 Research
 Open Access
 Published:
On properties of geodesic semilocal Epreinvex functions
Journal of Inequalities and Applications volume 2018, Article number: 353 (2018)
Abstract
The authors define a class of functions on Riemannian manifolds, which are called geodesic semilocal Epreinvex functions, as a generalization of geodesic semilocal Econvex and geodesic semi Epreinvex functions, and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming is considered, where the functions involved are geodesic Eηsemidifferentiability, sufficient optimality conditions are obtained. A dual is formulated and duality results are proved by using concepts of geodesic semilocal Epreinvex functions, geodesic pseudosemilocal Epreinvex functions, and geodesic quasisemilocal Epreinvex functions.
Introduction
Convexity and generalized convexity play a significant role in many fields, for example, in biological system, economy, optimization, and so on [1,2,3,4,5].
Generalized convex functions, labeled as semilocal convex functions, were introduced by Ewing [6] by using more general semilocal preinvexity and ηsemidifferentiability. After that optimality conditions for weak vector minima were given [7]. Also, optimality conditions and duality results for a nonlinear fractional involving ηsemidifferentiability were established [8].
Furthermore, some optimality conditions and duality results for semilocal Econvex programming were established [9]. Econvexity was extended to Epreinvexity [10]. Recently, semilocal Epreinvexity (SLEP) and some of its applications were introduced [11,12,13].
Generalized convex functions in manifolds, such as Riemannian manifolds, were studied by many authors; see [14,15,16,17]. Udrist [18] and Rapcsak [19] considered a generalization of convexity called geodesic convexity. In this setting, the linear space is replaced by a Riemannian manifold and the line segment by a geodesic one. In 2012, geodesic Econvex (GEC) sets and geodesic Econvex (GEC) functions on Riemannian manifolds were studied [20]. Moreover, geodesic semi Econvex (GsEC) functions were introduced [21]. Recently, geodesic strongly Econvex (GSEC) functions were introduced and some of their properties were discussed [22].
Geodesic semilocal Epreinvexity
Assume that ℵ is a complete ndimensional Riemannian manifold with Riemannian connection ▽. Let \(\kappa _{1}, \kappa _{2} \in \aleph \) and \(\gamma \colon [0,1]\longrightarrow \aleph \) be a geodesic joining the points \(\kappa _{1} \) and \(\kappa _{2} \), which means that \(\gamma _{\kappa _{1},\kappa _{2}}(0)= \kappa _{2}\) and \(\gamma _{\kappa _{1},\kappa _{2}}(1)=\kappa _{1} \).
Definition 2.1
A nonempty set \(B \subset \aleph \) is said to be

1.
a geodesic Einvex (GEI) with respect to η if there is exactly one geodesic \(\gamma _{E(\kappa _{1}), E(\kappa _{2})}: [0,1 ] \longrightarrow \aleph \) such that
$$\begin{aligned} \gamma _{E(\kappa _{1}), E(\kappa _{2})}(0)=E(\kappa _{2}), \qquad \acute{\gamma }_{E(\kappa _{1}), E(\kappa _{2})}=\eta \bigl(E(\kappa _{1}),E( \kappa _{2}) \bigr), \qquad \gamma _{E(\kappa _{1}), E(\kappa _{2})}(t)\in B, \end{aligned}$$\(\forall \kappa _{1},\kappa _{2}\in B\) and \(t\in [0,1]\).

2.
a geodesic local Einvex (GLEI) with respect to η if there is \(u(\kappa _{1},\kappa _{2})\in (0,1 ] \) such that \(\forall t\in [0,u (\kappa _{1},\kappa _{2})] \),
$$ \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\in B \quad \forall \kappa _{1},\kappa _{2}\in B. $$(1) 
3.
a geodesic local starshaped Econvex if there is a map E such that, corresponding to each pair of points \(\kappa _{1},\kappa _{2}\in A \), there is a maximal positive number \(u(\kappa _{1},\kappa _{2})\leq 1 \) such as
$$ \gamma _{E(\kappa _{1}),E(\kappa _{2})}\in A, \quad \forall t\in \bigl[0, u(\kappa _{1},\kappa _{2})\bigr]. $$(2)
Definition 2.2
A function \(f: A\subset \aleph \longrightarrow \mathbb{R} \) is said to be

1.
a geodesic Epreinvex (GEP) on \(A\subset \aleph \) with respect to η if A is a GEI set and
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f\bigl(E( \kappa _{1}) \bigr)+(1t)f\bigl(E(\kappa _{2})\bigr) , \quad \forall \kappa _{1},\kappa _{2}\in A, t\in [0,1]; $$ 
2.
a geodesic semi Epreinvex (GSEP) on A with respect to η if A is a GEI set and
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f(\kappa _{1})+(1t)f( \kappa _{2}) , \quad \forall \kappa _{1},\kappa _{2}\in A, t\in [0,1]. $$ 
3.
a geodesic local Epreinvex (GLEP) on \(A\subset \aleph \) with respect to η if, for any \(\kappa _{1},\kappa _{2}\in A \), there exists \(0< v(\kappa _{1},\kappa _{2})\leq u(\kappa _{1},\kappa _{2}) \) such that A is a GLEI set and
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f\bigl(E( \kappa _{1}) \bigr)+(1t)f\bigl(E(\kappa _{2})\bigr) , \quad \forall t\in \bigl[0,v( \kappa _{1},\kappa _{2})\bigr]. $$
Definition 2.3
A function \(f:\aleph \longrightarrow \mathbb{R} \) is a geodesic semilocal Econvex (GSLEC) on a geodesic local starshaped Econvex set \(B\subset \aleph \) if, for each pair of \(\kappa _{1},\kappa _{2}\in B \) (with a maximal positive number \(u(\kappa _{1},\kappa _{2})\leq 1 \) satisfying 2), there exists a positive number \(v(\kappa _{1},\kappa _{2})\leq u(\kappa _{1},\kappa _{2}) \) satisfying
Remark 2.1
Every GEI set with respect to η is a GLEI set with respect to η, where \(u(\kappa _{1},\kappa _{2})=1\), \(\forall \kappa _{1},\kappa _{2}\in \aleph \). On the other hand, their converses are not necessarily true, and we can see that in the next example.
Example 2.1
Put \(A= [ \left . 4,1 ) \right . \cup [1,4] \),
Hence A is a GLEI set with respect to η. However, when \(\kappa =3\), \(\iota =0 \), there is \(t_{1}\in [0,1] \) such that \(\gamma _{E(\kappa ),E(\iota )}(t_{1})=t_{1} \), then if \(t_{1}=1 \), we obtain \(\gamma _{E(\kappa ),E(\iota )}(t_{1})\notin A \).
Definition 2.4
A function \(f: \aleph \longrightarrow \mathbb{R} \) is GSLEP on \(B\subset \aleph \) with respect to η if, for any \(\kappa _{1},\kappa _{2}\in B \), there is \(0< v(\kappa _{1},\kappa _{2})\leq u(\kappa _{1},\kappa _{2})\leq 1 \) such that B is a GLEI set and
If
then f is GSLEP on B.
Remark 2.2
Any GSLEC function is a GSLEP function. Also, any GSEP function with respect to η is a GSLEP function. On the other hand, their converses are not necessarily true. The next example shows SLGEP, which is neither a GSLEC function nor a GSEP function.
Example 2.2
Assume that \(E: \mathbb{R}\longrightarrow \mathbb{R} \) is given as
and the map \(\eta : \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R} \) is defined as
also,
Assume that \(h: \mathbb{R}\longrightarrow \mathbb{R} \), where
and since \(\mathbb{R} \) is a geodesic local starshaped Econvex set and a geodesic local Einvex set with respect to η. Then h is a GSLEP on \(\mathbb{R} \) with respect to η. However, when \(m_{0}=2\), \(n_{0}=3 \) and for any \(v\in (0,1 ] \), there is a sufficiently small \(t_{0}\in (0,v ] \) such that
Then \(h(m) \) is not a GSLEC function on \(\mathbb{R} \).
Similarly, taking \(m_{1}=1\), \(n_{1}=4 \), we have
for some \(t_{1}\in [0,1] \).
Hence, \(h(m) \) is not a GSEP function on \(\mathbb{R} \) with respect to η.
Definition 2.5
A function \(h:S\subset \aleph \longrightarrow \mathbb{R} \), where S is a GLEI set, is said to be a geodesic quasisemilocal Epreinvex (GqSLEP) (with respect to η) if, for all \(\kappa _{1},\kappa _{2}\in S \) satisfying \(h(\kappa _{1})\leq h( \kappa _{2}) \), there is a positive number \(v(\kappa _{1},\kappa _{2}) \leq u(\kappa _{1},\kappa _{2}) \) such that
Definition 2.6
A function \(h:S\subset \aleph \longrightarrow \mathbb{R} \), where S is a GLEI set, is said to be a geodesic pseudosemilocal Epreinvex (GpSLEP) (with respect to η) if, for all \(\kappa _{1},\kappa _{2}\in S \) satisfying \(h(\kappa _{1})< h(\kappa _{2}) \), there are positive numbers \(v(\kappa _{1},\kappa _{2})\leq u( \kappa _{1},\kappa _{2}) \) and \(w(\kappa _{1},\kappa _{2}) \) such that
Remark 2.3
Every GSLEP on a GLEI set with respect to η is both a GqELEP function and a GpSLEP function.
Definition 2.7
A function \(h:S\longrightarrow \mathbb{R} \) is called a geodesic Eη semidifferentiable at \(\kappa ^{*} \in S \), where \(S\subset \aleph \) is a GLEI set with respect to η, if \(E(\kappa ^{*})=\kappa ^{*} \) and
exist for every \(\kappa \in S\).
Remark 2.4

1.
If \(\aleph =\mathbb{R}^{n} \), then the geodesic Eη semidifferentiable is Eηsemidifferentiable [11].

2.
If \(\aleph =\mathbb{R}^{n} \) and \(E=I \), then the geodesic Eηsemidifferentiable is the ηsemidifferentiability [23].

3.
If \(\aleph =\mathbb{R}^{n} \), \(E=I \), and \(\eta (\kappa ,\kappa ^{*})=\kappa \kappa ^{*} \), then the geodesic Eηsemidifferentiable is the semidifferentiability [11].
Lemma 2.1

1.
Assume that h is a GSLEP (Epreconcave) and a geodesic Eηsemidifferentiable at \(\kappa ^{*}\in S\subset \aleph \), where S is a GLEI set with respect to η. Then
$$ h(\kappa )h\bigl(\kappa ^{*}\bigr)\geqslant (\leq ) h'_{+}\bigl( \gamma _{\kappa ^{*},E(\kappa )}(t)\bigr),\quad \forall \kappa \in S. $$ 
2.
Let h be a GqSLEP (GpSLEP) and a geodesic Eηsemidifferentiable at \(\kappa ^{*}\in S\subset \aleph \), where S is a LGEI set with respect to η. Hence
$$ h(\kappa )\leq (< ) h\bigl(\kappa ^{*}\bigr)\quad \Rightarrow\quad h'_{+}\bigl( \gamma _{\kappa ^{*},E(\kappa )}(t)\bigr)\leq (< )0, \quad \forall \kappa \in S. $$
The above lemma is proved directly by using definitions (Definition 2.4, Definition 2.5, Definition 2.6, and Definition 2.4).
Theorem 2.1
Let \(f: S\subset \aleph \longrightarrow \mathbb{R} \) be a GLEP function on a GLEI set S with respect to η, then f is a GSLEP function iff \(f(E(\kappa ))\leq f( \kappa )\), \(\forall \kappa \in S \).
Proof
Assume that f is a GSLEP function on set S with respect to η, then \(\forall \kappa _{1},\kappa _{2}\in S \), there is a positive number \(v(\kappa _{1},\kappa _{2})\leq u(\kappa _{1},\kappa _{2}) \) where
By letting \(t=0 \), then \(f(E(\kappa _{1}))\leq f(\kappa _{1})\), \(\forall \kappa _{1}\in S \).
Conversely, consider that f is a GLEP function on a GLEI set S, then for any \(\kappa _{1},\kappa _{2}\in S \), there exist \(u(\kappa _{1},\kappa _{2}) \in (0,1 ] \) (1) and \(v(\kappa _{1},\kappa _{2}) \in (0,u(\kappa _{1},\kappa _{2}) ] \) such that
Since \(f(E(\kappa _{1})) \leq f(\kappa _{1})\), \(\forall \kappa _{1}\in S\), then
□
Definition 2.8
The set \(\omega = \lbrace (\kappa , \alpha ):\kappa \in B\subset \aleph , \alpha \in \mathbb{R} \rbrace \) is said to be a GLEI set with respect to η corresponding to ℵ if there are two maps η, E and a maximal positive number \(u((\kappa _{1},\alpha _{1}), (\kappa _{2}, \alpha _{2}))\leq 1 \) for each \((\kappa _{1},\alpha _{1}), (\kappa _{2}, \alpha _{2})\in \omega \) such that
Theorem 2.2
Let \(B\subset \aleph \) be a GLEI set with respect to η. Then f is a GSLEP function on B with respect to η iff its epigraph
is a GLEI set with respect to η corresponding to ℵ.
Proof
Suppose that f is a GSLEP on B with respect to η and \((\kappa _{1},\alpha _{1}), (\kappa _{2},\alpha _{2})\in \omega _{f} \), then \(\kappa _{1},\kappa _{2}\in B\), \(f(\kappa _{1})\leq \alpha _{1}\), \(f(\kappa _{2})\leq \alpha _{2} \). By applying Definition 2.1, we obtain \(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\in B\), \(\forall t\in [0, u(\kappa _{1},\kappa _{2}) ]\).
Moreover, there is a positive number \(v(\kappa _{1},\kappa _{2})\leq u( \kappa _{1},\kappa _{2}) \) such that
Conversely, if \(\omega _{f} \) is a GLEI set with respect to η corresponding to ℵ, then for any points \((\kappa _{1},f(\kappa _{1})) , (\kappa _{2},f(\kappa _{2}))\in \omega _{f}\), there is a maximal positive number \(u((\kappa _{1},f(\kappa _{1})), (\kappa _{2},f(\kappa _{2}))\leq 1 \) such that
That is, \(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \in B\),
Thus, B is a GLEI set and f is a GSLEP function on B. □
Theorem 2.3
If f is a GSLEP function on a GLEI set \(B\subset \aleph \) with respect to η, then the level \(K_{\alpha }= \lbrace \kappa _{1}\in B: f(\kappa _{1})\leq \alpha \rbrace \) is a GLEI set for any \(\alpha \in \mathbb{R} \).
Proof
For any \(\alpha \in \mathbb{R}\) and \(\kappa _{1},\kappa _{2} \in K_{\alpha } \), then \(\kappa _{1},\kappa _{2}\in B \) and \(f(\kappa _{1})\leq \alpha \), \(f(\kappa _{2})\leq \alpha \). Since B is a GLEI set, then there is a maximal positive number \(u(\kappa _{1},\kappa _{2})\leq 1 \) such that
In addition, since f is GSLEP, there is a positive number \(v(\kappa _{1},\kappa _{2})\leq u(y_{1},y_{2}) \) such that
That is, \(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\in K_{\alpha }\), \(\forall t\in [0,v(\kappa _{1},\kappa _{2}) ] \). Therefore, \(K_{\alpha } \) is a GLEI set with respect to η for any \(\alpha \in \mathbb{R} \). □
Theorem 2.4
Let \(f:B\subset \aleph \longrightarrow \mathbb{R} \), where B is a GLEI. Then f is a GSLEP function with respect to η if,f for each pair of points \(\kappa _{1},\kappa _{2}\in B \), there is a positive number \(v(\kappa _{1},\kappa _{2}) \leq u(\kappa _{1},\kappa _{2})\leq 1 \) such that
Proof
Let \(\kappa _{1},\kappa _{2}\in B \) and \(\alpha ,\beta \in \mathbb{R} \) such that \(f(\kappa _{1})<\alpha \) and \(f(\kappa _{2})<\beta \). Since B is GLEI, there is a maximal positive number \(u(\kappa _{1}, \kappa _{2})\leq 1 \) such that
In addition, there is a positive number \(v(\kappa _{1},\kappa _{2}) \leq u(\kappa _{1},\kappa _{2}) \), where
Conversely, let \((\kappa _{1},\alpha ) \in \omega _{f} \) and \((\kappa _{2},\beta ) \in \omega _{f} \), then \(\kappa _{1},\kappa _{2} \in B \), \(f(\kappa _{1})<\alpha \), and \(f(\kappa _{2})<\beta \). Hence, \(f(\kappa _{1})<\alpha +\varepsilon \) and \(f(\kappa _{2})<\beta + \varepsilon \) hold for any \(\varepsilon >0 \). According to the hypothesis for \(\kappa _{1},\kappa _{2}\in B \), there is a positive number \(v(\kappa _{1},\kappa _{2})\leq u(\kappa _{1},\kappa _{2})\leq 1 \) such that
Let \(\varepsilon \longrightarrow 0^{+} \), then
That is, \((\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) , t \alpha +(1t) \beta ) \in \omega _{f} \), \(\forall t\in [0,v(\kappa _{1},\kappa _{2}) ]\).
Therefore, \(\omega _{f} \) is a GLEI set corresponding to ℵ. From Theorem 2.2 it follows that f is a GSLEP on B with respect to η. □
Optimality criteria
In this section, let us consider the nonlinear fractional multiobjective programming problem
where \(K_{0}\subset \aleph \) is a GLEI set and \(g_{i}(\kappa )>0\), \(\forall \kappa \in K_{0} \), \(i\in P={1,2,\ldots , p} \).
Let \(f=(f_{1},f_{2},\ldots , f_{p})\), \(g=(g_{1},g_{2},\ldots ,g_{p}) \), and \(h=(h_{1},h_{2},\ldots ,h_{q}) \)
and denote that \(K= \lbrace \kappa :h_{j}(\kappa )\leq 0, j \in Q, \kappa \in K_{0} \rbrace \), the feasible set of problem (VFP).
For \(\kappa ^{*}\in K \), we put
We also formulate the nonlinear multiobjective programming problem as follows:
where \(\lambda =(\lambda _{1},\lambda _{2},\ldots ,\lambda _{p})\in \mathbb{R}^{p} \).
The following lemma connects the weak efficient solutions for (VFP) and (\(\mathrm{VFP}_{\lambda } \)).
Lemma 3.1
A point \(\kappa ^{*} \) is a weak efficient solution for (\(\mathrm{VFP}_{\lambda } \)) iff \(\kappa ^{*} \) is a weak efficient solution for (\(\mathrm{VFP}^{*}_{\lambda } \)), where \(\lambda ^{*}=(\lambda ^{*}_{1}, \ldots ,\lambda ^{*}_{p} )= (\frac{f_{1}(\kappa ^{*})}{g_{1}(\kappa ^{*})},\ldots ,\frac{f_{p}(\kappa ^{*})}{g_{p}(\kappa ^{*})} ) \).
Proof
Assume that there is a feasible point \(\kappa \in K \), where
⟹
⟹
which is a contradiction to the weak efficiency of \(\kappa ^{*} \) for (VFP).
Presently, let us take \(\kappa \in K \) as a feasible point such that
then \(f_{i}(\kappa )\lambda ^{*}_{i}g_{i}(\kappa )<0=f_{i}(\kappa ^{*})\lambda ^{*}_{i}g_{i}(\kappa ^{*})\), \(\forall i\in Q \), which is again a contradiction to the weak efficiency of \(\kappa ^{*} \) for (\(\mathrm{VFP} ^{*}_{\lambda } \)). □
Next, some sufficient optimality conditions for the problem (VFP) are established.
Theorem 3.1
Let \(\bar{\kappa }\in K\), \(E(\bar{\kappa })=\bar{ \kappa } \) and f, h be GSLEP and g be a geodesic semilocal Epreincave, and they are all geodesic Eη semidifferentiable at κ̄. Further, assume that there are \(\zeta ^{o}= (\zeta ^{o}_{i}, i=1,\ldots ,p )\in \mathbb{R}^{p} \) and \(\xi ^{o}= (\xi ^{o}_{j}, j=1,\ldots ,m )\in \mathbb{R} ^{m} \) such that
Then κ̄ is a weak efficient solution for (VFP).
Proof
By contradiction, let κ̄ be not a weak efficient solution for (VFP), then there exists a point \(\widehat{\kappa }\in K \) such that
By the above hypotheses and Lemma 3.1, we have
Multiplying (9) by \(\zeta ^{o}_{i} \) and (11) by \(\xi ^{o}_{j} \), we get
Since \(\widehat{\kappa }\in K, \xi ^{o}\geqslant 0 \) by (6) and (12), we have
Utilizing (7) and (13), there is at least \(i_{0} \) (\(1\leq i_{0}\leq p \)) such that
On the other hand, (5) and (10) imply
By using (14), (15), and \(g>0 \), we have
which is a contradiction to 8, then the proof of the theorem is completed. □
Similarly we can prove the next theorem.
Theorem 3.2
Consider that \(\bar{\kappa }\in B\), \(E(\bar{ \kappa })=\bar{\kappa } \) and f, h are geodesic Eηsemidifferentiable at κ̄. If there exist \(\zeta ^{o} \in \mathbb{R}^{n} \) and \(\xi ^{o}\in \mathbb{R}^{m} \) such that conditions (4)–(7) hold and \(\zeta ^{o}f(x)+\xi ^{o}h(x) \) is a GSLEP function, then κ̄ is a weak efficient solution for (VFP).
Theorem 3.3
Consider that \(\bar{\kappa }\in B\), \(E(\bar{ \kappa })=\bar{\kappa } \) and \(\lambda _{i}^{o}=\frac{f_{i}(\bar{ \kappa })}{g_{i}(\bar{\kappa })}\) (\(i\in P\)) are all pSLGEP functions and \(h_{j}(\kappa )\) (\(j\in \aleph (\bar{\kappa })\)) are all GqSLEP functions and f, g, h are all geodesic Eηsemidifferentiable at κ̄. If there are \(\zeta ^{o}\in \mathbb{R}^{p} \) and \(\xi ^{o}\in \mathbb{R}^{m} \) such that
then κ̄ is a weak efficient solution for (VFP).
Proof
Assume that κ̄ is not a weak efficient solution for (VFP). Therefore, there exists \(\kappa ^{*}\in B \), which yields
Then
which means that
By the pSLGEP of \(( f_{i}(\kappa )\lambda _{i}^{o}g_{i}(\kappa ) )\) (\(i\in P\)) and Lemma 2.1, we have
Utilizing \(\zeta ^{o}\geqslant 0 \), then
For \(h(\kappa ^{*})\leq 0 \) and \(h_{j}(\bar{\kappa })= 0\), \(j \in \aleph (\bar{\kappa }) \), we have \(h_{j}(\kappa ^{*})\leq h_{j}(\bar{ \kappa })\), \(\forall j\in \aleph (\bar{\kappa })\).
By the GqSLEP of \(h_{j} \) and Lemma 2.1, we have
Considering \(\xi ^{o}\geqslant 0 \) and \(\xi _{j}^{o}= 0\), \(j\in \aleph (\bar{\kappa })\), then
Hence, by (20) and (21), we have
which is a contradiction to relation (17) at \(\kappa ^{*} \in B \). Therefore, κ̄ is a weak efficient solution for (VFP). □
Theorem 3.4
Consider \(\bar{\kappa }\in B\), \(E(\bar{\kappa })=\bar{ \kappa } \) and \(\lambda _{i}^{o}=\frac{f_{i}(\bar{\kappa })}{g_{i}(\bar{ \kappa })}(i\in P) \). Also, assume that f, g, h are geodesic Eηsemidifferentiable at κ̄. If there are \(\zeta ^{o}\in \mathbb{R}^{p} \) and \(\xi ^{o}\in \mathbb{R}^{m} \) such that conditions (17)–(19) hold and \(\sum_{i=1}^{p} \zeta ^{o}_{i} (f_{i}(\kappa )\lambda ^{o}_{i}g_{i}(\kappa ) )+ \xi ^{o}_{\aleph (\bar{\kappa })}h_{\aleph (\bar{\kappa })}(\kappa ) \) is a GpSLEP function, then κ̄ is a weak efficient solution for (VFP).
Corollary 3.1
Let \(\bar{\kappa }\in B\), \(E(\bar{\kappa })=\bar{ \kappa } \) and \(\lambda _{i}^{o}=\frac{f_{i}(\bar{\kappa })}{g_{i}(\bar{ \kappa })}(i\in P) \). Further, let f, \(h_{\aleph (\bar{\kappa })}\) be all GSLEP functions, g be a geodesic semilocal Epreincave function, and f, g, h be all geodesic Eη semidifferentiable at κ̄. If there exist \(\zeta ^{o}\in \mathbb{R}^{p} \) and \(\xi ^{o}\in \mathbb{R}^{m} \) such that conditions (17)–(19) hold, then κ̄ is a weak efficient solution for (VFP).
The dual problem for (VFP) is formulated as follows:
where \(\zeta =(\zeta _{i}, i=1,2,\ldots , p)\geqslant 0\), \(\alpha =( \alpha _{i}, i=1,2,\ldots , p)> 0\), \(\beta =(\beta _{i}, i=1,2,\ldots , m)\geqslant 0\), \(\lambda \in K_{0}\).
Denote the feasible set problem (\(VFD \)) by \(K^{\prime} \).
Theorem 3.5
(General weak duality)
Let \(\kappa \in K \), \((\alpha ,\beta ,\lambda ,\zeta )\in K^{\prime} \), and \(E(\lambda )= \lambda \). If \(\sum_{i=1}^{p}\alpha _{i}(f_{i}\zeta _{i}g_{i}) \) is a GpSLEP function and \(\sum_{j=1}^{m}\beta _{j}h_{j} \) is a GqSLEP function and they are all geodesic Eηsemidifferentiable at λ, then \(\frac{f(\kappa )}{g(\kappa )}\nleq \zeta \).
Proof
From \(\alpha >0 \) and \((\alpha , \beta ,\lambda ,\zeta )\in K^{\prime} \), we have
By the GpSLEP of \(\sum_{i=1}^{p}\alpha _{i}(f_{i}\zeta _{i}g_{i}) \) and Lemma 2.1, we obtain
that is,
Also, from \(\beta \geqslant 0\) and \(\kappa \in K \), then
Using the GqSLEP of \(\sum_{j=1}^{m}\beta _{j}h_{j} \) and Lemma 2.1, one has
Then
Therefore,
This is a contradiction to \((\alpha ,\beta ,\lambda ,\zeta )\in K ^{\prime} \). □
Theorem 3.6
Consider that \(\kappa \in K \), \((\alpha , \beta ,\lambda ,\zeta )\in K^{\prime} \) and \(E(\lambda )=\lambda \). If \(\sum_{i=1}^{p}\alpha _{i}(f_{i}\zeta _{i}g_{i})+\sum_{j=1}^{m}\beta _{j}h_{j} \) is a GpSLEP function and a geodesic Eηsemidifferentiable at λ, then \(\frac{f(\kappa )}{g( \kappa )}\nleq \zeta \).
Theorem 3.7
(General converse duality)
Let \(\bar{\kappa } \in K \) and \((\kappa ^{*},\alpha ^{*}, \beta ^{*},\zeta ^{*})\in K^{\prime} \), \(E(\kappa ^{*})=\kappa ^{*} \), where \(\zeta ^{*}= \frac{f(\kappa ^{*})}{g( \kappa ^{*})}=\frac{f(\bar{\kappa })}{g(\bar{\kappa })}=(\zeta ^{*}_{i}, i=1,2,\ldots , p) \). If \(f_{i}\zeta _{i}^{*}g_{i} (i\in P)\), \(h _{j}(j\in \aleph )\) are all GSLEP functions and all geodesic Eηsemidifferentiable at \(\kappa ^{*} \), then κ̄ is a weak efficient solution for (VFP).
Proof
By using the hypotheses and Lemma 2.1, for any \(\kappa \in K \), we obtain
Utilizing the first constraint condition for (VFD), \(\alpha ^{*}>0, \beta ^{*}\geqslant 0\), \(\zeta ^{*}\geqslant 0 \), and the two inequalities above, we have
In view of \(h_{j}(\kappa )\leq 0\), \(\beta ^{*}_{j}\geqslant 0, \beta ^{*}_{j}h_{j}(\kappa ^{*})\geqslant (j\in \aleph ) \), and \(\zeta ^{*} _{i}= \frac{f_{i}(\kappa ^{*})}{g_{i}(\kappa ^{*})}\) (\(i\in P\)), then
Consider that κ̄ is not a weak efficient solution for (VFP). From \(\zeta ^{*}_{i}= \frac{f_{i}(\bar{\kappa })}{g_{i}(\bar{ \kappa })}\) (\(i\in P\)) and Lemma 3.1, it follows that κ̄ is not a weak efficient solution for (\(\mathrm{VFP}_{\zeta ^{*}} \)). Hence, \(\tilde{\kappa }\in K \) such that
Therefore \(\sum_{i=1}^{p}\alpha ^{*}_{i} (f_{i}(\tilde{\kappa }) \zeta _{i}^{*}g_{i}(\tilde{\kappa }) )<0 \). This is a contradiction to inequality (24). The proof of the theorem is completed. □
References
 1.
Grinalatt, M., Linnainmaa, J.T.: Jensen’s inquality, parameter uncertainty, and multiperiod investment. Rev. Asset Pricing Stud. 1(1), 1–34 (2011)
 2.
Kılıçman, A., Saleh, W.: Some inequalities for generalized sconvex functions. JP J. Geom. Topol. 17, 63–82 (2015)
 3.
Kılıçman, A., Saleh, W.: Generalized convex functions and their applications. In: Mathematical Analysis and Applications, Selected Topics, pp. 77–99 (2018)
 4.
Kılıçman, A., Saleh, W.: Preinvex functions and their applications. Symmetry 10(10), 493 (2018)
 5.
Ruel, J.J., Ayres, M.P.: Jensen’s inequality predicts effects of environmental variation. Trends Ecol. Evol. 14(9), 361–366 (1999)
 6.
Ewing, G.M.: Sufficient conditions for global minima of suitably convex functions from variational and control theory. SIAM Rev. 19, 202–220 (1977). https://doi.org/10.1137/1019037
 7.
Preda, V., StancuMinasian, I.M.: Duality in multiple objective programming involving semilocally preinvex and related functions. Glas. Mat. 32, 153–165 (1997)
 8.
Preda, V.: Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions. J. Math. Anal. Appl. 288, 365–382 (2003). https://doi.org/10.1016/S0022247X(02)004602
 9.
Hu, Q.J., Jian, J.B., Zheng, H.Y., Tang, C.M.: Semilocal Econvexity and semilocal Econvex programming. Bull. Aust. Math. Soc. 75, 59–74 (2007). https://doi.org/10.1017/S0004972700038983
 10.
Fulga, C., Preda, V.: Nonlinear programming with Epreinvex and local Epreinvex functions. Eur. J. Oper. Res. 192(3), 737–743 (2009)
 11.
Jiao, H., Liu, S.: Semilocal Epreinvexity and its applications in nonlinear multiple objective fractional programming. J. Inequal. Appl. 2011, 116 (2011)
 12.
Jiao, H., Liu, S., Pai, X.: A class of semilocal Epreinvex functions and its applications in nonlinear programming. J. Appl. Math. 2012, Article ID 629194 (2012)
 13.
Jiao, H.: A class of semilocal Epreinvex maps in Banach spaces with applications to nondifferentiable vector optimization. Int. J. Optim. Control Theor. Appl. 4(1), 1–10 (2013)
 14.
Agarwal, R.P., Ahmad, I., Iqbal, A., Ali, S.: Generalized invex sets and preinvex functions on Riemannian manifolds. Taiwan. J. Math. 16(5), 1719–1732 (2012)
 15.
Bartolo, R., Germinario, A., Sánchez, M.: Convexity of domains of Riemannian manifolds. Ann. Glob. Anal. Geom. 21(1), 63–83 (2002)
 16.
Ferrara, M., Mititelu, S.: Mond–Weir duality in vector programming with generalized invex functions on differentiable manifolds. Balk. J. Geom. Appl. 11(2), 80–87 (2006)
 17.
Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523–1560 (2011)
 18.
Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Springer, Boston (1994)
 19.
Rapcsák, T.: Smooth Nonlinear Optimization in \(\mathbb{R}^{n} \). Kluwer Academic, Dordrecht (1997)
 20.
Iqbal, A., Ali, S., Ahmad, I.: On geodesic Econvex sets, geodesic Econvex functions and Eepigraphs. J. Optim. Theory Appl. 55(1), 239–251 (2012)
 21.
Iqbal, A., Ahmad, I., Ali, S.: Some properties of geodesic semiEconvex functions. Nonlinear Anal., Theory Methods Appl. 74(17), 6805–6813 (2011)
 22.
Kılıçman, A., Saleh, W.: On geodesic strongly Econvex sets and geodesic strongly Econvex functions. J. Inequal. Appl. 2015, 297 (2015)
 23.
Niculescu, C.: Optimality and duality in multiobjective fractional programming involving ρsemilocally type Ipreinvex and related functions. J. Math. Anal. Appl. 335(1), 7–19 (2007)
Acknowledgements
The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.
Funding
Authors declare that no funding was received for this article.
Author information
Affiliations
Contributions
Both authors conceived of the study, participated in its design and coordination, drafted the manuscript and participated in the sequence alignment. All authors read and approved the final manuscript.
Corresponding author
Correspondence to Adem Kılıçman.
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Received
Accepted
Published
DOI
Keywords
 Generalized convexity
 Riemannian geometry
 Duality