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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.
1 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].
2 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. □
3 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. □
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Kılıçman, A., Saleh, W. On properties of geodesic semilocal Epreinvex functions. J Inequal Appl 2018, 353 (2018). https://doi.org/10.1186/s136600181944z
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DOI: https://doi.org/10.1186/s136600181944z
Keywords
 Generalized convexity
 Riemannian geometry
 Duality