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On properties of geodesic semilocal E-preinvex functions
- Adem Kılıçman^{1}Email authorView ORCID ID profile and
- Wedad Saleh^{2}
https://doi.org/10.1186/s13660-018-1944-z
© The Author(s) 2018
- Received: 6 September 2018
- Accepted: 10 December 2018
- Published: 20 December 2018
Abstract
The authors define a class of functions on Riemannian manifolds, which are called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions, and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming is considered, where the functions involved are geodesic E-η-semidifferentiability, sufficient optimality conditions are obtained. A dual is formulated and duality results are proved by using concepts of geodesic semilocal E-preinvex functions, geodesic pseudo-semilocal E-preinvex functions, and geodesic quasi-semilocal E-preinvex functions.
Keywords
- Generalized convexity
- Riemannian geometry
- Duality
1 Introduction
Convexity and generalized convexity play a significant role in many fields, for example, in biological system, economy, optimization, and so on [1–5].
Generalized convex functions, labeled as semilocal convex functions, were introduced by Ewing [6] by using more general semilocal preinvexity and η-semidifferentiability. After that optimality conditions for weak vector minima were given [7]. Also, optimality conditions and duality results for a nonlinear fractional involving η-semidifferentiability were established [8].
Furthermore, some optimality conditions and duality results for semilocal E-convex programming were established [9]. E-convexity was extended to E-preinvexity [10]. Recently, semilocal E-preinvexity (SLEP) and some of its applications were introduced [11–13].
Generalized convex functions in manifolds, such as Riemannian manifolds, were studied by many authors; see [14–17]. 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 E-convex (GEC) sets and geodesic E-convex (GEC) functions on Riemannian manifolds were studied [20]. Moreover, geodesic semi E-convex (GsEC) functions were introduced [21]. Recently, geodesic strongly E-convex (GSEC) functions were introduced and some of their properties were discussed [22].
2 Geodesic semilocal E-preinvexity
Assume that ℵ is a complete n-dimensional 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
- 1.a geodesic E-invex (GEI) with respect to η if there is exactly one geodesic \(\gamma _{E(\kappa _{1}), E(\kappa _{2})}: [0,1 ] \longrightarrow \aleph \) such that\(\forall \kappa _{1},\kappa _{2}\in B\) and \(t\in [0,1]\).$$\begin{aligned} \gamma _{E(\kappa _{1}), E(\kappa _{2})}(0)=E(\kappa _{2}), \qquad \acute{\gamma }_{E(\kappa _{1}), E(\kappa _{2})}=\eta \bigl(E(\kappa _{1}),E( \kappa _{2}) \bigr), \qquad \gamma _{E(\kappa _{1}), E(\kappa _{2})}(t)\in B, \end{aligned}$$
- 2.a geodesic local E-invex (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 E-convex 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
- 1.a geodesic E-preinvex (GEP) on \(A\subset \aleph \) with respect to η if A is a GEI set and$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f\bigl(E( \kappa _{1}) \bigr)+(1-t)f\bigl(E(\kappa _{2})\bigr) , \quad \forall \kappa _{1},\kappa _{2}\in A, t\in [0,1]; $$
- 2.a geodesic semi E-preinvex (GSEP) on A with respect to η if A is a GEI set and$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f(\kappa _{1})+(1-t)f( \kappa _{2}) , \quad \forall \kappa _{1},\kappa _{2}\in A, t\in [0,1]. $$
- 3.a geodesic local E-preinvex (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$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f\bigl(E( \kappa _{1}) \bigr)+(1-t)f\bigl(E(\kappa _{2})\bigr) , \quad \forall t\in \bigl[0,v( \kappa _{1},\kappa _{2})\bigr]. $$
Definition 2.3
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
Definition 2.4
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
Hence, \(h(m) \) is not a GSEP function on \(\mathbb{R} \) with respect to η.
Definition 2.5
Definition 2.6
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
Remark 2.4
- 1.
If \(\aleph =\mathbb{R}^{n} \), then the geodesic E-η- semidifferentiable is E-η-semidifferentiable [11].
- 2.
If \(\aleph =\mathbb{R}^{n} \) and \(E=I \), then the geodesic E-η-semidifferentiable is the η-semidifferentiability [23].
- 3.
If \(\aleph =\mathbb{R}^{n} \), \(E=I \), and \(\eta (\kappa ,\kappa ^{*})=\kappa -\kappa ^{*} \), then the geodesic E-η-semidifferentiable is the semidifferentiability [11].
Lemma 2.1
- 1.Assume that h is a GSLEP (E-preconcave) and a geodesic E-η-semidifferentiable 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 E-η-semidifferentiable 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
Definition 2.8
Theorem 2.2
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 \(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\in B\), \(\forall t\in [0, u(\kappa _{1},\kappa _{2}) ]\).
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
Theorem 2.4
Proof
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
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).
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
Next, some sufficient optimality conditions for the problem (VFP) are established.
Theorem 3.1
Proof
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 E-η-semidifferentiable 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
Proof
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 E-η-semidifferentiable 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 E-preincave function, and f, g, h be all geodesic E-η- 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).
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 E-η-semidifferentiable at λ, then \(\frac{f(\kappa )}{g(\kappa )}\nleq \zeta \).
Proof
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 E-η-semidifferentiable 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 E-η-semidifferentiable at \(\kappa ^{*} \), then κ̄ is a weak efficient solution for (VFP).
Proof
Declarations
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.
Authors’ 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.
Competing interests
The authors declare that they have no competing interests.
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.
Authors’ Affiliations
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