcal{R}$ is an $(E,F)$-preinvex. If $F\left({\overline{r}}_1\right)\in \mathcal{B}$ is a local optimal solution to the problem

$$\begin{array}{cc}\hfill & minimize\mathcal{H}\left(F\left(r_1\right)\right)\hfill \\ \hfill & subject\mathit{to}F\left(r_1\right)\in \mathcal{B},\hfill \end{array}$$

(7)

then $F\left({\overline{r}}_1\right)$ is a global minimum in Problem $\left(7\right)$.

Proof. 

Consider a local minimum $F\left({\overline{r}}_1\right)$. Then, there exists a neighborhood $C_{ϵ}\left(F\left({\overline{r}}_1\right)\right)$ such that

$$\mathcal{H}\left(F\left({\overline{r}}_1\right)\right)\le \mathcal{H}\left(E\left(r_1\right)\right),\forall E\left(r_1\right)\in \mathcal{B}\cap C_{ϵ}\left(F\left({\overline{r}}_1\right)\right).$$

(8)

Conversely, we assume that $F\left({\overline{r}}_1\right)$ is not a global minimum of $\mathcal{H}$. Then, there exists an element $E\left(s_1^{*}\right)\in \mathcal{B}$ s.t.

$$\mathcal{H}\left(E\left(r_1^{*}\right)\right)<\mathcal{H}\left(F\left({\overline{r}}_1\right)\right).$$

As $\mathcal{B}$ is a geodesic $(E,F)$-invex set, there exists exactly one geodesic ${\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}:[0,1]\to \mathcal{O}$ such that

$${\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}\left(0\right)=F\left(\overline{r_1}\right),{\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}^{\prime }\left(0\right)=G(E\left(r_1^{*}\right),F\left(\overline{r_1}\right)),{\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

Now, if we choose $ϵ>0$ to be small enough s.t. $d({\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}\left(s\right),F\left(\overline{r_1}\right))<ϵ$, then ${\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}\left(s\right)\in C_{ϵ}\left(F\left(\overline{r_1}\right)\right)$. As $\mathcal{H}$ represents $(E,F)$-preinvexity, we have

$$\begin{array}{ccc}\hfill \mathcal{H}\left({\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}\left(s\right)\right)& \le & s\mathcal{H}\left(E\left(r_1^{*}\right)\right)+(1-s)\mathcal{H}(F\left(\overline{r_1}\right)\hfill \\ & <& \mathcal{H}\left(F\left(\overline{r_1}\right)\right),\forall s\in (0,1).\hfill \end{array}$$

Therefore, for each ${\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}\left(s\right)\in \mathcal{B}\cap C_{ϵ}\left(F\left(\overline{r_1}\right)\right)$, which is a contradiction to Inequality (8), our result is proved. □

Definition 9.

Let the functions $E,F:\mathcal{O}\to \mathcal{O}$ be defined on Riemannian manifold $\mathcal{O}$, and let function $\mathcal{H}:\mathcal{O}\to (-\infty ,\infty ]$ be a lower semi-continuous function. An element σ in $T_{F\left(s_1\right)}\mathcal{O}$ is an $(E,F)$-proximal sub-gradient of $\mathcal{H}$ at $F\left(s_1\right)\in dom\left(\mathcal{H}\right)$, if ∃ the numbers $\mu >0$ and $\lambda >0$ s.t.

$$\mathcal{H}\left(E\left(r_1\right)\right)\ge \mathcal{H}\left(F\left(s_1\right)\right)+{⟨\sigma ,{exp}_{F\left(s_1\right)}^{-1}E\left(r_1\right)⟩}_{F\left(s_1\right)}-\lambda d{(E\left(r_1\right),F\left(s_1\right))}^2,$$

$\forall E\left(r_1\right)\in N(F\left(s_1\right),\mu )$, and where $dom\left(\mathcal{H}\right):=\{E\left(r_1\right)\in \mathcal{O}:\mathcal{H}\left(E\left(r_1\right)\right)<\infty \}$. The set of all $(E,F)$-proximal sub-gradients of $\mathcal{H}$ at $F\left(s_1\right)\in \mathcal{O}$ is denoted by ${\partial }_{(E,F)}\mathcal{H}\left(F\left(s_1\right)\right)$ and is called the $(E,F)$-proximal sub-differential of $\mathcal{H}$ at $F\left(s_1\right)$.

Theorem 5.

Let $\mathcal{B}$ be a non-empty open geodesic $(E,F)$-invex subset of the Cartan–Hadamard manifold $\mathcal{O}$ with $G(E\left(r_1\right),F\left(s_1\right))\ne 0$ for $E\left(r_1\right)\ne F\left(s_1\right)$. Also, let $\mathcal{H}:\mathcal{B}\to (-\infty ,\infty ]$ be a lower semi-continuous $(E,F)$-preinvex function. Assume the $F\left(s_1\right)\in$ domain to be $\left(\mathcal{H}\right)$ and $\sigma \in {\partial }_{(E,F)}\mathcal{H}\left(F\left(s_1\right)\right)$. Then, there exists a real number μ s.t.

$$\mathcal{H}\left(E\left(r_1\right)\right)\ge \mathcal{H}\left(F\left(s_1\right)\right)+{⟨\sigma ,G(E\left(r_1\right),F\left(s_1\right))⟩}_{F\left(s_1\right)},\forall E\left(r_1\right)\in \mathcal{B}\cap N(F\left(s_1\right),\mu )$$

(9)

Proof. 

Using the definition of the $(E,F)$-proximal sub-gradient ${\partial }_{(E,F)}\mathcal{H}\left(F\left(s_1\right)\right)$, the numbers $\mu >0$ and $\lambda >0$ exist s.t.

$$\mathcal{H}\left(E\left(r_1\right)\right)\ge \mathcal{H}\left(F\left(s_1\right)\right)+⟨\sigma ,{exp}_{F\left(s_1\right)}^{-1}E\left(r_1\right){⟩}_{F\left(s_1\right)}-\lambda d{(E\left(r_1\right),F\left(s_1\right))}^2,$$

(10)

$\forall E\left(r_1\right)\in N(F\left(s_1\right),\mu )$, specifically for $E\left(r_1\right)\in \mathcal{B}\cap N(F\left(s_1\right),\mu )$. As $\mathcal{B}$ is a geodesic $(E,F)$-invex set, there exists exactly one geodesic ${\beta }_{E\left(r_1^{*}\right),F\left(\overline{r_1}\right)}:[0,1]\to \mathcal{O}$ such that

$${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(0\right)=F\left(s_1\right),{\beta }_{E\left(r_1\right),F\left(s_1\right)}^{\prime }\left(0\right)=G(E\left(r_1\right),F\left(s_1\right)),{\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)\in \mathcal{B},\forall s\in [0,1].$$

As $\mathcal{O}$ is a Cartan–Hadamard manifold, ${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)={exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right)$ for any $s\in [0,1]$ (see [24] p. 776). Now, take $r_1={\displaystyle \frac{\mu }{\parallel G(E\left(r_1\right),F\left(s_1\right)){\parallel }_{F\left(s_1\right)}}}$. Therefore, ${\beta }_{E\left(r_1\right),F\left(s_1\right)}\left(s\right)={exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right)\in \mathcal{B}\cap N(F\left(s_1\right),\mu )$$,\forall s\in (0,r_1)$.

As $\mathcal{H}$ is geodesic $(E,F)$-preinvex, we have

$$\mathcal{H}({exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right))\le s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right),\forall s\in (0,r_1)$$

(11)

Combining Inequalities (10) and (11), we have

$$\begin{array}{ccc}\hfill \mathcal{H}({exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right)& \ge & \mathcal{H}\left(F\left(s_1\right)\right)-\lambda d{({exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right),F\left(s_1\right))}^2\hfill \\ & & +{⟨\sigma ,{exp}_{F\left(s_1\right)}^{-1}{exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right)⟩}_{F\left(s_1\right)}\hfill \\ & =& \mathcal{H}\left(F\left(s_1\right)\right)+{⟨\sigma ,sG(E\left(r_1\right),F\left(s_1\right))⟩}_{F\left(s_1\right)}\hfill \\ & & -\lambda d({exp}_{F\left(s_1\right)}{(sG(E\left(r_1\right),F\left(s_1\right)),F\left(s_1\right))}^2\hfill \end{array}$$

(12)

$\forall E\left(r_1\right)$ in $N(F\left(s_1\right),\mu )$. As $\mathcal{O}$ can be a Cartan–Hadamard manifold, we have

$$\begin{array}{ccc}\hfill d({exp}_{F\left(s_1\right)}{(sG(E\left(r_1\right),F\left(s_1\right)),F\left(s_1\right))}^2& =& \parallel sG(E\left(r_1\right),F\left(s_1\right)){\parallel }_{F\left(s_1\right)}^2\hfill \\ & =& s^2{\parallel G(E\left(r_1\right),F\left(s_1\right))\parallel }_{F\left(s_1\right)}^2.\hfill \end{array}$$

(13)

By combining Inequalities (11) and (12) with Equation (13), we have

$$\begin{array}{ccc}\hfill s\mathcal{H}\left(E\left(r_1\right)\right)+(1-s)\mathcal{H}\left(F\left(s_1\right)\right)& \ge & \mathcal{H}({exp}_{F\left(s_1\right)}\left(sG(E\left(r_1\right),F\left(s_1\right))\right)\hfill \\ & \ge & \mathcal{H}(F\left(s_1\right))+{⟨\sigma ,sG(E\left(r_1\right),F\left(s_1\right))⟩}_{F\left(s_1\right)}\hfill \\ & & -\lambda s^2{\parallel G(E\left(r_1\right),F\left(s_1\right))\parallel }_{F\left(s_1\right)}^2.\hfill \end{array}$$

⇒

$$s[\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}\left(F\left(s_1\right)\right]\ge s{⟨\sigma ,G(E\left(r_1\right),F\left(s_1\right))⟩}_{F\left(s_1\right)}-\lambda s^2\parallel G(E\left(r_1\right),F\left(s_1\right)){\parallel }_{F\left(s_1\right)}^2$$

⇒

$$\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}(F\left(s_1\right)\ge {⟨\sigma ,G(E\left(r_1\right),F\left(s_1\right))⟩}_{F\left(s_1\right)}-\lambda s{\parallel G(E\left(r_1\right),F\left(s_1\right))\parallel }_{F\left(s_1\right)}^2.$$

Now, taking limit $lims\to 0$ on both sides, we have

$$\mathcal{H}\left(E\left(r_1\right)\right)-\mathcal{H}(F\left(s_1\right)\ge {⟨\sigma ,G(E\left(r_1\right),F\left(s_1\right))⟩}_{F\left(s_1\right)},\forall E\left(r_1\right)\in \mathcal{B}\cap N(F\left(s_1\right),\mu ).$$

Therefore, the proof is completed. □