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].
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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
A nonempty set \(B \subset \aleph \) is said to be
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)
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Definition 2.2
A function \(f: A\subset \aleph \longrightarrow \mathbb{R} \) is said to be
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
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 \(A= [ \left . -4,-1 ) \right . \cup [1,4] \),
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 \(\gamma _{E(\kappa ),E(\iota )}(t_{1})\notin A \).
$$\begin{aligned} &E(\kappa ) = \textstyle\begin{cases} \kappa ^{2} & \mbox{if } \vert \kappa \vert \leq 2, \\ -1 & \mbox{if } \vert \kappa \vert > 2; \end{cases}\displaystyle \\ &\eta (\kappa ,\iota ) = \textstyle\begin{cases} \kappa -\iota & \mbox{if } \kappa \geqslant 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota \leq 0 , \\ -1-\iota & \mbox{if } \kappa >0, \iota \leq 0 \mbox{ or } \kappa \geqslant 0 ,\iota < 0, \\ 1-\iota & \mbox{if } \kappa < 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota >0; \end{cases}\displaystyle \\ &\gamma _{\kappa ,\iota }(t) = \textstyle\begin{cases} \iota +t(\kappa -l) & \mbox{if } \kappa \geqslant 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota \leq 0 , \\ \iota +t(-1-\iota ) & \mbox{if } \kappa >0, \iota \leq 0 \mbox{ or } \kappa \geqslant 0 ,\iota < 0, \\ \iota +t(1-\iota ) & \mbox{if } \kappa < 0, \iota \geqslant 0 \mbox{ or } \kappa \leq 0, \iota >0. \end{cases}\displaystyle \end{aligned}$$
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.
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq t f(\kappa _{1})+(1-t)f(\kappa _{2}) , \quad \forall t\in \bigl[0,v( \kappa _{1},\kappa _{2})\bigr]. $$
(3)
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\geqslant t f( \kappa _{1})+(1-t)f(\kappa _{2}) , \quad \forall t\in \bigl[0,v( \kappa _{1},\kappa _{2})\bigr], $$
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 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
Then \(h(m) \) is not a GSLEC function on \(\mathbb{R} \).
$$\begin{aligned} E(m) =& \textstyle\begin{cases} 0 & \mbox{if } m< 0, \\ 1 & \mbox{if } 1< m\leq 2, \\ m & \mbox{if } 0\leq m\leq 1 \mbox{ or } m>2; \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned} \eta (m,n) =& \textstyle\begin{cases} 0 & \mbox{if } m= n, \\ 1-m & \mbox{if } m\neq n ; \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned} \gamma _{m,n}(t) =& \textstyle\begin{cases} n &\mbox{if } m= n, \\ n+t(1-m) &\mbox{if } m\neq n. \end{cases}\displaystyle \end{aligned}$$
$$\begin{aligned} h(m) =& \textstyle\begin{cases} 0 & \mbox{if } 1< m\leq 2, \\ 1 & \mbox{if } m>2, \\ -m+1 & \mbox{if } 0\leq m\leq 1, \\ -m+2 & \mbox{if } m< 0; \end{cases}\displaystyle \end{aligned}$$
$$ h \bigl(\gamma _{E(m_{0}),E(n_{0})}(t_{0}) \bigr)=1>(1-t_{0})=t_{0}h(m _{0})+(1-t_{0})h(n_{0}) . $$
Similarly, taking \(m_{1}=1\), \(n_{1}=4 \), we have
for some \(t_{1}\in [0,1] \).
$$ h \bigl(\gamma _{E(m_{1}),E(n_{1})}(t_{1}) \bigr)=1>(1-t_{1})=t_{1}h(m _{1})+(1-t_{1})h(n_{1}) $$
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
exist for every \(\kappa \in S\).
$$ h'_{+} \bigl(\gamma _{\kappa ^{*},E(\kappa )}(t) \bigr)= \lim _{t\longrightarrow 0^{+}} \frac{1}{t} \bigl[h \bigl(\gamma _{\kappa ^{*},E( \kappa )}(t) \bigr) -h\bigl(\kappa ^{*}\bigr) \bigr] $$
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
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 \(f(E(\kappa _{1}))\leq f(\kappa _{1})\), \(\forall \kappa _{1}\in S \).
$$ f\bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\bigr)\leq tf(\kappa _{2})+(1-t)f( \kappa _{1}),\quad t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$
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
□
$$ f\bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\bigr)\leq tf\bigl(E(\kappa _{1}) \bigr)+(1-t)f\bigl(E( \kappa _{2})\bigr),\quad t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$
$$ f\bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\bigr)\leq tf(\kappa _{1})+(1-t)f( \kappa _{2}),\quad t\in \bigl[0,v(\kappa _{1},\kappa _{2})\bigr]. $$
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 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
$$ \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t),t\alpha _{1}+(1-t)\alpha _{2} \bigr)\in \omega ,\quad \forall t\in \bigl[0,u\bigl((\kappa _{1},\alpha _{1}), (\kappa _{2},\alpha _{2})\bigr) \bigr]. $$
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
is a GLEI set with respect to
η
corresponding to ℵ.
$$ \omega _{f}= \bigl\lbrace (\kappa _{1},\alpha ):\kappa _{1}\in B, f( \kappa _{1})\leq \alpha , \alpha \in \mathbb{R} \bigr\rbrace $$
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 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. □
$$ \bigl( \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t), tf(\kappa _{1}) +(1-t)f( \kappa _{2}) \bigr) \in \omega _{f},\quad \forall t\in \bigl[0, u\bigl( \bigl(\kappa _{1},f(\kappa _{1})\bigr),\bigl(\kappa _{2},f(\kappa _{2})\bigr)\bigr) \bigr]. $$
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr)\leq tf(\kappa _{1}) +(1-t)f( \kappa _{2}), \quad t\in \bigl[0,u\bigl(\bigl(\kappa _{1},f( \kappa _{1})\bigr),\bigl(\kappa _{2},f(\kappa _{2}) \bigr)\bigr) \bigr]. $$
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 \(\alpha \in \mathbb{R} \). □
$$ \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t)\in B, \quad \forall t\in \bigl[0,u(\kappa _{1},\kappa _{2}) \bigr] . $$
$$\begin{aligned} f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq & t f( \kappa _{1}) +(1-t)f(\kappa _{2}) \\ \leq & t\alpha +(1-t)\alpha \\ =& \alpha , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. \end{aligned}$$
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
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, \((\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) , t \alpha +(1-t) \beta ) \in \omega _{f} \), \(\forall t\in [0,v(\kappa _{1},\kappa _{2}) ]\).
$$ \gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \in B , \quad \forall t\in \bigl[0,u(\kappa _{1},\kappa _{2}) \bigr]. $$
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta +\varepsilon , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$
$$ f \bigl(\gamma _{E(\kappa _{1}),E(\kappa _{2})}(t) \bigr) \leq t \alpha +(1-t)\beta , \quad \forall t\in \bigl[0,v(\kappa _{1},\kappa _{2}) \bigr]. $$
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 \(g_{i}(\kappa )>0\), \(\forall \kappa \in K_{0} \), \(i\in P={1,2,\ldots , p} \).
$$\begin{aligned} (\mbox{VFP}) \textstyle\begin{cases} \mbox{minimize } \frac{f(\kappa )}{g(\kappa )}= (\frac{f_{1}(\kappa )}{g_{1}( \kappa )},\ldots ,\frac{f_{p}(\kappa )}{g_{p}(\kappa )} ), \\ \mbox{subject to } h_{j}(\kappa )\leq 0,\quad j\in Q={1,2,\ldots, q} \\ \kappa \in K_{0}; \end{cases}\displaystyle \end{aligned}$$
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:
where \(\lambda =(\lambda _{1},\lambda _{2},\ldots ,\lambda _{p})\in \mathbb{R}^{p} \).
$$\begin{aligned} (\mbox{VFP}_{\lambda }) \textstyle\begin{cases} \mbox{minimize } ( f_{1}(\kappa )-\lambda _{1}g_{1}(\kappa ),\ldots f_{p}(\kappa )-\lambda _{p}g_{p}(\kappa ) ), \\ \mbox{subject to } h_{j}(\kappa )\leq 0,\quad j\in Q={1,2,\ldots, q} \\ \kappa \in K_{0}; \end{cases}\displaystyle \end{aligned}$$
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).
$$ f_{i}(\kappa )-\lambda ^{*}_{i}g_{i}( \kappa )< f_{i}\bigl(\kappa ^{*}\bigr)-\lambda ^{*}_{i}g_{i}\bigl(\kappa ^{*}\bigr),\quad \forall i\in Q $$
$$ f_{i}(\kappa )< \frac{f_{i}(\kappa ^{*})}{g_{i}(\kappa ^{*})g_{i}(\kappa )} $$
$$ \frac{f_{i}(\kappa )}{g_{i}(\kappa )}< \frac{f_{i}(\kappa ^{*})}{g_{i}( \kappa ^{*})}, $$
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 (\(\mathrm{VFP} ^{*}_{\lambda } \)). □
$$ \frac{f_{i}(\kappa )}{g_{i}(\kappa )}< \frac{f_{i}(\kappa ^{*})}{g_{i}( \kappa ^{*})}= \lambda ^{*}_{i}, $$
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
Then
κ̄
is a weak efficient solution for (VFP).
$$\begin{aligned} &\zeta ^{o}_{i}f'_{i+} \bigl(\gamma _{\bar{\kappa },E(\widehat{\kappa })}(t) \bigr)+\xi ^{o}_{j} h'_{j+} \bigl(\gamma _{\bar{\kappa },E( \widehat{\kappa })}(t) \bigr)\geqslant 0\quad \forall \kappa \in K, t \in [0,1], \end{aligned}$$
(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)
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. □
$$ \frac{f_{i}(\widehat{\kappa })}{g_{i}(\widehat{\kappa })}< \frac{f_{i}(\bar{ \kappa })}{g_{i}(\bar{\kappa })},\quad i\in P. $$
(8)
$$\begin{aligned} &f_{i}(\widehat{\kappa })-f_{i}(\bar{\kappa })\geqslant f'_{i+} \bigl(\gamma _{\bar{\kappa },E(\widehat{\kappa })}(t) \bigr) ,\quad i\in P \end{aligned}$$
(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)
$$\begin{aligned} & \sum_{i=1}^{p} \zeta ^{o}_{i} \bigl(f_{i}(\widehat{\kappa })-f_{i}(\bar{ \kappa }) \bigr) + \sum_{j=1}^{m} \xi ^{o}_{j} \bigl(h_{j}( \widehat{\kappa })-h_{j}(\bar{\kappa }) \bigr) \\ &\quad \geqslant \zeta ^{o}_{i} f'_{i+} \bigl(\gamma _{\bar{\kappa },E( \widehat{\kappa })}(t) \bigr) +\xi ^{o}_{j} h'_{j+} \bigl(\gamma _{\bar{ \kappa },E(\widehat{\kappa })}(t) \bigr) \geqslant 0. \end{aligned}$$
(12)
$$ \sum_{i=1}^{p} \zeta ^{o}_{i} \bigl(f_{i}(\widehat{\kappa })-f_{i}(\bar{ \kappa }) \bigr)\geqslant 0. $$
(13)
$$ f_{i_{0}}(\widehat{\kappa })\geqslant f_{i_{0}}( \bar{\kappa }). $$
(14)
$$ g_{i}(\widehat{\kappa })\leq g_{i}(\bar{ \kappa }),\quad i\in P. $$
(15)
$$ \frac{f_{i_{0}}(\widehat{\kappa })}{g_{i_{0}}(\widehat{\kappa })} \geqslant \frac{f_{i_{0}}(\bar{\kappa })}{g_{i_{0}}(\bar{\kappa })}, $$
(16)
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
then
κ̄
is a weak efficient solution for (VFP).
$$\begin{aligned} &\sum_{i=1}^{p}\zeta _{i}^{o} \bigl(f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) -\lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \bigr) +\xi ^{o}h'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \geqslant 0 \end{aligned}$$
(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)
Proof
Assume that κ̄ is not a weak efficient solution for (VFP). Therefore, there exists \(\kappa ^{*}\in B \), which yields
Then
which means that
$$ \frac{f_{i}(\kappa ^{*})}{g_{i}(\kappa ^{*})}< \frac{f_{i}(\bar{\kappa })}{g _{i}(\bar{\kappa })}. $$
$$ f_{i}\bigl(\kappa ^{*}\bigr)-\lambda _{i}^{o}g_{i} \bigl(\kappa ^{*}\bigr)< 0, \quad i\in P, $$
$$ f_{i}\bigl(\kappa ^{*}\bigr)-\lambda _{i}^{o}g_{i} \bigl(\kappa ^{*}\bigr)< f_{i}(\bar{\kappa })-\lambda _{i}^{o}g_{i}(\bar{\kappa })< 0, \quad i\in P. $$
By the pSLGEP of \(( f_{i}(\kappa )-\lambda _{i}^{o}g_{i}(\kappa ) )\) (\(i\in P\)) and Lemma 2.1, we have
Utilizing \(\zeta ^{o}\geqslant 0 \), then
For \(h(\kappa ^{*})\leq 0 \) and \(h_{j}(\bar{\kappa })= 0\), \(j \in \aleph (\bar{\kappa }) \), we have \(h_{j}(\kappa ^{*})\leq h_{j}(\bar{ \kappa })\), \(\forall j\in \aleph (\bar{\kappa })\).
$$ f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) -\lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) , \quad i\in P. $$
$$ \sum_{i=1}^{p}\zeta _{i}^{o} \bigl(f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) -\lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \bigr)< 0. $$
(20)
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). □
$$ h_{j+} \bigl( \gamma _{\bar{\kappa },E(\kappa )}(t) \bigr) \leq 0, \quad \forall j \in \aleph (\bar{\kappa }). $$
$$ \sum_{j=1}^{m}\xi _{j}^{o}h'_{j+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) \leq 0. $$
(21)
$$\begin{aligned} \sum_{i=1}^{p}\zeta _{i}^{o} \bigl(f'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) - \lambda _{i}^{o}g'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) \bigr) +\xi ^{o}h'_{i+} \bigl( \gamma _{\bar{\kappa },E(\kappa ^{*})}(t) \bigr) < 0, \end{aligned}$$
(22)
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:
where \(\zeta =(\zeta _{i}, i=1,2,\ldots , p)\geqslant 0\), \(\alpha =( \alpha _{i}, i=1,2,\ldots , p)> 0\), \(\beta =(\beta _{i}, i=1,2,\ldots , m)\geqslant 0\), \(\lambda \in K_{0}\).
$$\begin{aligned} (\mathrm{VFD}) \textstyle\begin{cases} \mbox{minimize } (\zeta _{i}, i=1,2,\ldots , p ) , \\ \mbox{subject to } \sum_{i=1}^{p}\alpha _{i} (f'_{i+} ( \gamma _{\lambda ,E( \kappa )}(t) ) -\zeta _{i}g'_{i+} ( \gamma _{\lambda ,E( \kappa )}(t) ) ) +\sum_{j=1}^{m}\beta _{j}h'_{j+} ( \gamma _{\lambda ,E(\kappa )}(t) ) \geqslant 0 \\ \quad \kappa \in K_{0}, t\in [0,1], \\ \quad f_{i}(\lambda )-\zeta _{i}g_{i}(\lambda )\geqslant 0, \quad i\in P,\qquad \beta _{j}h_{j}(\lambda )\geqslant 0, \quad j\in \aleph ; \end{cases}\displaystyle \end{aligned}$$
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
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 \((\alpha ,\beta ,\lambda ,\zeta )\in K ^{\prime} \). □
$$ \sum_{i=1}^{p}\alpha _{i} \bigl(f_{i}(\kappa )-\zeta _{i}g_{i}(\kappa ) \bigr)< 0 \leq \sum_{i=1}^{p}\alpha _{i}\bigl(f_{i}(\lambda )-\zeta _{i}g_{i}( \lambda )\bigr). $$
$$ \Biggl( \sum_{i=1}^{p}\alpha _{i}(f_{i}-\zeta _{i}g_{i}) \Biggr)'_{+} \bigl(\gamma _{\lambda ,E(\kappa )}(t) \bigr) < 0, $$
$$ \sum_{i=1}^{p}\alpha _{i} (f'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) \bigr) -\zeta _{i}g'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) ] \bigr)< 0. $$
$$ \sum_{j=1}^{m}\beta _{j}h_{j}( \kappa )\leq 0 \leq \sum_{j=1}^{m}\beta _{j}h_{j}(\lambda ). $$
$$ \Biggl( \sum_{j=1}^{m}\beta _{j}h_{j} \Biggr)'_{+} \bigl(\gamma _{\lambda ,E( \kappa )}(t) \bigr) \leq 0. $$
$$ \sum_{j=1}^{m}\beta _{j}h'_{j+} \bigl(\gamma _{\lambda ,E(\kappa )}(t) \bigr) \leq 0. $$
$$ \sum_{i=1}^{p}\alpha _{i} \bigl(f'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) \bigr) - \zeta _{i}g'_{i+} \bigl( \gamma _{\lambda ,E( \kappa )}(t) \bigr) \bigr) +\sum_{j=1}^{m} \beta _{j}h'_{j+} \bigl( \gamma _{\lambda ,E(\kappa )}(t) \bigr) < 0. $$
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
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
$$\begin{aligned} & \sum_{i=1}^{p}\alpha ^{*}_{i} \bigl( \bigl( f_{i}(\kappa )-\zeta _{i}^{*}g_{i}(\kappa ) \bigr) - \bigl(f_{i}\bigl(\kappa ^{*}\bigr)-\zeta _{i} ^{*}g_{i}\bigl(\kappa ^{*}\bigr) \bigr) \bigr) + \sum_{j=1}^{m}\beta ^{*}_{j} \bigl(h_{j}(\kappa )-h_{j}\bigl(\kappa ^{*}\bigr) \bigr) \\ &\quad \geqslant \sum_{i=1}^{p} \bigl(f'_{i+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) - \zeta _{i}g'_{i+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) \bigr) \\ &\qquad {}+\sum_{j=1}^{m}\beta ^{*}_{j} h'_{j+} \bigl( \gamma _{\kappa ^{*},E(\kappa )}(t) \bigr) \\ &\quad \geqslant 0. \end{aligned}$$
(23)
$$ \sum_{i=1}^{p}\alpha ^{*}_{i} \bigl( f_{i}(\kappa )-\zeta _{i}^{*}g_{i}( \kappa ) \bigr)\geqslant 0 \quad \forall y\in Y . $$
(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
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. □
$$ f_{i}(\tilde{\kappa })-\zeta _{i}^{*}g_{i}( \tilde{\kappa }) < f_{i}(\bar{ \kappa })-\zeta _{i}^{*}g_{i}( \bar{\kappa }) = 0, \quad i\in P. $$
Acknowledgements
The authors would like to thank the referees for valuable suggestions and comments, which helped the authors to improve this article substantially.
Competing interests
The authors declare that they have no competing interests.
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