Optimality conditions and duality for E-differentiable multiobjective programming involving V-E-type I functions

Abstract

In this paper, we introduce a new concept of sets and a new class of functions called $\alpha$-E-invex sets and V-E-preinvex functions. Furthermore, a new concept of generalized convexity is introduced for (not necessarily) differentiable vector optimization problems. Namely, the concept of V-E-type I functions is defined for E-differentiable vector optimization problem. A number of sufficiency results are established under various types of (generalized) V-E-type I requirements. Moreover, several E-duality theorems in the sense of Mond–Weir are proved under appropriate (generalized) V-E-type I functions.

1 Introduction

Consider the following (not necessarily differentiable) multiobjective programming problem (VP):

$$\begin{aligned} \begin{array}{c} \text {minimize }f(x)=\left( f_{1}\left( x\right) ,...,f_{k}\left( x\right) \right) \\ \text {subject to }g_{j}(x)\leqq 0\text {, }j\in J=\left\{ 1,...,m\right\} , \end{array} \qquad \text {(VP)} \end{aligned}$$

where $f_{i}:R^{n}\rightarrow R$, $i\in I=\left\{ 1,...,k\right\}$, $g_{j}:R^{n}\rightarrow R$, $j\in J$, are not necessarily differentiable functions defined on $R^{n}.$ Let $\Omega :=\left\{ x\in R^{n}:g_{j}(x)\leqq 0\text {, }j\in J\right\}$ be the set of all feasible solutions of (VP). Further, we denote by $J\left( x\right)$ the set of inequality constraint indices that are active at a feasible solution x, that is, $J\left( x\right) =\left\{ j\in J:g_{j}(x)=0\right\} .$

Definition 1

A feasible point $\overline{x}$ is said to be a weak Pareto (weakly efficient) solution of (VP) if and only if there exists no feasible point x such that

$$\begin{aligned} f(x)<f(\overline{x})\text {.} \end{aligned}$$

Definition 2

A feasible point $\overline{x}$ is said to be a Pareto (efficient) solution of (VP) if and only if there exists no feasible point x such that

$$\begin{aligned} f(x)\le f(\overline{x})\text {.} \end{aligned}$$

During the last few years, vector optimization has grown remarkably in different directions in the settings of optimality conditions and duality theory. It has been improved by the applications of various types of generalizations of convexity under differentiable and nondifferentiable assumptions (see, for example, [1, 2, 4, 7,8,9,10,11,12,13,14, 16, 18, 19, 21, 26, 28], and others).

One of the generalized convex functions is an invex function, which was introduced by Hanson [17]. Hanson and Mond [15] considered a new generalization called type I function. Rueda and Hanson [16] introduced pseudo-type I and quasi-type I functions. Kaul et al. [21] obtained optimality and duality results for multiobjective nonlinear programming problems involving type I and generalized type I functions. Megahed et al. [22] introduced a new definition of an E-differentiable function which transforms a (not necessarily) differentiable function to a differentiable function. Abdulaleem [6] introduced the so-called E-type I functions for not necessarily differentiable multicriteria optimization problems, namely for E-differentiable multiobjective programming problems involving E-type I functions. Recently, Abdulaleem [3] introduced a new concept of generalized convexity. Namely, he defined the concept of E-differentiable V-E-invexity in the case of (not necessarily) differentiable vector optimization problems with E-differentiable functions.

In this paper, new classes of nonconvex E-differentiable multiobjective programming problems. Namely, the concept of the so-called V-E-type I functions for E-differentiable multiobjective programming problems is introduced. Moreover, several classes of generalized V-E-type I functions are also defined. Further, the sufficient optimality conditions are derived for the considered E-differentiable multiobjective programming problem under appropriate V-E-type I and/or generalized V-E-type I hypotheses. Optimality results established in the paper are illustrated by examples of E-differentiable multiobjective optimization problems with (generalized) V-E-type I functions. Furthermore, the so-called vector Mond–Weir E-duality is defined for E-differentiable problems. Then, several Mond–Weir E-duality results are established between the considered E-differentiable multicriteria optimization problem and its Mond–Weir vector dual problem also under appropriate (generalized) V-E-type I functions

2 New classes of nonconvex E-differentiable functions

Let $R^{n}$ be the n-dimensional Euclidean space and $R_{+}^{n}$ be its nonnegative orthant. The following convention for equalities and inequalities will be used in the paper. For any vectors $x=\left( x_{1},x_{2},...,x_{n}\right) ^{T}$ and $y=\left( y_{1},y_{2},...,y_{n}\right) ^{T}$ in $R^{n}$, we define:

• (i) $x=y$ if and only if $x_{i}=y_{i}$ for all $i=1,2,\cdots ,n$;

• (ii) $x>y$ if and only if $x_{i}>y_{i}$ for all $i=1,2,\cdots ,n$;

• (iii) $x\geqq y$ if and only if $x_{i}\geqq y_{i}$ for all $i=1,2,\cdots ,n$;

• (iv) $x\ge y$ if and only if $x_{i}\geqq y_{i}$ for all $i=1,2,\cdots ,n$ but $x\ne y$.

Now, we introduce the definition of a $\alpha$-E-invex set as a generalization of a $\alpha$-invex set given by Noor [25] and the definition of an E-invex set given by Abdulaleem [5].

Definition 3

Let $E:R^{n}\rightarrow R^{n}$. A set $S\subseteq R^{n}$ is said to be a $\alpha$-E-invex set, if and only if there exist $\eta : S\times S\rightarrow R^{n},$ $\alpha : S\times S \rightarrow R_{+}{\setminus }\{0\},$ such that the relation

$$\begin{aligned} E\left( x_{0}\right) +\lambda \alpha \left( E\left( x\right) ,E\left( x_{0}\right) \right) \eta \left( E\left( x\right) ,E\left( x_{0}\right) \right) \in S \end{aligned}$$

holds for all $x,x_{0}\in S$ and any $\lambda \in \left[ 0,1\right]$.

Remark 1

From Definition 3, there are the following special cases:

• a) If $E(x)=x$, then the definition of a $\alpha$-E-invex set reduces to the definition of a $\alpha$-invex set (see Noor [25]).

• b) If $\alpha \left( x, x_{0}\right) =1,$ $\eta (x, x_{0})=x-x_{0}$, then the definition of a $\alpha$-E-invex set reduces to the definition of an E-convex set (see Youness [29] ).

• c) If $\alpha \left( x, x_{0}\right) =1,$ then the definition of a $\alpha$-E-invex set reduces to the definition of an E-invex set (see Abdulaleem [5]).

• d) If $\alpha \left( x, x_{0}\right) =1,$ $E(x)=x$, then the definition of a $\alpha$-E-invex set reduces to the definition of an invex set (see Mohan and Neogy [23]).

• e) If $E(x)=x,$ $\alpha \left( x, x_{0}\right) =1$ and $\eta (x, x_{0})=x-x_{0}$, then the definition of a $\alpha$-E-invex set reduces to the definition of a convex set.

Now, we present an example of such a $\alpha$-E-invex set which is not $\alpha$-invex.

Example 1

Let $S=[1,9]\cup [-9,-1],$ $E(x) = {\left\{ \begin{array}{ll} x^{2} &{}\text{ if } \; -3\leqq x\leqq 3, \\ -1 &{}\text{ if } \; x<-3 \; \text {or}\; x>3.\\ \end{array}\right. },$ $\eta (x,x_{0}) = {\left\{ \begin{array}{ll} -1 &{}\text{ if } x<x_{0}, \\ -3-x_{0} &{}\text{ if } x\geqq x_{0}. \end{array}\right. },$ $\alpha (x,x_{0}) = {\left\{ \begin{array}{ll} x_{0}-x &{}\text{ if } x<x_{0}, \\ 1 &{}\text{ if } x\geqq x_{0}. \end{array}\right. }.$ Then, by Definition 3, S is a $\alpha$-E-invex set with respect to $\eta$ given above. However, it is not $\alpha$-invex set as can be seen by taking $x=-1$, $x_{0}=1$, and $\lambda =\frac{1}{2}$, we have

$$\begin{aligned} x_{0}+\lambda \alpha \left( x,x_{0}\right) \eta \left( x,x_{0}\right) =0 \not \in S. \end{aligned}$$

Hence, by the definition of a $\alpha$-invex set (see Noor [25]), it follows that S is not $\alpha$-invex.

Example 2

Let $S=[1,16]\cup [-16,-1]$ and $E:R\rightarrow R$ be an operator defined by

$$\begin{aligned} E(x) = {\left\{ \begin{array}{ll} x^{2} &{}\text{ if } \; 0\leqq x\leqq 4, \\ -x &{}\text{ if } -4\leqq x\leqq 0, \\ -1 &{}\text{ if } \; x<-4 \; \text {or}\; x>4.\\ \end{array}\right. } \end{aligned}$$

where $\alpha :S\times S\rightarrow R_{+}\setminus \{0\}$ and $\eta :S \times S\rightarrow R$ be defined by

$$\begin{aligned}{} & {} \eta (x,x_{0}) = {\left\{ \begin{array}{ll} x-x_{0} &{}\text{ if } x\geqq 0, \; x_{0}\geqq 0 \vee x\leqq 0, \; x_{0}\leqq 0, \\ -16-x_{0} &{}\text{ if } x> 0, \; x_{0}\leqq 0 \vee x\geqq 0, \; x_{0}< 0, \\ 16-16x_{0} &{}\text{ if } x< 0, \; x_{0}\geqq 0 \vee x \leqq 0, \; x_{0}> 0. \\ \end{array}\right. } \\{} & {} \alpha (x,x_{0}) = {\left\{ \begin{array}{ll} \frac{1}{16} &{}\text{ if } x< 0, \; x_{0}\geqq 0 \vee x \leqq 0, \; x_{0}> 0, \\ 1 &{}\text{ if } \text {otherwise}. \end{array}\right. } \end{aligned}$$

Then, by Definition 3, S is a $\alpha$-E-invex set with respect to the function $\eta$ given above. However, it is not E-convex as can be seen by taking $x=1$, $x_{0}=16$, and $\lambda =\frac{1}{2}$, we have

$$\begin{aligned} \lambda E\left( x\right) +\left( 1-\lambda \right) E\left( x_{0}\right) \not \in S. \end{aligned}$$

Hence, by the definition of an E-convex set (see Youness [29]), it follows that S is not E-convex. Also, S is not E-invex as can be seen by taking $x=10$, $x_{0}=4$, and $\lambda =\frac{1}{2}$, we have

$$\begin{aligned} E\left( x_{0}\right) +\lambda \eta \left( E\left( x\right) ,E\left( x_{0}\right) \right) \not \in S. \end{aligned}$$

Hence, by the definition of an E-invex set (see Abdulaleem [5]), it follows that S is not E-invex. Moreover, S is not an invex set with respect to the function $\eta$ given above.

Now, we introduce new concept of generalized convexity.

Definition 4

Let $E:R^{n}\rightarrow R^{n}$ and $S\subseteq R^{n}$ be a $\alpha$-E-invex set. The function $f:S\rightarrow R$ is said to be V-E-preinvex on S, if there exist $\eta : S\times S\rightarrow R^{n},$ $\alpha : S\times S \rightarrow R_{+}{\setminus }\{0\},$ such that the following inequality

$$\begin{aligned} f\left( E\left( x_{0}\right) +\lambda \alpha \left( E\left( x\right) ,E\left( x_{0}\right) \right) \eta \left( E\left( x\right) ,E\left( x_{0}\right) \right) \right) \leqq \left( 1-\lambda \right) f\left( E\left( x_{0}\right) \right) + \lambda f\left( E\left( x\right) \right) \end{aligned}$$

holds for all $x,x_{0}\in S$ and any $\lambda \in \left[ 0,1\right]$.

It is clear that every convex function is V-E-preinvex, but the converse is not true.

Definition 5

Let $E:R^{n}\rightarrow R^{n}$ and $S\subseteq R^{n}$ be a $\alpha$-E-invex set. The function $f:S\rightarrow R$ is said to be strictly V-E-preinvex on S, if there exist $\eta : S\times S\rightarrow R^{n},$ $\alpha : S\times S \rightarrow R_{+}{\setminus }\{0\},$ such that the following inequality

$$\begin{aligned} f\left( E\left( x_{0}\right) +\lambda \alpha \left( E\left( x\right) ,E\left( x_{0}\right) \right) \eta \left( E\left( x\right) ,E\left( x_{0}\right) \right) \right) < \left( 1-\lambda \right) f\left( E\left( x_{0}\right) \right) + \lambda f\left( E\left( x\right) \right) \end{aligned}$$

holds for all $x,x_{0}\in S,\; x\ne x_{0},$ and any $\lambda \in \left[ 0,1\right]$.

Definition 6

Let $E:R^{n}\rightarrow R^{n}$ and $S\subseteq R^{n}$ be a $\alpha$-E-invex set. The function $f:S\rightarrow R$ is said to be quasi V-E-preinvex on S,  if there exist $\eta : S\times S\rightarrow R^{n},$ $\alpha : S\times S \rightarrow R_{+}{\setminus }\{0\},$ such that the following inequality

$$\begin{aligned} f\left( E\left( x_{0}\right) +\lambda \alpha \left( E\left( x\right) ,E\left( x_{0}\right) \right) \eta \left( E\left( x\right) ,E\left( x_{0}\right) \right) \right) \leqq \max \{(f\circ E)(x),(f\circ E)(x_{0})\} \end{aligned}$$

holds for all $x,x_{0}\in S$ and any $\lambda \in \left[ 0,1\right]$.

Definition 7

[22] Let $E:R^{n}\rightarrow R^{n}$ and $f:S\rightarrow R$ be a (not necessarily) differentiable function at a given point $x_{0}\in S$. It is said that f is an E-differentiable function at $x_{0}$ if and only if $f\circ E$ is a differentiable function at $x_{0}$ (in the usual sense), that is,

$$\begin{aligned} \left( f\circ E\right) \left( x\right) =\left( f\circ E\right) \left( x_{0}\right) +\nabla \left( f\circ E\right) \left( x_{0}\right) \left( x-x_{0}\right) +\theta \left( x_{0},x-x_{0}\right) \left\| x-x_{0}\right\| , \end{aligned}$$

(1)

where $\theta \left( x_{0},x-x_{0}\right) \rightarrow 0$ as $x\rightarrow x_{0}$.

Let $E:R^{n}\rightarrow R^{n}$ be a given one-to-one and onto operator. Now, for the problem (VP), we define its associated differentiable vector optimization problem ($\hbox {VP}_{E}$) as follows:

$$\begin{aligned} \begin{array}{c} \text {minimize }f(E(x))= \left( f_{1}(E(x)),...,f_{k}(E(x)) \right) \\ \text {subject to } g_{j}(E(x))\leqq 0\text {, }j\in J=\left\{ 1,...,m\right\} . \end{array} \text {(VP}_{E}\text {)} \end{aligned}$$

We call the problem ($\hbox {VP}_{E}$) an E-vector optimization problem associated to (VP). Let

$$\begin{aligned} \Omega _{E}:=\left\{ x\in R^{n}:g_{j}(E(x))\leqq 0\text {, } j\in J\right\} \end{aligned}$$

be the set of all feasible solutions of ($\hbox {VP}_{E}$). Further, we denote by $J_{E}\left( x\right)$ the set of inequality constraint indices that are active at a feasible solution $x\in \Omega _{E}$, that is, $J_{E}\left( x\right) =\left\{ j\in J:\left( g_{j}\circ E\right) (x)=0\right\}$.

Definition 8

A feasible point $E(x_{0})$ is said to be a (weak E-Pareto solution) weakly E-efficient solution of (VP) if and only if there exists no feasible point E(x) such that

$$\begin{aligned} f(E(x))< f(E(x_{0}))\text {.} \end{aligned}$$

Definition 9

A feasible point $E(x_{0})$ is said to be an (E-Pareto solution) E-efficient solution of (VP) if and only if there exists no feasible point E(x) such that

$$\begin{aligned} f(E(x))\le f(E(x_{0}))\text {.} \end{aligned}$$

Lemma 1

[1] Let $E:R^{n}\rightarrow R^{n}$ be a one-to-one and onto. Then $E\left( \Omega _{E}\right) =\Omega$.

Lemma 2

[1] Let $x_{0}\in \Omega$ be a (weak Pareto solution) Pareto solution of the problem (VP). Then, there exists $\overline{z}\in \Omega _{E}$ such that $x_{0}=E\left( \overline{z}\right)$ and $\overline{z}$ is a (weak Pareto) Pareto solution of the problem ($\hbox {VP}_{E}$).

Lemma 3

[1] Let $\overline{z}\in \Omega _{E}$ be a (weak Pareto) Pareto solution of the problem ($\hbox {VP}_{E}$). Then $E\left( \overline{z}\right)$ is a (weak Pareto solution) Pareto solution of the problem (VP).

Now, we introduce new concept of generalized convexity for E-differentiable vector-valued functions.

Definition 10

Let $E:R^{n}\rightarrow R^{n}.$ (f, g) is said to be V-E-type I at $x_{0}\in \Omega _{E}$ if there exists functions $\eta :R^{n}\times R^{n}\rightarrow R^{n},$ $\alpha _{i},\beta _{j}: R^{n}\times R^{n}\rightarrow R_{+}{\setminus }\{0\},$ $i\in I,$ $j\in J,$ such that, for all $x\in \Omega _{E}$,

$$\begin{aligned}{} & {} f_{i}(E(x))-f_{i}(E(x_{0}))\geqq \alpha _{i} (E(x),E(x_{0})) \nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0})), \end{aligned}$$

(2)

$$\begin{aligned}{} & {} -g_{j}(E(x_{0}))\geqq \beta _{j} (E(x),E(x_{0})) \nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0})). \end{aligned}$$

(3)

If the inequalities (2, 3) are satisfied at each $x_{0}\in R^{n}$, then (f, g) are said to be V-E-type I on $R^{n}.$

Remark 2

From Definition 10, there are the following special cases:

• a) If f and g are differentiable functions and $E(x)\equiv x$ (E is an identity map), then the definition of V-E-type I functions reduces to the definition of V-type I functions introduced by Hanson et al. [18].

• b) If $\alpha _{i}(x,x_{0})=1$, $i=1,...,k,$ $\beta _{j}(x,x_{0})=1$, $j=1,...,m,$ f and g are differentiable functions and $E(x)\equiv x$ (E is an identity map), then the definition of V-E-type I functions reduces to the definition of type I functions introduced by Rueda and Hanson [27].

• c) If $\alpha _{i}(x,x_{0})=1$, $i=1,...,k,$ $\beta _{j}(x,x_{0})=1$, $j=1,...,m,$ then the definition of V-E-type I functions reduces to the definition of E-type I functions introduced by Abdulaleem [6].

Definition 11

Let $E:R^{n}\rightarrow R^{n}.$ (f, g) is said to be strictly V-E-type I at $x_{0}\in \Omega _{E}$ if there exists functions $\eta :R^{n}\times R^{n}\rightarrow R^{n},$ $\alpha _{i},\beta _{j}: R^{n}\times R^{n}\rightarrow R_{+}{\setminus }\{0\},$ $i\in I,$ $j\in J,$ such that, for all $x\in \Omega _{E}$,

$$\begin{aligned}{} & {} f_{i}(E(x))-f_{i}(E(x_{0}))> \alpha _{i} (E(x),E(x_{0})) \nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0})), \end{aligned}$$

(4)

$$\begin{aligned}{} & {} -g_{j}(E(x_{0}))\geqq \beta _{j} (E(x),E(x_{0})) \nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0})). \end{aligned}$$

(5)

If the inequalities (4, 5) are satisfied at each $x_{0}\in R^{n}$, then (f, g) are said to be strictly V-E-type I on $R^{n}.$

Now, we present an example of such V-E-type I functions which are not E-type I functions nor type I functions with respect to the same function $\eta$.

Example 3

$f:R\rightarrow R^{2},$ $g:R\rightarrow R$ defined by $f(x)=(e^{\root 3 \of {27x}}, 3e^{\root 3 \of {27x}}),$ $g(x)=-e^{\root 3 \of {x}}$ are V-E-type I functions with respect to $E(x)=x^{3},$ $\eta (E(x),E(x_{0})) = {\left\{ \begin{array}{ll} \frac{e^{3x}-e^{3x_{0}}}{e^{4x_{0}}} &{}\text{ if } x < x_{0}, \\ 0 &{}\text{ if } x\geqq x_{0}. \end{array}\right. },$     $\alpha _{i}(E(x),E(x_{0})) = {\left\{ \begin{array}{ll} \frac{1}{3}e^{x_{0}} &{}\text{ if } x < x_{0}, \\ 1 &{}\text{ if } x\geqq x_{0}. \end{array}\right. },$    $\beta (E(x),E(x_{0})) = {\left\{ \begin{array}{ll} \frac{e^{4}}{e^{3x_{0}}-e^{3x}} &{}\text{ if } x < x_{0}, \\ 1 &{}\text{ if } x\geqq x_{0}. \end{array}\right. }.$ However, (f, g) is not E-type I as can be seen by taking $x=\ln 2$, $x_{0}=\ln 6$, since the inequality

$$\begin{aligned} f_{i}(E(x))-f_{i}(E(x_{0}))< \nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0})) \end{aligned}$$

holds, it follows that (f, g) are not E-type I nor E-invex with respect to $\eta$ given above. Moreover, (f, g) is neither type I nor invex with respect to $\eta$ given above.

Now, we introduce various classes of generalized V-E-type I functions.

Definition 12

Let $E:R^{n}\rightarrow R^{n}.$ (f, g) is said to be quasi-V-E-type I at $x_{0}\in \Omega _{E}$ if there exists functions $\eta :R^{n}\times R^{n}\rightarrow R^{n},$ $\alpha _{i},\beta _{j}: R^{n}\times R^{n}\rightarrow R_{+}{\setminus }\{0\},$ $i\in I,$ $j\in J,$ such that, for all $x\in \Omega _{E}$,

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\alpha _{i}(E(x),E(x_{0})) \left( f_{i}(E(x))-f_{i}(E(x_{0}))\right) \leqq 0 \Rightarrow \sum _{i=1}^{k}\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))\leqq 0, \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \quad -\sum _{j=1}^{m}\beta _{j}(E(x),E(x_{0}))g_{j}(E(x_{0})) \leqq 0 \Rightarrow \sum _{j=1}^{m}\nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0}))\leqq 0. \end{aligned}$$

(7)

If (6, 7) are satisfied at each $x_{0}\in R^{n}$, then (f, g) are said to be quasi-V-E-type I on $R^{n}.$

Definition 13

Let $E:R^{n}\rightarrow R^{n}.$ (f, g) is said to be pseudo-V-E-type I at $x_{0}\in \Omega _{E}$ if there exists functions $\eta :R^{n}\times R^{n}\rightarrow R^{n},$ $\alpha _{i},\beta _{j}: R^{n}\times R^{n}\rightarrow R_{+}{\setminus }\{0\},$ $i\in I,$ $j\in J,$ such that, for all $x\in \Omega _{E}$,

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))\geqq 0 \Rightarrow \sum _{i=1}^{k}\alpha _{i}(E(x),E(x_{0})) \left( f_{i}(E(x))-f_{i}(E(x_{0}))\right) \geqq 0, \end{aligned}$$

(8)

$$\begin{aligned}{} & {} \sum _{j=1}^{m}\nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0}))\geqq 0 \Rightarrow -\sum _{j=1}^{m}\beta _{j}(E(x),E(x_{0}))g_{j}(E(x_{0})) \geqq 0. \end{aligned}$$

(9)

If (8, 9) are satisfied at each $x_{0}\in R^{n}$, then (f, g) are said to be pseudo-V-E-type I on $R^{n}.$

Now, we present an example of such a pseudo-V-E-type I function which is not V-E-type I functions.

Example 4

Let $E:R\rightarrow R,$ $f:R \rightarrow R^{2},$ and $g:R \rightarrow R$ be defined by

$$\begin{aligned} f(x)=\left( e^{2x^\frac{1}{3}},2e^{2x^\frac{1}{3}}\right) , \; \; g(x)=e^{3x^\frac{1}{3}}, \; \; E(x)=x^{3}. \end{aligned}$$

Therefore, (f, g) is pseudo-V-E-type I with respect to

$$\begin{aligned} \eta (x,x_{0}) = {\left\{ \begin{array}{ll} \frac{1}{(x^{\frac{2}{3}}-x_{0}^{\frac{2}{3}})e^{\root 3 \of {x_{0}}}} &{}\text{ if } x > x_{0}, \\ -1 &{}\text{ if } x \leqq x_{0}, \end{array}\right. } \end{aligned}$$

$\alpha _{i}(x,x_{0}) = {\left\{ \begin{array}{ll} x^{\frac{2}{3}}-x_{0}^{\frac{2}{3}} &{}\text{ if } x > x_{0}, \\ 3 &{}\text{ if } x \leqq x_{0}, \end{array}\right. }$      $\beta _{j}(x,x_{0}) = {\left\{ \begin{array}{ll} \left( x^{\frac{2}{3}}-x_{0}^{\frac{2}{3}}\right) e^{\root 3 \of {x_{0}}} &{}\text{ if } x > x_{0}, \\ 1 &{}\text{ if } x \leqq x_{0}. \end{array}\right. }$

However, (f, g) is not V-E-type I. Indeed, if we set $x=\frac{1}{2}$, $x_{0}=0$, then we have

$$\begin{aligned}{} & {} f(E(x))-f(E(x_{0}))< \alpha _{i}(x,x_{0})\nabla f(E(x_{0}))\eta (E(x),E(x_{0})), \\{} & {} -g(E(x_{0}))< \beta _{j}(x,x_{0}) \nabla g(E(x_{0}))\eta (E(x),E(x_{0})). \end{aligned}$$

Hence, by Definition 10, it follows that (f, g) is not V-E-type I.

Definition 14

Let $E:R^{n}\rightarrow R^{n}.$ (f, g) is said to be quasi-pseudo-V-E-type I at $x_{0}\in \Omega _{E}$ if there exists functions $\eta :R^{n}\times R^{n}\rightarrow R^{n},$ $\alpha _{i},\beta _{j}: R^{n}\times R^{n}\rightarrow R_{+}{\setminus }\{0\},$ $i\in I,$ $j\in J,$ such that, for all $x\in \Omega _{E}$,

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\alpha _{i}(E(x),E(x_{0})) \left( f_{i}(E(x))-f_{i}(E(x_{0}))\right) \leqq 0 \Rightarrow \sum _{i=1}^{k}\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))\leqq 0, \end{aligned}$$

(10)

$$\begin{aligned}{} & {} \sum _{j=1}^{m}\nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0}))\geqq 0 \Rightarrow -\sum _{j=1}^{m}\beta _{j}(E(x),E(x_{0}))g_{j}(E(x_{0})) \geqq 0. \end{aligned}$$

(11)

If (10, 11) are satisfied at each $x_{0}\in R^{n}$, then (f, g) are said to be quasi-pseudo-V-E-type I on $R^{n}.$ If (11) is replaced in the above definition by the inequality

$$\begin{aligned} \sum _{j=1}^{m}\nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0}))\geqq 0 \Rightarrow -\sum _{j=1}^{m}\beta _{j}(E(x),E(x_{0}))g_{j}(E(x_{0})) > 0 \end{aligned}$$

(12)

then we say (f, g) is strictly quasi-pseudo-V-E-type I at $x_{0}.$

Definition 15

Let $E:R^{n}\rightarrow R^{n}.$ (f, g) is said to be pseudo-quasi-V-E-type I at $x_{0}\in \Omega _{E}$ if there exists functions $\eta :R^{n}\times R^{n}\rightarrow R^{n},$ $\alpha _{i},\beta _{j}: R^{n}\times R^{n}\rightarrow R_{+}{\setminus }\{0\},$ $i\in I,$ $j\in J,$ such that, for all $x\in \Omega _{E}$,

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))\geqq 0 \Rightarrow \sum _{i=1}^{k}\alpha _{i}(E(x),E(x_{0})) \left( f_{i}(E(x))-f_{i}(E(x_{0}))\right) \geqq 0, \end{aligned}$$

(13)

$$\begin{aligned}{} & {} \quad -\sum _{j=1}^{m}\beta _{j}(E(x),E(x_{0}))g_{j}(E(x_{0})) \leqq 0 \Rightarrow \sum _{j=1}^{m}\nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0}))\leqq 0. \end{aligned}$$

(14)

If (13, 14) are satisfied at each $x_{0}\in R^{n}$, then (f, g) are said to be pseudo-quasi-V-E-type I on $R^{n}.$ If (13) is replaced in the above definition by the inequality

$$\begin{aligned} \sum _{i=1}^{k}\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))\geqq 0 \Rightarrow \sum _{i=1}^{k}\alpha _{i}(E(x),E(x_{0})) \left( f_{i}(E(x))-f_{i}(E(x_{0}))\right) > 0, \end{aligned}$$

(15)

then we say (f, g) is strictly pseudo-quasi-V-E-type I at $x_{0}.$

3 E-optimality conditions

In this section, we prove the sufficient E-optimality conditions for (weak E-Pareto) E-Pareto optimality in the considered E-differentiable vector optimization problem (VP) under assumption that the functions constituting it belong to the classes of E-differentiable functions defined in the preceding section.

Theorem 1

(E-Karush-Kuhn-Tucker necessary optimality conditions). Let $x_{0}\in \Omega _{E}$ be (a weak Pareto solution) a Pareto solution of the problem ($\hbox {VP}_{E}$) (and, thus, $E\left( x_{0}\right)$ be (a weak E-Pareto solution) an E-Pareto solution of the problem (VP)). Further, f, g are E-differentiable at $x_{0}$ and the E-Guignard constraint qualification [5] be satisfied at $x_{0}$. Then there exist Lagrange multipliers ${\lambda _{0} }\in R^{k}$, ${\mu _{0} }\in R^{m}$ such that

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\lambda _{0_{i}}\nabla \left( f_{i}\circ E\right) (x_{0})+\sum _{j=1}^{m}\mu _{0_{j}}\nabla \left( g_{j}\circ E\right) (x_{0})=0\text {,} \end{aligned}$$

(16)

$$\begin{aligned}{} & {} \mu _{0_{j}}\left( g_{j}\circ E\right) (x_{0})=0\text {, } j\in J\left( E\left( x_{0}\right) \right) \text {,} \end{aligned}$$

(17)

$$\begin{aligned}{} & {} {\lambda _{0} }\ge 0\text {, }{\mu _{0}}\geqq 0\text {.} \end{aligned}$$

(18)

Definition 16

$\left( x_{0},\lambda _{0},\mu _{0}\right) \in \Omega _{E} \times R^{k}\times R^{m}$ is said to be a Karush-Kuhn-Tucker point for the problem ($\hbox {VP}_{E}$) if the Karush-Kuhn-Tucker necessary optimality conditions (16–18) are satisfied at $x_{0}$ with Lagrange multiplier $\lambda _{0}$, $\mu _{0}$.

Theorem 2

Let $\left( x_{0},\lambda _{0},\mu _{0}\right) \in \Omega _{E} \times R^{k}\times R^{m}$ be a Karush-Kuhn-Tucker point of the problem ($\hbox {VP}_{E}$). If (f, g) is V-E-type I at $x_{0}$, then $x_{0}$ is a weak Pareto solution of the problem ($\hbox {VP}_{E}$) and, thus, $E(x_{0})$ is a weak E-Pareto solution of the problem (VP).

Proof

Suppose that $x_{0}\in \Omega _{E}$ is not a weak Pareto solution of the problem ($\hbox {VP}_{E}$) (and, thus, $E\left( x_{0}\right)$ is not a weak E-Pareto solution of the considered multiobjective programming problem (VP)). Then, there exists a feasible solution x for ($\hbox {VP}_{E}$) such that

$$\begin{aligned} f(E(x))< f(E(x_{0})). \end{aligned}$$

(19)

By $\lambda _{0_{i}}> 0$, $i\in I$, we have

$$\begin{aligned} \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x))< \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x_{0})). \end{aligned}$$

(20)

Since $(f_{i}, g_{j}),$ $i\in I,$ $j\in J$ is V-E-type I at $x_{0}$, we have for all $x\in \Omega _{E}$

$$\begin{aligned}{} & {} f_{i}(E(x))-f_{i}(E(x_{0}))\geqq \alpha _{i}(E(x),E(x_{0}))\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0})), \; \; \; i\in I \end{aligned}$$

(21)

$$\begin{aligned}{} & {} 0=-g_{j}(E(x_{0}))\geqq \beta _{j}(E(x),E(x_{0}))\nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0})). \; \; \; j\in J(E(x_{0})) \end{aligned}$$

(22)

Combining (19) and (21), we have

$$\begin{aligned} \alpha _{i}(E(x),E(x_{0}))\nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))<0. \; \; \; i\in I \end{aligned}$$

(23)

Since $\alpha _{i}(E(x),E(x_{0}))>0,\; i\in I,$ the above inequality yields

$$\begin{aligned} \nabla f_{i}(E(x_{0}))\eta (E(x),E(x_{0}))<0. \; \; \; i\in I \end{aligned}$$

(24)

Since $\beta _{j}(E(x),E(x_{0}))>0,\; j\in J\left( E\left( x_{0}\right) \right) , x\in \Omega _{E},$ (50) yields

$$\begin{aligned} \nabla g_{j}(E(x_{0}))\eta (E(x),E(x_{0}))\leqq 0. \; \; \; j\in J(E(x_{0})) \end{aligned}$$

(25)

By $\lambda _{0_{i}}> 0$, $i\in I,$ ${\mu }_{0_{j}}\geqq 0,$ $j\in J\left( E\left( x_{0}\right) \right)$, using (24, 25), we obtain that the inequality

$$\begin{aligned} \left[ \sum _{i=1}^{k}\lambda _{0_{i}}\nabla f_{i}( E \left( x_{0}\right) )+\sum _{j=1}^{m}\mu _{0_{j}}\nabla g_{j}(E(x_{0}))\right] \eta (E(x),E(x_{0}))<0 \end{aligned}$$

(26)

holds, contradicting (16). By assumption, $E:R^{n}\rightarrow R^{n}$ is a one-to-one and onto operator. Since $x_{0}$ is a weak Pareto solution of the problem ($\hbox {VP}_{E}$), by Lemma 3, $E\left( x_{0}\right)$ is a weak E-Pareto solution of the problem (VP). Thus, the proof of this theorem is completed. $\square$

Theorem 3

Let $\left( x_{0},\lambda _{0},\mu _{0}\right) \in \Omega _{E} \times R^{k}\times R^{m}$ be a Karush-Kuhn-Tucker point of the problem ($\hbox {VP}_{E}$). If $({\lambda _{0}}f, \mu _{0} g)$ is pseudo-quasi-V-E-type I with respect to $\eta$ at $x_{0}$, then $x_{0}$ is a Pareto solution of the problem ($\hbox {VP}_{E}$) and, thus, $E(x_{0})$ is an E-Pareto solution of the problem (VP).

Proof

Suppose contrary to the result, that $x_{0}$ is not a Pareto solution in problem ($\hbox {VP}_{E}$) (and, thus, $E\left( x_{0}\right)$ is not an E-Pareto solution of the considered multiobjective programming problem (VP)). Then, there exists a feasible solution $x^{\circ }$ for ($\hbox {VP}_{E}$) such that

$$\begin{aligned} f_{i}(E(x^{\circ }))< f_{i}(E(x_{0})), \; i\in I. \end{aligned}$$

(27)

Since $\lambda _{0_{i}}> 0$, $i\in I$, above inequalities give

$$\begin{aligned} \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x^{\circ }))< \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x_{0})). \end{aligned}$$

(28)

Using $\alpha _{i}(E(x^{\circ }),E(x_{0}))>0, i\in I,$ we get

$$\begin{aligned} \sum _{i=1}^{k}\alpha _{i}(E(x^{\circ }),E(x_{0}))\lambda _{0_{i}}f_{i}(E(x^{\circ }))< \sum _{i=1}^{k}\alpha _{i}(E(x^{\circ }),E(x_{0}))\lambda _{0_{i}}f_{i}(E(x_{0})). \end{aligned}$$

(29)

Also $\left( g_{j}\circ E\right) (x_{0})=0,$ $j \in J(E(x_{0})),$ $\beta _{j}(E(x^{\circ }),E(x_{0}))> 0$ yields

$$\begin{aligned} \sum _{j=1}^{m}\beta _{j}(E(x^{\circ }),E(x_{0})) \mu _{0_{j}}\left( g_{j}\circ E\right) (x_{0}) = 0. \end{aligned}$$

(30)

By $(\lambda _{0}f, \mu _{0} g)$ is pseudo-quasi-V-E-type I at $x_{0}$ and by inequalities (29) and (30) we have

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\lambda _{0_{i}}\nabla \left( f_{i}\circ E\right) \left( x_{0}\right) \eta \left( E\left( x^{\circ }\right) ,E\left( x_{0}\right) \right) < 0\text {, } \end{aligned}$$

(31)

$$\begin{aligned}{} & {} \sum _{j=1}^{m}\mu _{0_{j}} \nabla \left( g_{j}\circ E\right) (x_{0}) \eta \left( E\left( x^{\circ }\right) ,E\left( x_{0}\right) \right) \leqq 0 \text {.} \end{aligned}$$

(32)

Adding these above inequalities, we obtain the inequality

$$\begin{aligned} \bigg [ \sum _{i=1}^{k}\lambda _{0_{i}}\nabla f_{i}( E \left( x_{0}\right) ) +\sum _{j=1}^{m}\mu _{0_{j}}\nabla g_{j}\left( E\left( {x_{0}}\right) \right) \bigg ]\eta \left( E\left( x^{\circ } \right) ,E\left( x_{0}\right) \right) < 0\text {} \end{aligned}$$

(33)

holds, contradicting (16). Since $x_{0}$ is a Pareto solution of the problem ($\hbox {VP}_{E}$), by Lemma 3, $E\left( x_{0}\right)$ is an E-Pareto solution of the problem (VP). Thus, the proof of this theorem is completed. $\square$

Theorem 4

Let $\left( x_{0},\lambda _{0},\mu _{0}\right) \in \Omega _{E} \times R^{k}\times R^{m}$ be a Karush-Kuhn-Tucker point of the problem ($\hbox {VP}_{E}$). If $(\lambda _{0}f, \mu _{0} g)$ is pseudo-quasi-V-E-type I with respect to $\eta$ at $x_{0}$, then $x_{0}$ is a weak Pareto solution of the problem ($\hbox {VP}_{E}$) and, thus, $E(x_{0})$ is a weak E-Pareto solution of the problem (VP).

Proof

Suppose contrary to the result, that $x_{0}$ is not a weak Pareto solution in problem ($\hbox {VP}_{E}$) (and, thus, $E\left( x_{0}\right)$ is not a weak E-Pareto solution of the considered multiobjective programming problem (VP)). Then, there exists a feasible solution $x^{\circ }$ for ($\hbox {VP}_{E}$) such that

$$\begin{aligned} f_{i}(E(x^{\circ }))< f_{i}(E(x_{0})), \; \; \; i\in I \end{aligned}$$

(34)

Since $\lambda _{0_{i}}> 0$, $i\in I$, above inequality give

$$\begin{aligned} \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x^{\circ }))< \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x_{0})). \end{aligned}$$

(35)

The remaining part of the proof is similar to that of Theorem 3. $\square$

Theorem 5

Let $\left( x_{0},\lambda _{0},\mu _{0}\right) \in \Omega _{E} \times R^{k}\times R^{m}$ be a Karush-Kuhn-Tucker point of the problem ($\hbox {VP}_{E}$). If $(\lambda _{0}f, \mu _{0} g)$ is quasi-pseudo-V-E-type I with respect to $\eta$, then $x_{0}$ is a Pareto solution of the problem ($\hbox {VP}_{E}$) and, thus, $E(x_{0})$ is an E-Pareto solution of the problem (VP).

Proof

Suppose contrary to the result, that $x_{0}$ is not a Pareto solution in problem ($\hbox {VP}_{E}$) (and, thus, $E\left( x_{0}\right)$ is not an E-Pareto solution of the considered multiobjective programming problem (VP)). Then, there exists a feasible solution $\widehat{x}$ for ($\hbox {VP}_{E}$) such that

$$\begin{aligned} f_{i}(E(\widehat{x}))\le f_{i}(E(x_{0})), \; \; \; i\in I \end{aligned}$$

(36)

Since $\lambda _{0_{i}}> 0$, $i\in I$, above inequality give

$$\begin{aligned} \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(\widehat{x}))\le \sum _{i=1}^{k}\lambda _{0_{i}}f_{i}(E(x_{0})). \end{aligned}$$

(37)

Using $\alpha _{i}(E(\widehat{x}),E(x_{0}))>0, i\in I,$ we get

$$\begin{aligned} \sum _{i=1}^{k}\alpha _{i}(E(\widehat{x}),E(x_{0}))\lambda _{0_{i}}f_{i}(E(\widehat{x}))< \sum _{i=1}^{k}\alpha _{i}(E(\widehat{x}),E(x_{0}))\lambda _{0_{i}}f_{i}(E(x_{0})). \end{aligned}$$

(38)

Also $\left( g_{j}\circ E\right) (x_{0})=0,$ $j \in J(E(x_{0})),$ $\beta _{j}(E(\widehat{x}),E(x_{0}))> 0$ yields

$$\begin{aligned} -\sum _{j=1}^{m}\beta _{j}(E(\widehat{x}),E(x_{0}))\mu _{0_{j}} \left( g_{j}\circ E\right) (x_{0})\leqq 0. \end{aligned}$$

(39)

By $(\lambda _{0}f, \mu _{0} g)$ is quasi-pseudo-V-E-type I with respect to $\eta$ at $x_{0}$ and by inequalities (38) and (39) we have

$$\begin{aligned}{} & {} \sum _{i=1}^{k}\lambda _{0_{i}}\nabla \left( f_{i}\circ E\right) \left( x_{0}\right) \eta \left( E\left( \widehat{x}\right) ,E\left( x_{0}\right) \right) \leqq 0\text {, } \end{aligned}$$

(40)

$$\begin{aligned}{} & {} \sum _{j=1}^{m}\mu _{0_{j}} \nabla \left( g_{j}\circ E\right) (x_{0}) \eta \left( E\left( \widehat{x}\right) ,E\left( x_{0}\right) \right) \leqq 0\text {.} \end{aligned}$$

(41)

Adding these above inequalities, we obtain the inequality

$$\begin{aligned} \bigg [ \sum _{i=1}^{k}\lambda _{0_{i}}\nabla f_{i}( E \left( x_{0}\right) ) +\sum _{j=1}^{m}\mu _{0_{j}}\nabla g_{j} \left( E\left( {x_{0}}\right) \right) \bigg ]\eta \left( E\left( \widehat{x} \right) ,E\left( x_{0}\right) \right) \leqq 0\text {} \end{aligned}$$

(42)

holds, contradicting (16). By assumption, $E:R^{n}\rightarrow R^{n}$ is a one-to-one and onto operator. Since $x_{0}$ is a Pareto solution of the problem ($\hbox {VP}_{E}$), by Lemma 3, $E\left( x_{0}\right)$ is an E-Pareto solution of the problem (VP). Thus, the proof of this theorem is completed. $\square$

Example 5

Consider the following nondifferentiable vector optimization problem

$$\begin{aligned} \begin{array}{c} f(x)=\left( f_{1}(x), f_{2}(x) \right) =\left( \root 3 \of {x}e^{\root 3 \of {x}},\; 2\root 3 \of {x}e^{\root 3 \of {x}}\right) \rightarrow V\text {-}\min \\ \text {s.t.} \; \; g(x)=1-e^{\root 3 \of {x}}\leqq 0. \end{array} \text { (VP1}) \end{aligned}$$

Note that $\Omega =\{ x \in R: \;1-e^{\root 3 \of {x}}\leqq 0 \}$ is the set of all feasible solutions of the problem (VP1). Further, note that the functions constituting problem (VP1) are nondifferentiable at $x_{0}=0$. Let $E\left( x\right) = x^{3}.$ For the considered vector optimization problem (VP1), we define its associated constrained E-vector optimization problem ($\hbox {VP}_{E}$1) as follows

$$\begin{aligned} \begin{array}{c} f(E(x))=\left( f_{1}(E(x)), f_{2}(E(x)) \right) =\left( {x}e^{{x}},\; 2{x}e^{{x}}\right) \rightarrow V\text {-}\min \\ \text {s.t.} \; \; g(E(x))=1-e^{{x}}\leqq 0. \end{array} \text {(VP}_{E}1\text {)} \end{aligned}$$

Note that $\Omega _{E} =\{ x \in R: \;1-e^{{x}}\leqq 0 \}$ is the set of all feasible solutions of the problem ($\hbox {VP}_{E}$1) and $\eta (E(x),E(x_{0}))={x}+x_{0},$ $\alpha (E(x),E(x_{0}))=e^{x}+x_{0}$ and $\beta (E(x),E(x_{0}))=2e^{x}-x_{0}.$ Then, by Definition 10, it can be shown that the objective functions f and the constraint function g are V-E-type I at $x_{0}$ on $\Omega _{E}$. Thus, all hypotheses of Theorem 2 are fulfilled and, therefore, we conclude that $x_{0}=0$ is a weak Pareto solution of the E-vector optimization problem ($\hbox {VP}_{E}$1) and, thus, $E(x_{0})$ is a weak E-Pareto solution of the considered multiobjective programming problem (VP1).

4 Mond–Weir E-duality

In this section, for the problem ($\hbox {VP}_E$), we define its vector E-dual problem ($\hbox {VD}_{E}$) in the sense of Mond–Weir [15].

$$\begin{aligned}{} & {} f(E(z))=(f_{1}(E(z)),...,f_{p}(E(z))) \rightarrow V-\max \end{aligned}$$

(43)

$$\begin{aligned}{} & {} \text {subject to}\; \; \; \; \sum _{i=1}^{p}{\lambda }_{i}\nabla f_{i}(E(z))+ \sum _{j=1}^{m}{\mu }_{j}\nabla g_{j}(E(z))=0, \end{aligned}$$

(44)

$$\;\;\;\;\;\;\;\;\;\;\;\; \begin{aligned}{} & {} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum _{j=1}^{m}{\mu }_{j} g_{j}(E(z))\geqq 0,\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;(\hbox {VD}_{E}) \end{aligned}$$

(45)

$$\begin{aligned}{} & {} {\lambda } \in {R}^{p},{\lambda }\ge 0\text { } \text {, }{\mu }\in {R}^{m},{\mu }\geqq 0\text {.} \end{aligned}$$

(46)

Further, let

$$\begin{aligned} \begin{array}{c} W _{E}=\bigg \{\left( {z},{\lambda },{\mu }\right) \in {R}^{n}\times {R}^{p}\times {R}^{m}: \sum _{i=1}^{p}{\lambda }_{i}\nabla f_{i}(E(z))+ \sum _{j=1}^{m}{\mu }_{j}\nabla g_{j}( E(z))=0, \\ \sum _{j=1}^{m}{\mu }_{j} g_{j}(E(z))\geqq 0, \text { }{\lambda }\ge 0 \text {, }{\mu }\geqq 0\bigg \} \end{array} \end{aligned}$$

be the set of all feasible solutions of the problem ($\hbox {VD}_{E}$). Let us denote, $Y_{E}=\{z\in R^{n}:\left( {z},{\lambda }, {\mu }\right) \in W_{E}\}.$

Theorem 6

(Weak duality between ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$)). Let $x\in \Omega _{E}$ and $\left( {z},{\lambda _{0} },{\mu _{0} }\right) \in W_{E}$ such that (f, g) is V-E-type I at z. Then

$$\begin{aligned} f( E(x))\nless f( E(z))\text {.} \end{aligned}$$

(47)

Proof

Let $x\in \Omega _{E}$ and $\left( {z},{\lambda _{0} },\mu _{0}\right) \in W_{E}.$ We proceed by contradiction. Suppose, contrary to the result, that the inequality

$$\begin{aligned} f( E(x))< f( E(z)) \end{aligned}$$

(48)

holds. By assumption, x and $\left( {z},{\lambda _{0} },{{\mu }_{0} }\right)$ are feasible solutions of ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$), respectively. Since (f, g) is V-E-type I with respect to $\eta$ at x and by Definition 10, the following inequalities

$$\begin{aligned}{} & {} f_{i}(E(x))-f_{i}(E(z))\geqq \alpha _{i}(E(x),E(z))\nabla f_{i}(E(z))\eta (E(x),E(z)), \; \; \; i\in I \end{aligned}$$

(49)

$$\begin{aligned}{} & {} -g_{j}(E(z))\geqq \beta _{j}(E(x),E(z))\nabla g_{j}(E(z))\eta (E(x),E(z)). \; \; \; j\in J \end{aligned}$$

(50)

hold, respectively. Combining (48, 49), we get

$$\begin{aligned} \alpha _{i}(E(x),E(z))\nabla f_{i}(E(z))\eta (E(x),E(z))<0, \; \; \; i\in I \end{aligned}$$

(51)

Since $\alpha _{i}(E(x),E(z))>0,\; i\in I,$ the above inequalities yield

$$\begin{aligned} \nabla f_{i}(E(z))\eta (E(x),E(z))<0, \; \; \; i\in I \end{aligned}$$

(52)

Multiplying (52) by the corresponding Lagrange multipliers and then adding both sides of the obtained inequalities, we get that the following inequality

$$\begin{aligned} \left[ \sum _{i=1}^{p}\lambda _{0_{i}}\nabla f_{i}( E \left( z\right) )\right] \eta (E(x),E(z))<0. \end{aligned}$$

(53)

Using the condition (17), together with $z\in \Omega _{E}$ and (50), we get

$$\begin{aligned} \beta _{j}(E(x),E(z))\nabla g_{j}(E(z))\eta (E(x),E(z))\leqq 0. \; \; \; j\in J \end{aligned}$$

(54)

Since $\beta _{j}(E(x),E(z))>0,\; j\in J,$ the above inequalities yield

$$\begin{aligned} \nabla g_{j}(E(z))\eta (E(x),E(z))\leqq 0. \; \; \; j\in J \end{aligned}$$

(55)

Multiplying (55) by the corresponding Lagrange multipliers and then adding both sides of the obtained inequalities, we get that the following inequality

$$\begin{aligned} \left[ \sum _{j=1}^{m}\mu _{0_{j}}\nabla g_{j}(E(z))\right] \eta (E(x),E(z))\leqq 0. \end{aligned}$$

(56)

Adding (53) and (56), we obtain that the inequality

$$\begin{aligned} \left[ \sum _{i=1}^{p}\lambda _{0_{i}}\nabla f_{i}( E \left( z\right) )+\sum _{j=1}^{m}\mu _{0_{j}}\nabla g_{j}(E(z))\right] \eta (E(x),E(z))<0 \end{aligned}$$

(57)

holds, contradicting (44). This means that the proof of weak duality theorem between the problems ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$) is completed. $\square$

Theorem 7

(Weak E-duality between (VP) and (VD$_{E}$)). Let $E\left( x\right) \in \Omega$ and $\left( {z},{\lambda _{0} },{\mu _{0} }\right) \in W_{E}.$ Further, assume that all hypotheses of Theorem 6 are fulfilled. Then, weak E-duality between (VP) and ($\hbox {VD}_{E}$) holds, that is,

$$\begin{aligned} f(E (x))\nless f(E(z)). \end{aligned}$$

Proof

Let $E\left( x\right) \in \Omega$ and $\left( {z},{\lambda _{0} },{\mu _{0}}\right) \in W_{E}.$ Then, by Lemma 1. it follows that $x \in \Omega _{E}$. Since all hypotheses of Theorem 6 are fulfilled, the weak E-duality theorem between the problems (VP) and ($\hbox {VD}_{E}$) follows directly from Theorem 6. $\square$

Theorem 8

(Weak duality between ($\hbox {VP}_{E}$) and (VD$_{E}$)). Let $x\in \Omega _{E}$ and $\left( {z},{\lambda },{\mu }\right) \in W_{E}$ such that $(\lambda f, \mu g)$ is pseudo-quasi-V-E-type I at z. Then

$$\begin{aligned} f( E(x))\nless f( E(z))\text {.} \end{aligned}$$

(58)

Proof

Let $x\in \Omega _{E}$ and $\left( {z},{\lambda },{\mu }\right) \in W_{E}.$ If $x = z$, then the weak duality trivially holds. Now, we prove the weak duality theorem when $x\ne z$. We proceed by contradiction. Suppose, contrary to the result, that the inequality

$$\begin{aligned} f( E(x))< f( E(z)) \end{aligned}$$

(59)

holds. By the feasibility of $\left( {z},{\lambda },{\mu }\right) \in W_{E}$, $\alpha _{i}\left( E\left( x\right) ,E\left( z\right) \right) >0$ and $\lambda >0$, the above inequality yields

$$\begin{aligned} \sum _{i=1}^{p}{\lambda }_{i}\alpha _{i}\left( E\left( x\right) ,E\left( z\right) \right) ( f_{i}(E(x))-f_{i}(E(z)))<0. \end{aligned}$$

(60)

By $\left( {z},{\lambda },{\mu }\right) \in W_{E}$, we have ${\mu }_{j} g_{j}(E(z))=0,$ for all $j\in J$ implies that

$$\begin{aligned} \sum _{j=1}^{m}{\mu }_{j}\alpha _{i}\left( E\left( x\right) ,E\left( z\right) \right) g_{j}(E(z))=0. \end{aligned}$$

(61)

Since $(\lambda f, {\mu } g)$ is pseudo-quasi-V-E-type I with respect to $\eta$ at z and together with inequalities (61), it follows that

$$\begin{aligned} \sum _{j=1}^{m}{\mu }_{j}\nabla g_{j}(E(z)) \eta \left( E\left( x\right) ,E\left( z\right) \right) \leqq 0. \end{aligned}$$

(62)

Using (62) and $\left( {z},{\lambda },{\mu }\right) \in W_{E}$ we have

$$\begin{aligned} \sum _{i=1}^{p}{\lambda }_{i}\nabla f_{i}( E(\left( z\right) ) \eta \left( E\left( x\right) ,E\left( z\right) \right) \geqq 0\text {, } \end{aligned}$$

(63)

Since $(\lambda f, {\mu } g)$ is pseudo-quasi-V-E-type I with respect to $\eta$ at z and by inequalities (63), the following inequalities

$$\begin{aligned} \sum _{i=1}^{p}{\lambda }_{i}\alpha _{i}\left( E\left( x\right) ,E\left( z\right) \right) ( f_{i}(E(x))-f_{i}(E(z)))\geqq 0. \end{aligned}$$

(64)

hold, which contradicts (60). This means that the proof of the Mond–Weir weak duality theorem between the problems ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$) is completed. $\square$

Theorem 9

(Weak E-duality between (VP) and (VD$_{E}$)). Let $E\left( x\right) \in \Omega$ and $\left( {z},{\lambda },{\mu }\right) \in W_{E}.$ Further, assume that all hypotheses of Theorem 8 are fulfilled. Then, Mond–Weir weak E-duality between (VP) and ($\hbox {VD}_{E}$) holds, that is,

$$\begin{aligned} f(E (x))\nless f(E(z)). \end{aligned}$$

Proof

Let $E\left( x\right) \in \Omega$ and $\left( {z},{\lambda },{\mu }\right) \in W_{E}.$ Then, by Lemma 1. it follows that $x \in \Omega _{E}$. Since all hypotheses of Theorem 8 are fulfilled, the Mond–Weir weak E-duality theorem between the problems (VP) and ($\hbox {VD}_{E}$) follows directly from Theorem 8. $\square$

Theorem 10

(Weak duality between ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$)). Let $x\in \Omega _{E}$ and $\left( {z},{\lambda },{\mu }\right) \in W_{E}$ such that $(\lambda f, \mu g)$ is strictly pseudo-quasi-V-E-type I at z. Then

$$\begin{aligned} f( E(x))\nless f( E(z))\text {.} \end{aligned}$$

(65)

Theorem 11

(Weak E-duality between (VP) and ($\hbox {VD}_{E}$)). Let $E\left( x\right) \in \Omega$ and $\left( {z},{\lambda },{\mu }\right) \in W_{E}.$ Further, assume that all hypotheses of Theorem 10 are fulfilled. Then, weak E-duality between (VP) and ($\hbox {VD}_{E}$) holds, that is,

$$\begin{aligned} f(E (x))\nless f(E(z)). \end{aligned}$$

Theorem 12

(Mond–Weir strong duality between ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$) and also strong E-duality between (VP) and ($\hbox {VD}_{E}$)). Let $x_{0}\in \Omega _{E}$ be a weak Pareto solution (Pareto solution) of the problem ($\hbox {VP}_{E}$) (and, thus, $E(x_{0})\in \Omega$ be a weak E-Pareto solution (E-Pareto solution) of the problem (VP)). Further, assume that the Abadie constraint qualification ($\hbox {ACQ}_{E}$) be satisfied at $x_{0}.$ Then there exist ${\lambda }\in R^{p}$, ${\mu }\in R^{m}$, ${\mu }\geqq 0$ such that $\left( x_{0},{\lambda },{\mu } \right) \in W_{E}.$ If all hypotheses of (Theorem 8) Theorem 10 are satisfied, then $\left( x_{0},\lambda ,{\mu } \right)$ is a (weak) efficient solution of a maximum type in the problem ($\hbox {VD}_{E}$).In other words, if $E(x_{0})\in \Omega$ is a (weak) E-Pareto solution of the problem (VP), then $\left( x_{0},\lambda ,{\mu } \right)$ is a (weak) efficient solution of a maximum type in the dual problem ($\hbox {VD}_{E}$).

Proof

Since $x_{0}\in \Omega _{E}$ is a weak Pareto solution of the problem ($\hbox {VP}_{E}$) and the Abadie constraint qualification ($\hbox {ACQ}_{E}$) is satisfied at $x_{0}$, there exist ${\lambda }\in R^{p}$, ${\mu }\in R^{m}$, ${\mu }\geqq 0$ such that

$$\begin{aligned} \sum _{i=1}^{p}{\lambda }_{i}\nabla \left( f_{i}\circ E\right) ( x_{0})+\sum _{j=1}^{m}{\mu }_{j}\nabla \left( g_{j}\circ E\right) (x_{0})=0\text {,} \\ {\mu }_{j}\left( g_{j}\circ E\right) (x_{0})=0\text {, } j\in J, \\ {\lambda }\ge 0\text {, }{\mu }\geqq 0. \end{aligned}$$

Thus, $\left( x_{0},\lambda ,{\mu } \right)$ is a feasible solution for ($\hbox {VD}_{E}$). If $\left( x_{0},\lambda ,{\mu } \right)$ is not a (weak) efficient solution for ($\hbox {VD}_{E}$), then there exists a feasible solution $\left( \widetilde{x},\widetilde{\lambda },\widetilde{\mu } \right)$ of ($\hbox {VD}_{E}$) such that $f(E(\widetilde{x}))<f(E(x_{0})),$ which contradicts the Theorem 8. Hence $\left( x_{0},\lambda ,{\mu } \right)$ is a (weak) efficient solution for ($\hbox {VD}_{E}$).

Moreover, we have, by Lemma 1, that $E\left( x_{0}\right) \in \Omega$. Since $x_{0}\in \Omega _{E}$ is a weak Pareto solution of the problem ($\hbox {VP}_{E}$), by Lemma 3, it follows that $E\left( x_{0}\right)$ is a weak E-Pareto solution in the problem (VP). Then, by the Mond–Weir strong duality between ($\hbox {VP}_{E}$) and ($\hbox {VD}_{E}$), we conclude that also the Mond–Weir strong E-duality holds between the problems (VP) and ($\hbox {VD}_{E}$). This means that if $E\left( x_{0}\right) \in \Omega$ is a weak E-Pareto solution of the problem (VP), there exist $\lambda \in R^{p}$, ${\mu }\in R^{m}$, ${\mu }\geqq 0$ such that $\left( x_{0},\lambda ,{\mu }\right)$ is a weakly efficient solution of a maximum type in the Mond–Weir dual problem ($\hbox {VD}_{E}$). $\square$

5 Concluding remarks

In this paper, new classes of (not necessarily) differentiable nonconvex multiobjective programming problems have been considered. Specifically, the concept of V-E-type I and/or generalized V-E-type I have been introduced for (not necessarily) differentiable multiobjective programming problems. Further, the sufficiency of the E-Karush-Kuhn-Tucker optimality conditions have been established for these (not necessarily) differentiable vector optimization problems under (generalized) V-E-type I hypotheses. Additionally, the vector Mond–Weir E-dual problems have been formulated for such E-differentiable multiobjective programming problems. Furthermore, various E-duality theorems between the considered E-differentiable vector optimization problem and its Mond–Weir vector dual problem have been proved under (generalized) V-E-type I hypotheses.

Nevertheless, there are still some interesting topics that warrant further research. It would be worthwhile to explore whether similar results can be proven for other classes of E-differentiable vector optimization problems. We intend to investigate these questions in future papers.