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On properties of geodesic semilocal E-preinvex functions

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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.

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 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 [11,12,13].

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 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].

Geodesic semilocal E-preinvexity

Assume that 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

A nonempty set \(B \subset \aleph \) is said to be

  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

    $$\begin{aligned} \gamma _{E(\kappa _{1}), E(\kappa _{2})}(0)=E(\kappa _{2}), \qquad \acute{\gamma }_{E(\kappa _{1}), E(\kappa _{2})}=\eta \bigl(E(\kappa _{1}),E( \kappa _{2}) \bigr), \qquad \gamma _{E(\kappa _{1}), E(\kappa _{2})}(t)\in B, \end{aligned}$$

    \(\forall \kappa _{1},\kappa _{2}\in B\) and \(t\in [0,1]\).

  2. 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. 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

A function \(f: A\subset \aleph \longrightarrow \mathbb{R} \) is said to be

  1. 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. 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. 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

A function \(f:\aleph \longrightarrow \mathbb{R} \) is a geodesic semilocal E-convex (GSLEC) on a geodesic local starshaped E-convex 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

$$ f \bigl(\gamma _{E(\kappa _{1}), E(\kappa _{2})}(t) \bigr)\leq t f( \kappa _{1})+(1-t)f( \kappa _{2}) , \quad \forall t\in \bigl[0,v(\kappa _{1}, \kappa _{2})\bigr]. $$

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 \(A= [ \left . -4,-1 ) \right . \cup [1,4] \),

$$\begin{aligned} &E(\kappa ) = \textstyle\begin{cases} \kappa ^{2} & \mbox{if } \vert \kappa \vert \leq 2, \\ -1 & \mbox{if } \vert \kappa \vert > 2; \end{cases}\displaystyle \\ &\eta (\kappa ,\iota ) = \textstyle\begin{cases} \kappa -\iota & \mbox{if } \kappa \geqslant 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota \leq 0 , \\ -1-\iota & \mbox{if } \kappa >0, \iota \leq 0 \mbox{ or } \kappa \geqslant 0 ,\iota < 0, \\ 1-\iota & \mbox{if } \kappa < 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota >0; \end{cases}\displaystyle \\ &\gamma _{\kappa ,\iota }(t) = \textstyle\begin{cases} \iota +t(\kappa -l) & \mbox{if } \kappa \geqslant 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota \leq 0 , \\ \iota +t(-1-\iota ) & \mbox{if } \kappa >0, \iota \leq 0 \mbox{ or } \kappa \geqslant 0 ,\iota < 0, \\ \iota +t(1-\iota ) & \mbox{if } \kappa < 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota >0. \end{cases}\displaystyle \end{aligned}$$

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

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f(\kappa _{1})+(1-t)f(\kappa _{2}) , \quad \forall t\in \bigl[0,v( \kappa _{1},\kappa _{2})\bigr]. $$
(3)

If

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\geqslant t f( \kappa _{1})+(1-t)f(\kappa _{2}) , \quad \forall t\in \bigl[0,v( \kappa _{1},\kappa _{2})\bigr], $$

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

$$\begin{aligned} E(m) =& \textstyle\begin{cases} 0 & \mbox{if } m< 0, \\ 1 & \mbox{if } 1< m\leq 2, \\ m & \mbox{if } 0\leq m\leq 1 \mbox{ or } m>2; \end{cases}\displaystyle \end{aligned}$$

and the map \(\eta : \mathbb{R}\times \mathbb{R}\longrightarrow \mathbb{R} \) is defined as

$$\begin{aligned} \eta (m,n) =& \textstyle\begin{cases} 0 & \mbox{if } m= n, \\ 1-m & \mbox{if } m\neq n ; \end{cases}\displaystyle \end{aligned}$$

also,

$$\begin{aligned} \gamma _{m,n}(t) =& \textstyle\begin{cases} n &\mbox{if } m= n, \\ n+t(1-m) &\mbox{if } m\neq n. \end{cases}\displaystyle \end{aligned}$$

Assume that \(h: \mathbb{R}\longrightarrow \mathbb{R} \), where

$$\begin{aligned} h(m) =& \textstyle\begin{cases} 0 & \mbox{if } 1< m\leq 2, \\ 1 & \mbox{if } m>2, \\ -m+1 & \mbox{if } 0\leq m\leq 1, \\ -m+2 & \mbox{if } m< 0; \end{cases}\displaystyle \end{aligned}$$

and since \(\mathbb{R} \) is a geodesic local starshaped E-convex set and a geodesic local E-invex 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

$$ h \bigl(\gamma _{E(m_{0}),E(n_{0})}(t_{0}) \bigr)=1>(1-t_{0})=t_{0}h(m _{0})+(1-t_{0})h(n_{0}) . $$

Then \(h(m) \) is not a GSLEC function on \(\mathbb{R} \).

Similarly, taking \(m_{1}=1\), \(n_{1}=4 \), we have

$$ h \bigl(\gamma _{E(m_{1}),E(n_{1})}(t_{1}) \bigr)=1>(1-t_{1})=t_{1}h(m _{1})+(1-t_{1})h(n_{1}) $$

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 quasi-semilocal E-preinvex (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

$$ h \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq h(\kappa _{2}),\quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$

Definition 2.6

A function \(h:S\subset \aleph \longrightarrow \mathbb{R} \), where S is a GLEI set, is said to be a geodesic pseudo-semilocal E-preinvex (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

$$ h \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq h(\kappa _{2})-t w( \kappa _{1},\kappa _{2}),\quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$

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 E-η- semidifferentiable at \(\kappa ^{*} \in S \), where \(S\subset \aleph \) is a GLEI set with respect to η, if \(E(\kappa ^{*})=\kappa ^{*} \) and

$$ h'_{+} \bigl(\gamma _{\kappa ^{*},E(\kappa )}(t) \bigr)= \lim _{t\longrightarrow 0^{+}} \frac{1}{t} \bigl[h \bigl(\gamma _{\kappa ^{*},E( \kappa )}(t) \bigr) -h\bigl(\kappa ^{*}\bigr) \bigr] $$

exist for every \(\kappa \in S\).

Remark 2.4

  1. 1.

    If \(\aleph =\mathbb{R}^{n} \), then the geodesic E-η- semidifferentiable is E-η-semidifferentiable [11].

  2. 2.

    If \(\aleph =\mathbb{R}^{n} \) and \(E=I \), then the geodesic E-η-semidifferentiable is the η-semidifferentiability [23].

  3. 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. 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. 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

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

$$ f\bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\bigr)\leq tf(\kappa _{2})+(1-t)f( \kappa _{1}),\quad t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$

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

$$ f\bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\bigr)\leq tf\bigl(E(\kappa _{1}) \bigr)+(1-t)f\bigl(E( \kappa _{2})\bigr),\quad t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$

Since \(f(E(\kappa _{1})) \leq f(\kappa _{1})\), \(\forall \kappa _{1}\in S\), then

$$ f\bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\bigr)\leq tf(\kappa _{1})+(1-t)f( \kappa _{2}),\quad t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$

 □

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 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

$$ \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t),t\alpha _{1}+(1-t)\alpha _{2} \bigr)\in \omega ,\quad \forall t\in \bigl[0,u\bigl((\kappa _{1},\alpha _{1}), (\kappa _{2},\alpha _{2})\bigr) \bigr]. $$

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

$$ \omega _{f}= \bigl\lbrace (\kappa _{1},\alpha ):\kappa _{1}\in B, f( \kappa _{1})\leq \alpha , \alpha \in \mathbb{R} \bigr\rbrace $$

is a GLEI set with respect to η corresponding to .

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

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t), t\alpha _{1}+(1-t) \alpha _{2} \bigr)\in \omega _{f},\quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$

Conversely, if \(\omega _{f} \) is a GLEI set with respect to η corresponding to , 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

$$ \bigl( \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t), tf(\kappa _{1}) +(1-t)f( \kappa _{2}) \bigr) \in \omega _{f},\quad \forall t\in \bigl[0, u\bigl( \bigl(\kappa _{1},f(\kappa _{1})\bigr),\bigl(\kappa _{2},f(\kappa _{2})\bigr)\bigr) \bigr]. $$

That is, \(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \in B\),

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq tf(\kappa _{1}) +(1-t)f( \kappa _{2}), \quad t\in \bigl[0,u\bigl(\bigl(\kappa _{1},f( \kappa _{1})\bigr),\bigl(\kappa _{2},f(\kappa _{2}) \bigr)\bigr) \bigr]. $$

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

$$ \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\in B, \quad \forall t\in \bigl[0,u(\kappa _{1},\kappa _{2}) \bigr] . $$

In addition, since f is GSLEP, there is a positive number \(v(\kappa _{1},\kappa _{2})\leq u(y_{1},y_{2}) \) such that

$$\begin{aligned} f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq & t f( \kappa _{1}) +(1-t)f(\kappa _{2}) \\ \leq & t\alpha +(1-t)\alpha \\ =& \alpha , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. \end{aligned}$$

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

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$

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

$$ \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \in B , \quad \forall t\in \bigl[0,u(\kappa _{1},\kappa _{2}) \bigr]. $$

In addition, there is a positive number \(v(\kappa _{1},\kappa _{2}) \leq u(\kappa _{1},\kappa _{2}) \), where

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$

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

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta +\varepsilon , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$

Let \(\varepsilon \longrightarrow 0^{+} \), then

$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$

That is, \((\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) , t \alpha +(1-t) \beta ) \in \omega _{f} \), \(\forall t\in [0,v(\kappa _{1},\kappa _{2}) ]\).

Therefore, \(\omega _{f} \) is a GLEI set corresponding to . 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

$$\begin{aligned} (\mbox{VFP}) \textstyle\begin{cases} \mbox{minimize } \frac{f(\kappa )}{g(\kappa )}= (\frac{f_{1}(\kappa )}{g_{1}( \kappa )},\ldots ,\frac{f_{p}(\kappa )}{g_{p}(\kappa )} ), \\ \mbox{subject to } h_{j}(\kappa )\leq 0,\quad j\in Q={1,2,\ldots, q} \\ \kappa \in K_{0}; \end{cases}\displaystyle \end{aligned}$$

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

$$ Q\bigl(\kappa ^{*}\bigr)= \bigl\lbrace j:h_{j}\bigl(\kappa ^{*}\bigr)= 0, j\in Q \bigr\rbrace , \qquad L\bigl(\kappa ^{*}\bigr)=\frac{Q}{Q(\kappa ^{*})}. $$

We also formulate the nonlinear multiobjective programming problem as follows:

$$\begin{aligned} (\mbox{VFP}_{\lambda }) \textstyle\begin{cases} \mbox{minimize } ( f_{1}(\kappa )-\lambda _{1}g_{1}(\kappa ),\ldots f_{p}(\kappa )-\lambda _{p}g_{p}(\kappa ) ), \\ \mbox{subject to } h_{j}(\kappa )\leq 0,\quad j\in Q={1,2,\ldots, q} \\ \kappa \in K_{0}; \end{cases}\displaystyle \end{aligned}$$

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

$$ f_{i}(\kappa )-\lambda ^{*}_{i}g_{i}( \kappa )< f_{i}\bigl(\kappa ^{*}\bigr)-\lambda ^{*}_{i}g_{i}\bigl(\kappa ^{*}\bigr),\quad \forall i\in Q $$

$$ f_{i}(\kappa )< \frac{f_{i}(\kappa ^{*})}{g_{i}(\kappa ^{*})g_{i}(\kappa )} $$

$$ \frac{f_{i}(\kappa )}{g_{i}(\kappa )}< \frac{f_{i}(\kappa ^{*})}{g_{i}( \kappa ^{*})}, $$

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

$$ \frac{f_{i}(\kappa )}{g_{i}(\kappa )}< \frac{f_{i}(\kappa ^{*})}{g_{i}( \kappa ^{*})}= \lambda ^{*}_{i}, $$

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 E-preincave, and they are all geodesic E-η- 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

$$\begin{aligned} &\zeta ^{o}_{i}f'_{i+} \bigl(\gamma _{\bar{\kappa },E(\widehat{\kappa })}(t) \bigr)+\xi ^{o}_{j} h'_{j+} \bigl(\gamma _{\bar{\kappa },E( \widehat{\kappa })}(t) \bigr)\geqslant 0\quad \forall \kappa \in K, t \in [0,1], \end{aligned}$$
(4)
$$\begin{aligned} &g'_{i+} \bigl(\gamma _{\bar{\kappa },E(\kappa )}(t) \bigr)\leq 0,\quad \forall \kappa \in K, i\in P, \end{aligned}$$
(5)
$$\begin{aligned} &\xi ^{o}h(\bar{\kappa })=0 \end{aligned}$$
(6)
$$\begin{aligned} &\zeta ^{o}\geqslant 0 ,\qquad \xi ^{o}\geqslant 0. \end{aligned}$$
(7)

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

$$ \frac{f_{i}(\widehat{\kappa })}{g_{i}(\widehat{\kappa })}< \frac{f_{i}(\bar{ \kappa })}{g_{i}(\bar{\kappa })},\quad i\in P. $$
(8)

By the above hypotheses and Lemma 3.1, we have

$$\begin{aligned} &f_{i}(\widehat{\kappa })-f_{i}(\bar{\kappa })\geqslant f'_{i+} \bigl(\gamma _{\bar{\kappa },E(\widehat{\kappa })}(t) \bigr) ,\quad i\in P \end{aligned}$$
(9)
$$\begin{aligned} &g_{i}(\widehat{\kappa })-g_{i}(\bar{\kappa })\leq g'_{i+} \bigl(\gamma _{\bar{ \kappa },E(\widehat{\kappa })}(t) \bigr) ,\quad i \in P \end{aligned}$$
(10)
$$\begin{aligned} &h_{i}(\widehat{\kappa })-h_{i}(\bar{\kappa })\geqslant h'_{j+} \bigl(\gamma _{\bar{\kappa },E(\widehat{\kappa })}(t) \bigr) ,\quad j\in Q. \end{aligned}$$
(11)

Multiplying (9) by \(\zeta ^{o}_{i} \) and (11) by \(\xi ^{o}_{j} \), we get

$$\begin{aligned} & \sum_{i=1}^{p} \zeta ^{o}_{i} \bigl(f_{i}(\widehat{\kappa })-f_{i}(\bar{ \kappa }) \bigr) + \sum_{j=1}^{m} \xi ^{o}_{j} \bigl(h_{j}( \widehat{\kappa })-h_{j}(\bar{\kappa }) \bigr) \\ &\quad \geqslant \zeta ^{o}_{i} f'_{i+} \bigl(\gamma _{\bar{\kappa },E( \widehat{\kappa })}(t) \bigr) +\xi ^{o}_{j} h'_{j+} \bigl(\gamma _{\bar{ \kappa },E(\widehat{\kappa })}(t) \bigr) \geqslant 0. \end{aligned}$$
(12)

Since \(\widehat{\kappa }\in K, \xi ^{o}\geqslant 0 \) by (6) and (12), we have

$$ \sum_{i=1}^{p} \zeta ^{o}_{i} \bigl(f_{i}(\widehat{\kappa })-f_{i}(\bar{ \kappa }) \bigr)\geqslant 0. $$
(13)

Utilizing (7) and (13), there is at least \(i_{0} \) (\(1\leq i_{0}\leq p \)) such that

$$ f_{i_{0}}(\widehat{\kappa })\geqslant f_{i_{0}}( \bar{\kappa }). $$
(14)

On the other hand, (5) and (10) imply

$$ g_{i}(\widehat{\kappa })\leq g_{i}(\bar{ \kappa }),\quad i\in P. $$
(15)

By using (14), (15), and \(g>0 \), we have

$$ \frac{f_{i_{0}}(\widehat{\kappa })}{g_{i_{0}}(\widehat{\kappa })} \geqslant \frac{f_{i_{0}}(\bar{\kappa })}{g_{i_{0}}(\bar{\kappa })}, $$
(16)

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 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

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 E-η-semidifferentiable at κ̄. If there are \(\zeta ^{o}\in \mathbb{R}^{p} \) and \(\xi ^{o}\in \mathbb{R}^{m} \) such that

$$\begin{aligned} &\sum_{i=1}^{p}\zeta _{i}^{o} \bigl(f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) -\lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \bigr) +\xi ^{o}h'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \geqslant 0 \end{aligned}$$
(17)
$$\begin{aligned} &\xi ^{o}h(\bar{\kappa })=0, \end{aligned}$$
(18)
$$\begin{aligned} &\zeta ^{o}\geqslant 0,\qquad \xi ^{o}\geqslant 0, \end{aligned}$$
(19)

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

$$ \frac{f_{i}(\kappa ^{*})}{g_{i}(\kappa ^{*})}< \frac{f_{i}(\bar{\kappa })}{g _{i}(\bar{\kappa })}. $$

Then

$$ f_{i}\bigl(\kappa ^{*}\bigr)-\lambda _{i}^{o}g_{i} \bigl(\kappa ^{*}\bigr)< 0, \quad i\in P, $$

which means that

$$ f_{i}\bigl(\kappa ^{*}\bigr)-\lambda _{i}^{o}g_{i} \bigl(\kappa ^{*}\bigr)< f_{i}(\bar{\kappa })-\lambda _{i}^{o}g_{i}(\bar{\kappa })< 0, \quad i\in P. $$

By the pSLGEP of \(( f_{i}(\kappa )-\lambda _{i}^{o}g_{i}(\kappa ) )\) (\(i\in P\)) and Lemma 2.1, we have

$$ f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) -\lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) , \quad i\in P. $$

Utilizing \(\zeta ^{o}\geqslant 0 \), then

$$ \sum_{i=1}^{p}\zeta _{i}^{o} \bigl(f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) -\lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \bigr)< 0. $$
(20)

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

$$ h_{j+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \leq 0, \quad \forall j \in \aleph (\bar{\kappa }). $$

Considering \(\xi ^{o}\geqslant 0 \) and \(\xi _{j}^{o}= 0\), \(j\in \aleph (\bar{\kappa })\), then

$$ \sum_{j=1}^{m}\xi _{j}^{o}h'_{j+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) \leq 0. $$
(21)

Hence, by (20) and (21), we have

$$\begin{aligned} \sum_{i=1}^{p}\zeta _{i}^{o} \bigl(f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) - \lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) \bigr) +\xi ^{o}h'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) < 0, \end{aligned}$$
(22)

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 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).

The dual problem for (VFP) is formulated as follows:

$$\begin{aligned} (\mathrm{VFD}) \textstyle\begin{cases} \mbox{minimize } (\zeta _{i}, i=1,2,\ldots , p ) , \\ \mbox{subject to } \sum_{i=1}^{p}\alpha _{i} (f'_{i+} ( \gamma _{\lambda ,E( \kappa )}(t) ) -\zeta _{i}g'_{i+} ( \gamma _{\lambda ,E( \kappa )}(t) ) ) +\sum_{j=1}^{m}\beta _{j}h'_{j+} ( \gamma _{\lambda ,E(\kappa )}(t) ) \geqslant 0 \\ \quad \kappa \in K_{0}, t\in [0,1], \\ \quad f_{i}(\lambda )-\zeta _{i}g_{i}(\lambda )\geqslant 0, \quad i\in P,\qquad \beta _{j}h_{j}(\lambda )\geqslant 0, \quad j\in \aleph ; \end{cases}\displaystyle \end{aligned}$$

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 E-η-semidifferentiable at λ, then \(\frac{f(\kappa )}{g(\kappa )}\nleq \zeta \).

Proof

From \(\alpha >0 \) and \((\alpha , \beta ,\lambda ,\zeta )\in K^{\prime} \), we have

$$ \sum_{i=1}^{p}\alpha _{i} \bigl(f_{i}(\kappa )-\zeta _{i}g_{i}(\kappa ) \bigr)< 0 \leq \sum_{i=1}^{p}\alpha _{i}\bigl(f_{i}(\lambda )-\zeta _{i}g_{i}( \lambda )\bigr). $$

By the GpSLEP of \(\sum_{i=1}^{p}\alpha _{i}(f_{i}-\zeta _{i}g_{i}) \) and Lemma 2.1, we obtain

$$ \Biggl( \sum_{i=1}^{p}\alpha _{i}(f_{i}-\zeta _{i}g_{i}) \Biggr)'_{+} \bigl(\gamma _{\lambda ,E(\kappa )}(t) \bigr) < 0, $$

that is,

$$ \sum_{i=1}^{p}\alpha _{i} (f'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) \bigr) -\zeta _{i}g'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) ] \bigr)< 0. $$

Also, from \(\beta \geqslant 0\) and \(\kappa \in K \), then

$$ \sum_{j=1}^{m}\beta _{j}h_{j}( \kappa )\leq 0 \leq \sum_{j=1}^{m}\beta _{j}h_{j}(\lambda ). $$

Using the GqSLEP of \(\sum_{j=1}^{m}\beta _{j}h_{j} \) and Lemma 2.1, one has

$$ \Biggl( \sum_{j=1}^{m}\beta _{j}h_{j} \Biggr)'_{+} \bigl(\gamma _{\lambda ,E( \kappa )}(t) \bigr) \leq 0. $$

Then

$$ \sum_{j=1}^{m}\beta _{j}h'_{j+} \bigl(\gamma _{\lambda ,E(\kappa )}(t) \bigr) \leq 0. $$

Therefore,

$$ \sum_{i=1}^{p}\alpha _{i} \bigl(f'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) \bigr) - \zeta _{i}g'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) \bigr) \bigr) +\sum_{j=1}^{m} \beta _{j}h'_{j+} \bigl( \gamma _{\lambda ,E(\kappa )}(t) \bigr) < 0. $$

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 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

By using the hypotheses and Lemma 2.1, for any \(\kappa \in K \), we obtain

$$\begin{aligned} &\bigl( f_{i}(\kappa )-\zeta _{i}^{*}g_{i}( \kappa ) \bigr) - \bigl(f _{i}\bigl(\kappa ^{*}\bigr)-\zeta _{i}^{*}g_{i}\bigl(\kappa ^{*}\bigr) \bigr)\geqslant f'_{i+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) -\zeta _{i}g'_{i+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) \\ &h_{j}(y)-h_{j}\bigl(\kappa ^{*}\bigr)\geqslant h'_{j+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr). \end{aligned}$$

Utilizing the first constraint condition for (VFD), \(\alpha ^{*}>0, \beta ^{*}\geqslant 0\), \(\zeta ^{*}\geqslant 0 \), and the two inequalities above, we have

$$\begin{aligned} & \sum_{i=1}^{p}\alpha ^{*}_{i} \bigl( \bigl( f_{i}(\kappa )-\zeta _{i}^{*}g_{i}(\kappa ) \bigr) - \bigl(f_{i}\bigl(\kappa ^{*}\bigr)-\zeta _{i} ^{*}g_{i}\bigl(\kappa ^{*}\bigr) \bigr) \bigr) + \sum_{j=1}^{m}\beta ^{*}_{j} \bigl(h_{j}(\kappa )-h_{j}\bigl(\kappa ^{*}\bigr) \bigr) \\ &\quad \geqslant \sum_{i=1}^{p} \bigl(f'_{i+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) - \zeta _{i}g'_{i+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) \bigr) \\ &\qquad {}+\sum_{j=1}^{m}\beta ^{*}_{j} h'_{j+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) \\ &\quad \geqslant 0. \end{aligned}$$
(23)

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

$$ \sum_{i=1}^{p}\alpha ^{*}_{i} \bigl( f_{i}(\kappa )-\zeta _{i}^{*}g_{i}( \kappa ) \bigr)\geqslant 0 \quad \forall y\in Y . $$
(24)

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

$$ f_{i}(\tilde{\kappa })-\zeta _{i}^{*}g_{i}( \tilde{\kappa }) < f_{i}(\bar{ \kappa })-\zeta _{i}^{*}g_{i}( \bar{\kappa }) = 0, \quad i\in P. $$

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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The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.

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Correspondence to Adem Kılıçman.

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Keywords

  • Generalized convexity
  • Riemannian geometry
  • Duality