On vector variational E-inequalities and differentiable vector optimization problem

Abstract

In this paper, in order to characterize optimality of a class of nonconvex differentiable multiobjective programming problems, we introduce two new types of vector variational-like inequalities, namely, a weak variational E-inequality and a vector variational E-inequality. Namely, under (strictly) E-convexity, we prove relationships between the aforesaid vector variational-like inequalities and differentiable vector optimization problems. Further, under (strictly) pseudo-E-convexity, we are in position to identify vector critical E-points, weak E-Pareto (E-Pareto) solutions of differentiable vector optimization problems and solutions of these introduced vector variational-like inequalities.

1 Introduction

Optimization theory is a fundamental discipline within the field of mathematics that deals with the study of finding the best possible solution among a set of feasible options. It plays a central role in various fields, ranging from engineering and operations research to economics and data science. The overarching objective of optimization theory is to develop mathematical models, algorithms, and techniques to optimize or maximize desirable outcomes while considering constraints (see, for example, [1,2,3,4], and others).

During the last few years, different types of vector variational inequalities have received great interest from many authors in the areas of optimization theory. This is a consequence of the fact that vector variational inequalities are applied in mathematical programming, operations research, vector approximation problems, equilibrium problems, transportation, engineering, economics, control theory, and others. Since variational inequalities were first introduced by Giannessi [5] in a finite-dimensional space, the theory of vector variational inequalities has shown many various applications in vector optimization, for example, traffic equilibrium problems (see, for example, [6,7,8,9,10,11]). A useful and important generalization of the variational inequalities is called the variational-like inequalities, which have been studied extensively in recent years (see, for example, [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28], and others). Vector variational-like inequalities (for short, VVI) were first introduced by Ansari [29]. Since then, various kind of vector variational-like inequalities have been studied by many authors in different directions. Some recent results established in multiobjective programming have shown that optimality conditions proved for some classes of vector optimization problems can be characterized by vector variational inequalities (see, for example, [23, 30,31,32,33,34,35,36,37,38], and others).

In recent years, Li and You [39] proved the equivalence between solutions of two types of vector variational-like inequalities and a multiobjective programming problem and also the existence results for the discussed variational-like inequalities by using the KKM-Fan theorem. Studying the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving $(G,\alpha )$-invex functions, Jayswal and Choudhury [15] established the equivalence among solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and a weak efficient solution of a vector optimization problem under $(G,\alpha )$-invexity hypotheses. In [40], Irfan et al. studied new types of generalized nonsmooth exponential type vector variational-like inequality problems involving Mordukhovich limiting subdifferential operator. Based also on the Fan-KKM theorem, they established some relationships between generalized nonsmooth exponential type vector variational-like inequality problems and vector optimization problems under some invexity assumptions. Using Michel-Penot subdifferential, Laha and Singh [41] established various results related to weak Pareto solutions of the considered multiobjective programming problem with V-r-invex functions and its corresponding weak versions of the vector variational-like inequalities. Upadhyay and Mishra [42] studied generalized Minty and Stampacchia vector variational-like inequalities with their weaker forms and, under strong invexity hypothesis, they established the equivalence among solutions of generalized vector variational-like inequalities and efficient minimizers of order s in the nonsmooth multiobjective programming problem. Very recently, Ram and Bhardwa [43] established not only the relationships between vector variational-like inequality problems and vector optimization problems under $(G,\alpha )$-univexity hypotheses, but also the equivalence among vector critical points, weak efficient points of vector optimization problems and solutions of weak vector variational-like inequality problems.

In this paper, we introduce two new vector variational-like inequalities to characterize optimality for a new class of nonconvex differentiable vector optimization problems. Namely, based on the effect of a given operator $E:R^{n}\rightarrow R^{n},$ we define a vector variational E-inequality and a weak vector variational E-inequality which are an extension of the well-known vector variational inequalities existing in the literature. We prove the existence of the solvability of the weak vector variational E-inequality under appropriate compactness hypotheses. Further, we investigate relationships between the sets of solutions in both aforesaid vector variational E-inequalities and in some nonconvex differentiable vector optimization problems. Therefore, the results established in this paper can be helpful for studying a characterization of the sets of (weak) E-Pareto solutions in a new class of (not necessarily convex) differentiable vector optimization problems. Namely, it is possible to find aforesaid (weak) E-Pareto solutions in differentiable multicriteria optimization problems with E-convex functions by the help of solutions sets in the (weak) vector variational E-inequalities introduced in the paper. Thus, we characterize the aforesaid (weakly) efficiency notions for such a class of nonconvex vector optimization problems. We illustrate this result by an example of such a nonconvex multiobjective programming problem. Moreover, we show that, under invexity hypotheses (with respect to $\eta$), it is not possible to characterize (weakly) efficiency for the investigated nonconvex multiobjective programming problem by the help of vector variational-like inequalities formulated with the aforesaid vector-valued functions $\eta$. This is a consequence of the fact that the functions involved in such multicriteria optimization problems are not invex with respect to any vector-valued function $\eta$. Hence, vector variational-like inequalities introduced in the paper allow to characterize a new class of nonconvex smooth multiobjective programming problems, that is, with E-convex functions. The results, which show the relations among solutions sets in the introduced vector variational-like inequalities and differentiable vector optimization problems, are illustrated by the relationship diagram. We also introduce the definition of a vector critical E-point in the considered differentiable vector optimization problem. Then, under (strictly) pseudo-E-convexity, we prove the equivalence of vector critical E-point, (weak) E-Pareto solutions and solutions in (weak) vector variational E-inequalities. Hence, we present a characterization of optimality by the aforesaid vector variational inequalities for yet another class of nonconvex multiobjective programming problems, that is, with (strictly) pseudo-E-convex functions. To the best of our knowledge, there is no a characterization of optimality in the literature for a vector variational E-inequality E-inequality and a weak vector variational E-inequality.

2 Preliminaries

Throughout this paper, the following conventions vectors $x=\left( x_{1},x_{2},\ldots ,x_{n}\right) ^{T}$ and $y=\left( y_{1},y_{2},\ldots ,y_{n}\right) ^{T}$ in $R^{n}$ will be followed:

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

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

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

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

• (v) $x\ngtr y$ is the negation of $x > y.$

Further, denote by $R_{+}^{n}=\left\{ y\in R^{n}:y\geqq 0\right\}$ and $R_{++}^{n}=\left\{ y\in R^{n}:y>0\right\}$ the nonnegative orthant and interior of nonnegative orthant of $R^{n}$, respectively.

Classes of nonconvex sets and nonconvex functions, called E-convex sets and E-convex functions, respectively, were introduced and studied by Youness [44]. This kind of generalized convexity is based on the effect of an operator $E:R^{n}\rightarrow R^{n}$ on the sets and the domains of functions. Now, for convenience, we recall the aforesaid definitions.

Definition 1

[44] A set $S\subseteq R^{n}$ is said to be an E-convex set (with respect to an operator $E:R^{n}\rightarrow R^{n}$) if and only if, the following relation

$$\begin{aligned} E\left( u\right) +\lambda \left( E\left( x\right) -E\left( u\right) \right) \in S \end{aligned}$$

(1)

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

Note that every convex set is E-convex (if E is the identity map), but the converse is not true. If $M\subseteq R^{n}$ is an E-convex set, then $E\left( S\right) \subseteq S$. If $E\left( S\right)$ is a convex set and $E\left( S\right) \subseteq S$, then S is E-convex (see, Youness [45]).

Let S be a nonempty E-convex subset of $R^{n}$.

Definition 2

[44] A real-valued function $f:S\rightarrow R$ is said to be E-convex (with respect to an operator $E:R^{n}\rightarrow R^{n}$) on S if and only if, the following inequality

$$\begin{aligned} f\left( \lambda E\left( x\right) +\left( 1-\lambda \right) E\left( u\right) \right) \leqq \lambda f\left( E\left( x\right) \right) +\left( 1-\lambda \right) f\left( E\left( u\right) \right) \end{aligned}$$

(2)

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

It is clear that every convex function is E-convex (if E is the identity map).

Definition 3

A real-valued function $f:S\rightarrow R$ is said to be strictly E-convex (with respect to an operator $E:R^{n}\rightarrow R^{n}$) on S if and only if, the following inequality

$$\begin{aligned} f\left( \lambda E\left( x\right) +\left( 1-\lambda \right) E\left( u\right) \right) <\lambda f\left( E\left( x\right) \right) +\left( 1-\lambda \right) f\left( E\left( u\right) \right) \end{aligned}$$

(3)

holds for all $x,u\in S$, $E(x)\ne E(u)$, and any $\lambda \in \left( 0,1\right)$.

Throughout this paper, $E:R^{n}\rightarrow R^{n}$ be a one-to-one and onto operator.

Proposition 1

Let $f:R^{n}\rightarrow R$ be a differentiable function and S be a nonempty E-convex subset of $R^{n}.$ If f is an E-convex function at $u\in S$ on S,  then the following inequality

$$\begin{aligned} f\left( E\left( x\right) \right) -f\left( E\left( u\right) \right) \geqq \nabla f\left( E\left( u\right) \right) \left( E\left( x\right) -E\left( u\right) \right) \end{aligned}$$

(4)

holds for all $x\in S$.

Proof

Let $E:R^{n}\rightarrow R^{n}$ be a given operator. Then, assume that S is an E-convex subset of $R^{n}$, $f:S\rightarrow R$ is an E-convex function on S and $u\in S$. By Definition 2, it follows that the inequality

$$\begin{aligned} f\left( \lambda E\left( x\right) +\left( 1-\lambda \right) E\left( u\right) \right) \leqq \lambda f\left( E\left( x\right) \right) +\left( 1-\lambda \right) f\left( E\left( u\right) \right) \end{aligned}$$

holds for all $x,u\in S$ and any $\lambda \in \left[ 0,1\right]$. Thus, the above inequality yields

$$\begin{aligned} f\left( E\left( x\right) \right) -f\left( E\left( u\right) \right) \geqq \frac{f\left( E\left( u\right) +\lambda \left( E\left( x\right) -E\left( u\right) \right) \right) -f\left( E\left( u\right) \right) }{\lambda }. \end{aligned}$$

Letting $\lambda \rightarrow 0$, we obtain the inequality (4). $\square$

Now, we introduce a definition of E-monotonicity.

Definition 4

Let $E:R^{n}\rightarrow R^{n}$ and S be an E-convex subset of $R^{n}$. Then, a differentiable E-convex function $f:R^{n}\rightarrow R^{p}$ is said to be (strictly) E-monotone on S if and only if, the inequality

$$\begin{aligned} (\nabla f\left( E\left( x\right) \right) -\nabla f\left( E\left( u\right) \right) ) \left( E\left( x\right) -E\left( u\right) \right) \geqq 0\quad \left( >\right) \end{aligned}$$

(5)

holds for all $x,\ u\in S$.

Now, we present the necessary condition for differentiable (strictly) E-convexity.

Proposition 2

Let S be a subset of $R^{n}$, $E:R^{n}\rightarrow R^{n}$ be a given operator and $f:R^{n}\rightarrow R^{p}$ be a differentiable function on S. If f is an E-convex function on S,  then the inequality

$$\begin{aligned} (\nabla f\left( E\left( x\right) \right) -\nabla f\left( E\left( u\right) \right) ) \left( E\left( x\right) -E\left( u\right) \right) \geqq 0 \end{aligned}$$

(6)

holds for all $x,\ u\in M.$

Proposition 3

Let S be a subset of $R^{n}$, $E:R^{n}\rightarrow R^{n}$ be a given operator and $f:R^{n}\rightarrow R^{p}$ be a differentiable function on S. If f is a strictly E-convex function on S,  then the inequality

$$\begin{aligned} (\nabla f\left( E\left( x\right) \right) -\nabla f\left( E\left( u\right) \right) ) \left( E\left( x\right) -E\left( u\right) \right) > 0 \end{aligned}$$

(7)

holds for all $x,\ u\in S,\; (E(x)\ne E(u))$.

Now, we introduce the definition of a differentiable vector-valued E-convex function.

Definition 5

Let $E:R^{n}\rightarrow R^{n}$ be a given operator. Further, let S be a nonempty E-convex subset of $R^{n}$ and $f:R^{n}\rightarrow R^{p}$ be a differentiable function. Then f is a vector-valued E-convex (strictly E-convex) function at ${\overline{x}}\in S$ on S,  if the following inequalities

$$\begin{aligned} f_{i}\left( E\left( x\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) \geqq \nabla f_{i}\left( E\left( {\overline{x}}\right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) ,\quad \left( >\right) , i=1,\ldots ,p \end{aligned}$$

(8)

hold for all $x\in S$. If inequalities (8) are satisfied for any ${\overline{x}}\in S$, then f is said to be an E-convex vector-valued function on S.

Now, we give the definition of differentiable generalized E-convex functions.

Definition 6

Let $E:R^{n}\rightarrow R^{n}$, S be a nonempty E-convex subset of $R^{n}$ and $f:R^{n}\rightarrow R^{p}$ be a differentiable function. f is said to be a pseudo-E-convex function at ${\overline{x}}\in S$ on S,  if the following relation

$$\begin{aligned} f_{i}\left( E\left( x\right) \right)<f_{i}\left( E\left( {\overline{x}} \right) \right) \Longrightarrow \nabla f_{i}\left( E\left( {\overline{x}} \right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) <0, ~i=1,\ldots ,p \end{aligned}$$

(9)

holds for all $x\in S$. If (9) is satisfied for each ${\overline{x}}\in S$, then f is said to be a pseudo-E-convex function on S.

Definition 7

Let $E:R^{n}\rightarrow R^{n}$, S be a nonempty E-convex subset of $R^{n}$ and $f:R^{n}\rightarrow R^{p}$ be a differentiable function. f is said to be a strictly pseudo-E-convex function at ${\overline{x}}\in S$ on S,  if the following relation

$$\begin{aligned} f_{i}\left( E\left( x\right) \right) \leqq f_{i}\left( E\left( {\overline{x}} \right) \right) \Longrightarrow \nabla f_{i}\left( E\left( {\overline{x}} \right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) <0,~i=1,\ldots ,p \end{aligned}$$

(10)

holds for all $x\in S$. If (10) is satisfied for each ${\overline{x}} \in S$, then f is said to be a strictly pseudo-E-convex function on S.

Every differentiable E-convex function is differentiable pseudo-E-convex, but the converse is not true. A strictly pseudo-E-convex function is pseudo-E-convex, but the converse is not true.

Now, we present an example of such a differentiable pseudo-E-convex function which is not E-convex.

Example 1

Let $f:R \rightarrow R^{2}$ be defined by $f(x)=((x-1)^{3}+x,\; x^3+x-1)$ and $E:R\rightarrow R$ be an operator defined by $E(x)=x+1.$ Further, assume that $\left( f_{1}\circ E\right) \left( x\right) <\left( f_{1}\circ E\right) \left( u\right)$. Thus, we have

$$\begin{aligned} \left( f_{1}\circ E\right) \left( x\right) =x^{3}+x<u^{3}+u=\left( f_{1}\circ E\right) \left( u\right) . \end{aligned}$$

This implies that $x<u$ for all $x,u\in R$. Moreover, we have

$$\begin{aligned} \nabla \left( f_{1}\circ E\right) \left( u\right) \left( E\left( x\right) -E\left( u\right) \right) =(3u^{2}+1)(x-u)<0. \end{aligned}$$

Therefore, by Definition 6, $f_{1}$ is differentiable pseudo-E-convex on R. But, if we set $x=1$, $u=-1$, then we have

$$\begin{aligned} f_{1}(E(x))-f_{1}(E(u))=4<8 =\nabla f_{1}(E(u))(E(x)-E(u)). \end{aligned}$$

Hence, by Definition 5, it follows that $f_{1}$ is not E-convex.

Moreover, assume that $\left( f_{2}\circ E\right) \left( x\right) <\left( f_{2}\circ E\right) \left( u\right)$. Thus, we have

$$\begin{aligned} \left( f_{2}\circ E\right) \left( x\right) =(x+1)^3+x<(u+1)^3+u=\left( f_{2}\circ E\right) \left( u\right) . \end{aligned}$$

This implies that $x<u$ for all $x,u\in R$. Moreover, we have

$$\begin{aligned} \nabla \left( f_{2}\circ E\right) \left( u\right) \left( E\left( x\right) -E\left( u\right) \right) =(3(u+1)^{2}+1)(x-u)<0. \end{aligned}$$

Therefore, by Definition 6, $f_{2}$ is a differentiable pseudo-E-convex function on R. But, if we set $x=0$, $u=-2$, then we have

$$\begin{aligned} f_{2}(E(x))-f_{2}(E(u))=4<12 =\nabla f_{2}(E(u))(E(x)-E(u)). \end{aligned}$$

Hence, by Definition 5, it follows that $f_{2}$ is not E-convex. Then $f_{2}$ is a differentiable pseudo E-convex function, but it is not E-convex.

Definition 8

Let $E:R^{n}\rightarrow R^{n}$ be a given operator. It is said that ${\overline{x}}\in S$ is an E-minimizer of $f:S\rightarrow R$ if the inequality

$$\begin{aligned} f\left( E\left( {\overline{x}}\right) \right) \leqq f\left( E\left( x\right) \right) \end{aligned}$$

holds for all $x\in S$.

Proposition 4

Let $f:S\rightarrow R$ be a differentiable E-convex function at a given point ${\overline{x}}\in S.$ If $\nabla f\left( E\left( {\overline{x}}\right) \right) =0,$ then ${\overline{x}}\in S$ is an E-minimizer of f.

Proof

This result follows directly from Proposition 1 and Definition 8. $\square$

We now show that an E-minimizer of a differentiable E-convex function on an E-convex set S can be characterized by the generalized inequality.

Lemma 1

Let $f:S\rightarrow R$ be a differentiable E-convex function at ${\overline{x}}\in S$ , where S is an E-convex subset of $R^{n}$. Then, ${\overline{x}}\in S$ is an E-minimizer of f on S if and only if, the following inequality

$$\begin{aligned} \left\langle \nabla f\left( E\left( {\overline{x}}\right) \right) ,E\left( x\right) -E\left( {\overline{x}}\right) \right\rangle \geqq 0 \end{aligned}$$

(11)

holds for all $x\in S$.

Proof

Let ${\overline{x}}\in S$ be an E-minimizer of f on S. Then, by Definition 8, the inequality

$$\begin{aligned} f(E\left( {\overline{x}}\right) )\leqq f( E\left( x\right) ). \end{aligned}$$

(12)

holds for all $x\in S$. Since S is an E-convex set, by Definition 1, the following relation

$$\begin{aligned} E\left( {\overline{x}}\right) +\lambda \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) \in S \end{aligned}$$

(13)

holds for all $x,{\overline{x}}\in S$ and any $\lambda \in \left[ 0,1\right] ,$ therefore, we have

$$\begin{aligned} f(E\left( {\overline{x}}\right) )\leqq f(E\left( {\overline{x}}\right) +\lambda \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) ). \end{aligned}$$

(14)

By dividing the above inequality by $\lambda ,$ we have

$$\begin{aligned} \frac{f(E\left( {\overline{x}}\right) +\lambda \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) )-f(E\left( {\overline{x}}\right) )}{\lambda } \geqq 0. \end{aligned}$$

(15)

Letting $\lambda \rightarrow 0$, we obtain the inequality (11).

Conversely, since $f:S\rightarrow R$ is an E-convex function on S and ${\overline{x}}\in S$. By Definition 2, it follows that the inequality

$$\begin{aligned} f\left( \lambda E\left( x\right) +\left( 1-\lambda \right) E\left( {\overline{x}}\right) \right) \leqq \lambda f\left( E\left( x\right) \right) +\left( 1-\lambda \right) f\left( E\left( {\overline{x}}\right) \right) \end{aligned}$$

holds for all $x,{\overline{x}}\in S$ and any $\lambda \in \left[ 0,1\right]$. Thus, the above inequality yields

$$\begin{aligned} f\left( E\left( x\right) \right) -f\left( E\left( {\overline{x}}\right) \right) \geqq \frac{f\left( E\left( {\overline{x}}\right) +\lambda \left( E\left( x\right) -E\left( u\right) \right) \right) -f\left( E\left( {\overline{x}}\right) \right) }{\lambda }. \end{aligned}$$

Letting $\lambda \rightarrow 0$, we obtain the inequality

$$\begin{aligned} f(E(x))-f(E\left( {\overline{x}}\right) ) \geqq \left\langle \nabla f\left( E\left( {\overline{x}}\right) \right) ,E\left( x\right) -E\left( {\overline{x}}\right) \right\rangle . \end{aligned}$$

(16)

Then, by (11), the inequality

$$\begin{aligned} f(E\left( {\overline{x}}\right) )\leqq f( E\left( x\right) ) \end{aligned}$$

(17)

holds for all $x \in S.$ Hence, by Proposition 4, we conclude that ${\overline{x}}\in S$ is an E-minimizer of f. $\square$

Now, we give the KKM-Fan theorem (Fan [46]) (see, for example, [30, 47, 48]) which is a version of the Knaster-Kuratowski-Mazurkiewicz theorem (Knaster et al. [49]) (see also Chen and Yang [31]), that we use in proving the solvability of new vector variational-like inequalities introduced in the next section.

Theorem 1

Let Q be a subset of the topological vector space Y. Let assume that, for each $y\in Q$ , a closed set $F\left( y\right)$ in Y be given such that $F\left( y\right)$ is compact for at least one $y\in Q$. If the convex hull of every finite set $\left\{ y_{1},\ldots ,y_{r}\right\}$ of Y is contained in $\bigcup _{i=1}^{r}F\left( y_{i}\right)$ , then $\bigcup_{y\in Y}F\left( y\right) \ne 0$.

3 New vector variational-like inequalities

In this section, we introduce two new vector variational-like inequalities, that is, a weak vector variational E-inequality problem and a vector variational E-inequality problem. In order to do this, let $E:R^{n}\rightarrow R^{n}$ be a given operator, S be an E-convex subset of $R^{n}$ and $F:S\rightarrow R^{pn}$ be a matrix-valued function.

Now, we formulate the aforesaid vector variational E-inequality problems.

A vector variational E-inequality problem (E-VVIP) is to find a point ${\overline{x}} \in S$ such that there exists no other $x \in S$ satisfying

$$\begin{aligned} \left\langle F\left( E\left( {\overline{x}}\right) \right) ,E\left( x\right) -E\left( {\overline{x}}\right) \right\rangle \le 0. \end{aligned}$$

(18)

A weak vector variational E-inequality problem (E-WVVIP) is to find a point ${\overline{x}}\in S$ such that there exists no other $x\in S$ satisfying

$$\begin{aligned} \left\langle F\left( E\left( {\overline{x}}\right) \right) ,E\left( x\right) -E\left( {\overline{x}}\right) \right\rangle <0. \end{aligned}$$

(19)

The weak vector variational E-inequality problem (E-WVVIP) and the vector variational E-inequality problem (E-VVIP) introduced above can also be formulated as follows:

The vector variational E-inequality problem (E-VVIP) is to find a point ${\overline{x}} \in S$ such that

$$\begin{aligned} \left\langle F\left( E\left( {\overline{x}}\right) \right) ,E\left( x\right) -E\left( {\overline{x}}\right) \right\rangle \notin -R_{+}^{p}\backslash \left\{ 0\right\} , \; \forall x\in S. \end{aligned}$$

(20)

The weak vector variational E-inequality problem (E-WVVIP) is to find a point ${\overline{x}}\in S$ such that

$$\begin{aligned} \left\langle F\left( E\left( {\overline{x}}\right) \right) ,E\left( x\right) -E\left( {\overline{x}}\right) \right\rangle \notin -intR_{+}^{p}, \; \forall x\in S. \end{aligned}$$

(21)

Remark 1

It is obvious that (E-VVIP) implies (E-WVVIP).

Remark 2

Note that the vector variational inequality considered in the literature (see, for example, [11, 34, 50]) is a particular case of the above introduced vector variational E-inequality problem (E-VVIP), taking $E\left( x\right) \equiv x$.

Now, we consider the existence of a solution of the weak vector variational E-inequality problem (E-WVVIP).

Theorem 2

Let $E:R^{n}\rightarrow R^{n}$ be a given continuous operator, S be a compact E-convex subset of $R^{n}$. Further, we assume that $F:S\rightarrow R^{pn}$ is a continuous function. If the set $\Omega \left( u\right) =\left\{ E\left( y\right) \in S:\left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle <0\right\}$ is E-convex for each fixed $u\in S$ , then the weak vector variational E-inequality problem (E-WVVIP) is solvable. Moreover, the solution set of (E-WVVIP) is compact.

Proof

For $E\left( y\right) \in S$, we define the following set $\Gamma \left( E\left( y\right) \right) =\left\{ E\left( u\right) \in S:\right.$ $\left. \left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle \nless 0\right\}$. Let $\left\{ E\left( u_{1}\right) ,\ldots ,E\left( u_{r}\right) \right\} \subset S$, $\lambda _{i}\in R$, $i=1,\ldots ,r$, with $\lambda _{i}\geqq 0$ and, moreover, $\sum _{i=1}^{r}\lambda _{i}=1$. We show that the convex hull of every finite subset $\left\{ E\left( u_{1}\right) ,\ldots ,E\left( u_{r}\right) \right\}$ of S is contained in the corresponding union $\bigcup_{i=1}^{r}\Gamma \left( E\left( u_{i}\right) \right)$, that is, $\sum _{i=1}^{r}\lambda _{i}E\left( u_{i}\right) \in \bigcup_{i=1}^{r}\Gamma \left( E\left( u_{i}\right) \right)$. We proceed by contradiction. Suppose, contrary to the result, that $E\left( u\right) =\sum _{i=1}^{r}\lambda _{i}E\left( u_{i}\right) \notin \bigcup_{i=1}^{r}\Gamma \left( E\left( u_{i}\right) \right)$. This means that

$$\begin{aligned} \left\langle F\left( E\left( u\right) \right) ,E\left( u_{i}\right) -E\left( u\right) \right\rangle <0,\quad \forall i=1,\ldots ,r. \end{aligned}$$

(22)

By assumption, the set $\Omega \left( y\right) =\left\{ E\left( y\right) \in S:\left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle <0\right\}$ is E-convex for each $y\in S$. By the E-convexity of $\Omega \left( y\right)$ and (22), we have that the following relations

$$\begin{aligned} 0=\left\langle F\left( E\left( u\right) \right) ,E\left( u\right) -E\left( u\right) \right\rangle \leqq \sum _{i=1}^{r}\lambda _{i}\left\langle F\left( E\left( u\right) \right) ,E\left( u_{i}\right) -E\left( u\right) \right\rangle <0 \end{aligned}$$

hold, which is impossible. Thus, we have that

$$\begin{aligned} E\left( u\right) =\sum _{i=1}^{r}\lambda _{i}E\left( u_{i}\right) \in \bigcup \limits _{i=1}^{r}\Gamma \left( E\left( u_{i}\right) \right) . \end{aligned}$$

Now, we claim that, for each $E\left( y\right) \in S$, the set $\Gamma \left( E\left( y\right) \right)$ is closed. Indeed, let $E\left( y\right) \in S$ and let a sequence $\left\{ E\left( u_{k}\right) \right\} \subset \Gamma \left( E\left( y\right) \right)$ satisfy

$$\begin{aligned} \left\| E\left( u_{k}\right) \rightarrow E\left( u\right) \right\| \rightarrow 0 \text { as }k\rightarrow \infty . \end{aligned}$$

(23)

Since F is a continuous function, one has that

$$\begin{aligned} F\left( E\left( u_{k}\right) \right) \rightarrow F\left( E\left( u\right) \right) \text { as }k\rightarrow \infty . \end{aligned}$$

(24)

Thus,

$$\begin{aligned}{} & {} \left\| \left\langle F\left( E\left( u_{k}\right) \right) ,E\left( y\right) -E\left( u_{k}\right) \right\rangle -\left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle \right\| \\{} & {} \quad \leqq \left\| \left\langle F\left( E\left( u_{k}\right) \right) ,E\left( y\right) -E\left( u_{k}\right) \right\rangle -\left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u_{k}\right) \right\rangle \right\| \nonumber \\{} & {} \qquad +\left\| \left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u_{k}\right) \right\rangle -\left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle \right\| \nonumber \\{} & {} \quad \leqq \left\| F\left( E\left( u_{k}\right) \right) -F\left( E\left( u\right) \right) \right\| \left\| E\left( y\right) -E\left( u_{k}\right) \right\| \nonumber \\{} & {} \qquad +\left\| F\left( E\left( u\right) \right) \right\| \left\| E\left( u_{k}\right) -E\left( u\right) \right\| \rightarrow 0\text { as } k\rightarrow \infty, \nonumber \end{aligned}$$

(25)

where the above convergence to 0 follows from (23) and (24). Since $\left\langle F\left( E\left( u_{k}\right) \right) ,\right.$ $\left. E\left( y\right) -E\left( u_{k}\right) \right\rangle \nless 0$ for each k or, in other words, $\left\langle F\left( E\left( u_{k}\right) \right) ,E\left( y\right) -E\left( u_{k}\right) \right\rangle \in R^{p}\backslash \left( -intR_{+}^{p}\right)$ and the set $R^{p}\backslash \left( -intR_{+}^{p}\right)$ is closed, therefore, (24) implies

$$\begin{aligned} \left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle \in R^{p}\backslash \left( -intR_{+}^{p}\right) , \end{aligned}$$

or, in other words,

$$\begin{aligned} \left\langle F\left( E\left( u\right) \right) ,E\left( y\right) -E\left( u\right) \right\rangle \nless 0. \end{aligned}$$

Thus, we have proved that the set $\Gamma \left( E\left( y\right) \right)$ is closed. Further, $\Gamma \left( E\left( y\right) \right)$ is a nonempty set due to the fact that $E\left( y\right) \in \Gamma \left( E\left( y\right) \right)$. Hence, since the set $\Gamma \left( E\left( y\right) \right)$ is closed, it is compact for each $E\left( y\right) \in S$. By using the Knaster-Kuratowski-Mazurkiewicz theorem (Knaster et al. [49]) (see also Theorem 1), we get that

$$\begin{aligned} \bigcup \limits _{E\left( y\right) \in S}\Gamma \left( E\left( y\right) \right) \ne \varnothing . \end{aligned}$$

(26)

By (26), we conclude that there exists at least one point ${\overline{u}} \in \bigcup \limits _{E\left( y\right) \in S}\Gamma \left( E\left( y\right) \right)$ such that

$$\begin{aligned} \left\langle F\left( E\left( {\overline{u}}\right) \right) ,E\left( y\right) -E\left( {\overline{u}}\right) \right\rangle <0,\quad \forall E\left( y\right) \in S. \end{aligned}$$

Thus, we have proved that the weak vector variational E-inequality problem (E-WVVIP) is solvable.

Now, we show that its solution set is compact. The set $\Gamma \left( E\left( y\right) \right)$ is closed for each $E\left( y\right) \in S$. Since the intersection of closed sets is closed, therefore, the solution set of (E-WVVIP), that is, the set $\bigcup_{E\left( y\right) \in S}\Gamma \left( E\left( y\right) \right)$ is also closed. Thus, it is a compact set. This completes the proof of this theorem. $\square$

4 Differentiable vector optimization problems and (weak) vector variational E-inequalities

In this section, we consider the multicriteria optimization problem defined by

$$\begin{aligned} \begin{array}{c} V\text {-min } \ f\left( x\right) \\ \text {subject to }x\in S, \end{array} \quad \text {(MOP)} \end{aligned}$$

where S is a nonempty subset of $R^{n}$ and $f:R^{n}\rightarrow R^{p}$ is a vector-valued function.

Let $E:R^{n}\rightarrow R^{n}$ be a given operator. Further, we assume that S is an E-convex subset of $R^{n}$ and $f:R^{n}\rightarrow R^{p}$ is a differentiable function on $R^{n}$. For such multicriterion optimization problems, the concept of a weak E-Pareto solution and the concept of an E-Pareto solution (see Antczak and Abdulaleem [51]) are defined as follows:

Definition 9

A feasible point ${\overline{x}}$ is said to be an E-Pareto solution (an E-efficient solution) for (MOP) if and only if there exists no other feasible point x such that

$$\begin{aligned} f(E\left( x\right) )\le f(E\left( {\overline{x}}\right) ). \end{aligned}$$

Definition 10

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

$$\begin{aligned} f(E\left( x\right) )<f(E\left( {\overline{x}}\right) ). \end{aligned}$$

We denote the set of all weak E-Pareto points in (MOP) by WE-$P\left( f,S\right)$ and the set of all E-Pareto points in (MOP) by E-$P\left( f,S\right)$.

Then, we now define two multicriteria optimization problems as follows:

The weak vector optimization problem (WVOP) is to find the points of the set WE-$P\left( f,S\right)$ in

$$\begin{aligned} \begin{array}{c} WV\text {-min } \ f\left( x\right) \\ \text {subject to }x\in S. \end{array} \quad \text {(WVOP)} \end{aligned}$$

The vector optimization problem (VOP) is to find the points of the set E-$P\left( f,S\right)$ in

$$\begin{aligned} \begin{array}{c} V\text {-min } \ f\left( x\right) \\ \text {subject to }x\in S. \end{array} \quad \text {(VOP)} \end{aligned}$$

Clearly, that ${\overline{x}}\in E$-$P\left( f,S\right)$ implies that ${\overline{x}}\in WE$-$P\left( f,S\right)$. However, if we assume that the objective function f is strictly E-convex, then it can be verified that ${\overline{x}}\in WE$-$P\left( f,S\right)$ is equivalent to ${\overline{x}}\in E$-$P\left( f,S\right)$.

Now, we prove the connection between an E-Pareto solution in the vector optimization problem (VOP) and an E-solution of the vector variational E-inequality problem (E-VVIP).

Theorem 3

Let $f: R^{n}\rightarrow R^{p}$ be a differentiable function and S be a nonempty E-convex subset of $R^{n}$. If $F=\nabla f,$ ${\overline{x}}$ solves the vector variational E-inequality problem (E-VVIP) and f is an E-convex function at ${\overline{x}}$ on S , then ${\overline{x}}$ is an E-Pareto solution in the vector optimization problem (VOP).

Proof

We proceed by contradiction. Suppose, contrary to the result that $\overline{ x}\in S$ is not an E-Pareto solution in the vector optimization problem (VOP). Then, by Definition 9, there exists other ${\widetilde{x}}\in S$ such that

$$\begin{aligned} f\left( E\left( {\widetilde{x}}\right) \right) \le f\left( E\left( \overline{x }\right) \right) . \end{aligned}$$

(27)

By assumption, f is a differentiable E-convex function at ${\overline{x}}$ on S. Hence, by Definition 5, the inequalities

$$\begin{aligned} f_{i}\left( E\left( x\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) \geqq \nabla f_{i}\left( E\left( {\overline{x}}\right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) ,~ i=1,\ldots ,p \end{aligned}$$

(28)

hold for all $x\in S$. Therefore, they are also satisfied for $x=\widetilde{x }\in S$. Combining (27) and (28), we get that the inequality

$$\begin{aligned} \nabla f\left( E\left( {\overline{x}}\right) \right) \left( E\left( \widetilde{ x}\right) -E\left( {\overline{x}}\right) \right) \le 0 \end{aligned}$$

holds. Since $F=\nabla f$, the inequality above implies that the inequality

$$\begin{aligned} F\left( E\left( {\overline{x}}\right) \right) \left( E\left( {\widetilde{x}} \right) -E\left( {\overline{x}}\right) \right) \le 0 \end{aligned}$$

holds. This is a contradiction to the assumption that ${\overline{x}}$ solves the vector variational E-inequality problem (E-VVIP). $\square$

Now, we prove the converse result to that formulated in Theorem 3.

Theorem 4

Let $f:R^{n}\rightarrow R^{p}$ be a differentiable function and S be a nonempty E-convex subset of $R^n$. If $F=\nabla f,$ ${\overline{x}}$ solves vector optimization problem (VOP) (that is, ${\overline{x}}$ is an E-Pareto solution in (VOP)) and $-f$ is an E-convex function at ${\overline{x}}$ on S , then ${\overline{x}}$ solves the vector variational E-inequality problem (E-VVIP).

Proof

Assume that ${\overline{x}}$ solves the vector optimization problem (VOP), that is, ${\overline{x}}$ is an E-Pareto solution in (VOP). We proceed by contradiction. Suppose, contrary to the result, that ${\overline{x}}$ doesn’t solve the vector variational E-inequality problem (E-VVIP). Hence, by the formulation of (E-VVIP), it follows that there exists other ${\widetilde{x}}\in S$ such that

$$\begin{aligned} F\left( E\left( {\overline{x}}\right) \right) \left( E\left( {\widetilde{x}} \right) -E\left( {\overline{x}}\right) \right) \le 0. \end{aligned}$$

Since $F=\nabla f$, the aforesaid inequality gives

$$\begin{aligned} \nabla f\left( E\left( {\overline{x}}\right) \right) \left( E\left( {\widetilde{x}}\right) -E\left( {\overline{x}}\right) \right) \le 0. \end{aligned}$$

(29)

From the assumption, $-f$ is an E-convex function at ${\overline{x}}$ on S. Hence, by Definition 5, the inequalities

$$\begin{aligned} f_{i}\left( E\left( x\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) \leqq \nabla f_{i}\left( E\left( {\overline{x}}\right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) ,~ i=1,\ldots ,p \end{aligned}$$

(30)

hold for all $x\in S$. Therefore, they are also satisfied for $x=\widetilde{x }\in S$. Combining (29) and (30), we get that the inequalities

$$\begin{aligned}{} & {} f_{i}\left( E\left( {\widetilde{x}}\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) \leqq 0,~i=1,\ldots ,p,\\{} & {} f_{i}\left( E\left( {\widetilde{x}}\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) < 0\text { for at least one }i\in \left\{ 1,\ldots ,p\right\} \end{aligned}$$

hold. This is a contradiction, by Definition 9, to the assumption that ${\overline{x}}$ solves the vector optimization problem (VOP), that is, that ${\overline{x}}\in E$-$P\left( f,S\right)$. $\square$

Let us now look for the conditions under which we are in position to connect solutions of the weak vector variational E-inequality problem (E-VVIP) with weak E-Pareto solutions in (VOP).

Theorem 5

Let $f:R^{n}\rightarrow R^{p}$ be a differentiable function and S be a nonempty E-convex subset of $R^{n}$. If $F=\nabla f,$ ${\overline{x}}$ solves the weak vector variational E-inequality problem (E-WVVIP), then ${\overline{x}}$ is a weak E-Pareto solution in the vector optimization problem (VOP).

Theorem 6

Assume that S is an E-convex set and $F=\nabla f$. If ${\overline{x}}\in S$ is a weak E-Pareto solution in the weak vector optimization problem (WVOP), then ${\overline{x}}$ solves the weak vector variational E-inequality problem (E-WVVIP).

Proof

Let S be an E-convex set. Hence, by Definition 1, the following relation

$$\begin{aligned} E\left( {\overline{x}}\right) +\alpha \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) \in S \end{aligned}$$

(31)

holds for all $x,{\overline{x}}\in S$ and any $\alpha \in \left[ 0,1\right]$. By assumption, ${\overline{x}}\in S$ is a weak E-Pareto solution in the weak vector optimization problem (WVOP). Hence, by Definition 10 and (31), there does not exist other $x\in S$ such that, for any $\alpha \in \left( 0,1\right)$,

$$\begin{aligned} f\left( E\left( {\overline{x}}\right) +\alpha \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) \right) -f\left( E\left( {\overline{x}}\right) \right) <0. \end{aligned}$$

Thus, dividing the above inequality by $\alpha >0$, we get

$$\begin{aligned} \frac{f\left( E\left( {\overline{x}}\right) +\alpha \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) \right) -f\left( E\left( {\overline{x}} \right) \right) }{\alpha }<0. \end{aligned}$$

(32)

Letting $\alpha \rightarrow 0$, we obtain that the inequality $\nabla f\left( E\left( {\overline{x}}\right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) <0$ holds. Since, by assumption, $F=\nabla f$, (31) gives that there does not exist $x\in S$ such that the following inequality $F\left( E\left( {\overline{x}}\right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) <0$ would be satisfied. This means that ${\overline{x}}$ solves the vector variational E-inequality problem (E-WVVIP). $\square$

Now, under pseudo-E-convexity, we prove the converse result.

Theorem 7

Let $f:R^{n}\rightarrow R^{p}$ be a differentiable function and ${\overline{x}}\in S$ be a solution of the weak vector variational E-inequality problem (E-WVVIP). If f is pseudo-E-convex at ${\overline{x}}$ on S and $F=\nabla f$ , then ${\overline{x}}$ is also a weak E-Pareto solution in the weak vector optimization problem (WVOP).

Proof

Let ${\overline{x}}\in S$ be a solution of the weak vector variational E-inequality problem (E-VVIP). We proceed by contradiction. Suppose, contrary to the result, that ${\overline{x}}$ is not a weak E-Pareto solution in the weak vector optimization problem (WVOP). Hence, by Definition 10, there does exist other ${\widetilde{x}}\in S$ such that

$$\begin{aligned} f\left( E\left( {\widetilde{x}}\right) \right) <f\left( E\left( {\overline{x}} \right) \right) . \end{aligned}$$

(33)

By Definition 6, (33) yields that the inequality

$$\begin{aligned} \nabla f\left( E\left( {\overline{x}}\right) \right) \left( E\left( \widetilde{ x}\right) -E\left( {\overline{x}}\right) \right) <0 \end{aligned}$$

(34)

holds. Since $F=\nabla f$, (34) implies that the inequality $F\left( E\left( {\overline{x}}\right) \right) \left( E\left( {\widetilde{x}}\right) -E\left( {\overline{x}}\right) \right) <0$ holds, contradicting the assumption that ${\overline{x}}\in S$ is a solution of the weak vector variational E-inequality problem (E-WVVIP). $\square$

Theorem 8

Let $f:R^{n}\rightarrow R^{p}$ be a differentiable function and S be a nonempty E-convex subset of $R^{n}$. If $F=\nabla f,$ $-f$ is a strictly E-convex function at ${\overline{x}}$ on S and ${\overline{x}}$ solves the weak vector optimization problem (WVOP), then ${\overline{x}}$ solves the vector variational E-inequality problem (E-VVIP).

Proof

Assume that ${\overline{x}}$ solves the weak vector optimization problem (WVOP), that is, ${\overline{x}}$ is a weak E-Pareto solution in (WVOP). We proceed by contradiction. Suppose, contrary to the result, that ${\overline{x}}$ doesn’t solve the vector variational E-inequality problem (E-VVIP). Hence, by the formulation of (E-VVIP), it follows that there exists other ${\widetilde{x}}\in S$ such that

$$\begin{aligned} F\left( E\left( {\overline{x}}\right) \right) \left( E\left( {\widetilde{x}} \right) -E\left( {\overline{x}}\right) \right) \le 0. \end{aligned}$$

Since $F=\nabla f$, the aforesaid inequality gives

$$\begin{aligned} \nabla f\left( E\left( {\overline{x}}\right) \right) \left( E\left( \widetilde{ x}\right) -E\left( {\overline{x}}\right) \right) \le 0. \end{aligned}$$

(35)

From the assumption, $-f$ is a strictly E-convex function at ${\overline{x}}$ on S. Hence, by Definition 5, the inequalities

$$\begin{aligned} f_{i}\left( E\left( x\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) < \nabla f_{i}\left( E\left( {\overline{x}}\right) \right) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) ,~ i=1,\ldots ,p \end{aligned}$$

(36)

hold for all $x\in S,\; (x\ne {\overline{x}})$. Therefore, they are also satisfied for $x={\widetilde{x}}\in S$. Combining (35) and (36), we get that the inequalities

$$\begin{aligned} f_{i}\left( E\left( {\widetilde{x}}\right) \right) -f_{i}\left( E\left( {\overline{x}} \right) \right) < 0,~i=1,\ldots ,p \end{aligned}$$

hold. Then, by Definition 10, this is a contradiction to the assumption that ${\overline{x}}$ solves the weak vector optimization problem (WVOP), that is, that ${\overline{x}}\in WE$-$P\left( f,S\right)$. $\square$

Weakening the condition of strictly E-convexity imposed on $-f$ in above theorem, the following result is true.

Theorem 9

Let $f:R^{n}\rightarrow R^{p}$ be a differentiable function and S be a nonempty E-convex subset of $R^{n}$. If $F=\nabla f,$ $-f$ is an E-convex function at ${\overline{x}}$ on S and ${\overline{x}}$ solves the vector optimization problem (VOP), then ${\overline{x}}$ solves the weak vector variational E-inequality problem (E-WVVIP).

Remark 3

We have shown that if f is pseudo-E-convex at ${\overline{x}}$ on S and $F=\nabla f$, then there is the equivalence between solutions of the weak vector optimization problem (WVOP) and the weak vector variational E-inequality problem (E-WVVIP). This means that (WVOP) and (E-VVIP) have the same sets of solutions. Obviously, a solution of (E-VVIP) is also a solution of (E-WVVIP), but the converse is not true in a general case. Moreover, a solution of (VOP) is also a solution of (WVOP), but the converse is not true in a general case. Nevertheless, under an extra assumption, the converse result is true.

Theorem 10

Let $f:R^{n}\rightarrow R^{p}$ be a differentiable function and S be a nonempty E-convex subset of $R^{n}$. Further, assume that f is a strictly E-convex function at ${\overline{x}}\in S$ on S and ${\overline{x}}$ is a weak E-Pareto solution in the weak vector optimization problem (WVOP). Then ${\overline{x}}$ is an E-Pareto solution in the vector optimization problem (VOP).

Proof

Assume that ${\overline{x}}$ is a weak E-Pareto solution in the weak vector optimization problem (WVOP).

We proceed by contradiction. Suppose, contrary to the result, that $\overline{ x}\in S$ is not an E-Pareto solution in (VOP). Then, by Definition 9, there exists other ${\widetilde{x}}\in S$ such that

$$\begin{aligned} f\left( E\left( {\widetilde{x}}\right) \right) \le f\left( E\left( \overline{x }\right) \right) . \end{aligned}$$

(37)

By assumption, f is a strictly E-convex function at ${\overline{x}}$ on S. Hence, by Definition 5 and (37), it follows that

$$\begin{aligned} 0 \geq f\left( E\left( {\widetilde{x}}\right) \right) -f\left( E\left( {\overline{x}} \right) \right) > \nabla f\left( E\left( {\overline{x}}\right) \right) \left( E\left( {\widetilde{x}}\right) -E\left( {\overline{x}}\right) \right) . \end{aligned}$$

(38)

By (38), we conclude that $E({\overline{x}})\in S$ does not solve the weak vector variational E-inequality problem (E-WVVIP). Then, by Theorem 6, we get a contradiction. Hence, ${\overline{x}}$ is an E-Pareto solution in (VOP). $\square$

It is known in optimization theory that important and useful generalizations of the variational inequalities, called the variational-like inequalities, have been studied extensively in recent years because they also allowed to characterize/find (weakly) efficient solutions of some classes of nonconvex vector optimization problems in which the involved functions are invex (with respect to the vector-valued $\eta$). The formulations of such vector variational-like inequalities are different, but they contain the aforesaid vector-valued function $\eta$. This shows that the concept of invexity plays the same role for variational-like inequalities as classical convexity plays for classical variational inequalities (similar to that ones introduced by Giannessi [5]). However, there are vector optimization problems in which the functions involved are not invex (with respect to any function $\eta$). Hence, they cannot be characterized by the aforesaid variational-like inequalities which are formulated just by such a function $\eta$. It turned out that the solvability of some of such vector optimization problems can be characterized by variational inequalities introduced in our paper. To illustrate this fact, we now give the example of such a nonconvex vector optimization problem in which the functions involved are E-convex, but they are not invex.

Example 2

Consider the following nonconvex vector optimization problem defined by

$$\begin{aligned} \begin{array}{ll} \text {V-min} &{}\quad f(x_{1},x_{2})=(f_{1}(x_{1},x_{2}), f_{2}(x_{1},x_{2}))=(x_{1}^{3},x_{1}^{2}+x_{2})\\ \text {s.t.}&{}\quad (x_{1},x_{2})\in S=\{(x_{1},x_{2})\in R^{2}: x_{1}\geqq -1, \;\; x_{2}\geqq 0\} \end{array} \quad \text {(VOP1)} \end{aligned}$$

Note that ${\overline{x}}=(0,0)$ is a feasible solution in (VOP1). Moreover, by Definition 9, ${\overline{x}}$ is a Pareto solution in (VOP1). It can be shown that $f_1$ is not an invex function at ${\overline{x}}$ on S since ${\overline{x}}$ is a stationary point of $f_1,$ but it is not its minimizer (see, Theorem 1 [52]). However, by Definition 5, it follows that $f=(f_1,f_2 )$ is E-convex at ${\overline{x}}=(0,0)$ on S with respect to the operator $E:R^2\rightarrow R^2,$ where $E(x_1,x_2 )=(x_1^{2},x_2 ).$ This means that all assumptions of Theorem 3 are fulfilled and, therefore, ${\overline{x}}=(0,0)$, is an E-Pareto solution in (VOP1). One of the aforesaid assumptions means that ${\overline{x}}=(0,0)$ is an efficient solution in the vector variational E-inequality problem (E-VVIP). Thus, the aforesaid vector variational E-inequality proved useful in finding a solution in (VOP1). Further, we cannot use variational-like inequalities defining in the following forms (see, for example, [32, 53, 54]):

• Find a point ${\overline{x}} \in S$ such $\left\langle F\left( {\overline{x}} \right) ,\eta (x, {\overline{x}}) \right\rangle \notin -R_{+}^{p}\backslash \left\{ 0\right\}$ for all $x\in S$ ($\hbox {VVI}_{\eta }$)

• Find a point ${\overline{x}} \in S$ such $\left\langle F\left( {\overline{x}} \right) ,\eta (x, {\overline{x}}) \right\rangle \notin - int R_{+}^{p}$ for all $x\in S$ ($\hbox {WVVI}_{\eta }$)

and prove the equivalence solutions in (VVI) (or (WVVI)) and solutions in (VOP) (or (WVOP)) under suitable invexity hypotheses (with respect to the given vector-valued function $\eta$). This is a consequence of the mentioned fact that the functions involved might not be invex (with respect to any function $\eta$) as in the considered example. However, we can use the results established in our paper in this example. Namely, it is possible to show that f is E-convex at ${\overline{x}}$ on S with respect to the operator $E:R^2\rightarrow R^2,$ defined above by $E(x_1,x_2 )=(x_1^{2},x_2 ).$ Then, if we set $F =\nabla f$ and if we define the vector variational E-inequality problem (E-VVIP) as follows $\begin{bmatrix} 0\\ x_{1}^{2}+x_{2} \end{bmatrix} \notin - R_{+}^{2}\backslash \left\{ 0\right\}$, then, by Theorem 3, it follows that ${\overline{x}}=(0,0)$ is a E-Pareto solution in (VOP1).

Therefore, through the aforementioned example, we strive to emphasize the significance and novelty of our results, setting them apart from those already present in the literature.

Now, we introduce the definition of a vector critical E-point for the vector optimization problem (MOP) (thus, also for (WVOP) and (VOP)).

Definition 11

Let $E:R^{n}\rightarrow R^{n}$ be a given operator. A point ${\overline{x}}\in S$ is said to be a vector critical E-point (E-VCP) in the vector optimization problem (MOP) if there does exist a vector ${\overline{\lambda }} \in R^{p}$ with ${\overline{\lambda }}\ge 0$ such that ${\overline{\lambda }} \nabla f\left( E\left( {\overline{x}}\right) \right) =0$.

Remark 4

As it follows from the above definition, a vector critical E-point (E-VCP) of (WVOP) is that there exists a non-negative linear combination of the gradient vectors $\nabla f_{i}\left( E\left( {\overline{x}}\right) \right)$, $i=1,\ldots ,p$, equal to 0.

Theorem 11

All vector E-critical points are weak E-Pareto solutions of (WVOP) if, and only if, the vector-valued function f is pseudo-E-convex on S.

Proof

Firstly, we prove that if all vector E-critical points are weak E-Pareto solutions of (WVOP), then the vector-valued function f is pseudo-E-convex on S. Let ${\overline{x}}$ be a weak E-Pareto solutions of (WVOP). Then, by Definition 10, the system

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

(39)

has no solution in $x\in S$. On the other hand, if ${\overline{x}}$ is a vector E-critical point of (WVOP), then there exists ${\overline{\lambda }} \in R^{p}$ with ${\overline{\lambda }}\ge 0$ such that ${\overline{\lambda }} \nabla f\left( E\left( {\overline{x}}\right) \right) =0$. Applying the Gordan’s theorem of the alternative Mangasarian [55], the system

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

(40)

has no solution at $d\in R^n$, $d\ne 0.$. Thus, the systems (39) and (40) are equivalent. This means that if there exists a solution $x\in S$ of (39), such that

$$\begin{aligned} (f\circ E)(x)< (f\circ E)({\overline{x}}), \end{aligned}$$

(41)

then there exists $\left( E\left( x\right) -E\left( {\overline{x}}\right) \right) \in R^n$ which is a solution of (40). Therefore, one has

$$\begin{aligned} \nabla (f\circ E)({\overline{x}}) \left( E\left( x\right) -E\left( {\overline{x}}\right) \right) <0. \end{aligned}$$

(42)

Thus, by Definition 6, the vector-valued function f is pseudo-E-convex on S.

On the other hand, let $E({\overline{x}})$ be a vector E-critical point of (WVOP). Then, there exists ${\overline{\lambda }}\in R^{p}$ with ${\overline{\lambda }}\ge 0$ such that ${\overline{\lambda }}^{T} \nabla (f\circ E)({\overline{x}})=0.$ We proceed by contradiction. Suppose that ${\overline{x}}$ is not a weak E-Pareto solution of (WVOP). Then, by Definition 10, there exists other point ${\widetilde{x}}\in R^{n}$ such that

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

(43)

Since f is a pseudo-E-convex function, by Definition 6 and (43), we get

$$\begin{aligned} \nabla f(E({\overline{x}}))(E({\widetilde{x}})-E({\overline{x}}))<0. \end{aligned}$$

(44)

Thus, the inequality

$$\begin{aligned} {\overline{\lambda }}^{T} \nabla f(E({\overline{x}}))(E({\widetilde{x}})-E({\overline{x}}))<0 \; \; \text {for any} \; {\overline{\lambda }} \ge 0 \end{aligned}$$

holds, contradicting the definition of a vector E-critical point. The proof of this theorem is completed. $\square$

Note that, by Theorems 6, 7 and 11, the following results are true:

Corollary 1

Let S be an E-convex set and $F=\nabla f$. Further, assume that the objective function f is pseudo-E-convex on S. Then, vector critical E-points, weak E-Pareto solutions of (WVOP) and solutions of the weak vector variational E-inequality problem (E-WVVIP) are equivalent.

Corollary 2

Let S be an E-convex set and $F=\nabla f$. Further, assume that the objective function f is strictly pseudo-E-convex on S. Then, vector critical E-points, E-Pareto solutions of (VOP) and solutions of the vector variational E-inequality problem (E-VVIP) are equivalent.

Then, we are able to conclude the results established in the paper as follows:

Remark 5

Since (E-WVVIP) implies (WVOP) and (WVOP) implies (E-VVIP), therefore, (E-WVVIP) implies (E-VVIP) if we assume that f is an E-convex function.

Remark 6

Since (E-WVVIP) implies (WVOP) and (WVOP) implies (VOP), therefore, (E-WVVIP) implies (VOP) if we assume that f is a pseudo-E-convex function.

Remark 7

Since (E-VVIP) implies (E-WVVIP) and (E-WVVIP) implies (WVOP), therefore, (E-VVIP) implies (WVOP) if we assume that f is a strictly E-convex function.

Remark 8

Since f is strictly E-convex, therefore, (WVOP) is equivalent to (VOP).

Now, we summarize the findings so far with the following diagram (Fig. 1):

Fig. 1

Relationships between (VOP), (WVOP), (E-VVIP) and (E-WVVIP)

5 Applications

We now present an example of an economic optimization problem in which the functions involved are E-convex. Then, we characterize its optimality by using one of aforesaid vector variational-like inequalities.

Example 3

Firm A and Firm B are two manufacturing firms producing agricultural fertilizers. Both firms have entered into a contract to supply 100 bags of fertilizer at the end of the first month, 100 bags at the end of the second month, and 100 bags at the end of the third month. The cost of producing x bags of fertilizer for Firm A is given by $8x^3$ dollars, and for Firm B is given by $4x^2$ dollars. The firms can produce additional bags of fertilizer in any month and carry them over to subsequent months. However, it costs 40 dollars per bag for any bags of fertilizer carried over from one month to the next. Assuming that there is no initial inventory, determine the number of bags of fertilizer to be produced in each month for both firms to minimize the total cost. Let $x_{1}$, $x_{2}$, and $x_{3}$ represent the number of bags of fertilizer produced in the first, second, and third month, respectively. The total cost to be minimized is given by

$$\begin{aligned} \text {Total cost} = \text {production cost} + \text {holding cost} \end{aligned}$$

Hence, the vector optimization problem which is a model of the considered economic extremum problem is defined by

$$\begin{aligned} & f(x)=(f_{1}(x),f_{2}(x))\\ &\qquad=(8x_{1}^{3}+8x_{2}^{3}+8x_{3}^{3} +40(x_{1}-100)+40(x_{1}+x_{2}-200),\\& \qquad \quad 4x_{1}^{2}+4x_{2}^{2}+4x_{3}^{2} +40(x_{1}-100)+40(x_{1}+x_{2}-200)) \rightarrow V-\min \\ & {\text {subject to}} \qquad \quad \left( x_{1},x_{2},x_{3}\right) \in S =\left\{ \left( x_{1},x_{2},x_{3}\right) \in R^{3}:\;100-x_{1}\leqq 0 \, \wedge \right. \\ & \left. \qquad \qquad \qquad \quad 200-x_{1}-x_{2}\leqq 0 \; \wedge\; 300-x_{1}-x_{2}-x_{3}\leqq 0 \right\}. \quad {\rm{VOP2}}\end{aligned}$$

Let $E:R^3\rightarrow R^3$ be defined by $E\left( x_1,x_2,x_3\right) =\left( \frac{1}{2}x_1,\frac{1}{2}x_2,\frac{1}{2}x_3\right) .$ Note that ${\bar{x}}=\left( 100,100,100\right)$ is a feasible solution in (VOP2). We use the vector variational E-inequality to verify that ${\bar{x}}$ is an E-Pareto solution in (VOP2). Thus, if we set $F\left( E\left( {\bar{x}}\right) \right) =\left( \nabla f_1\left( E\left( {\bar{x}}\right) \right) ,\nabla f_2\left( E\left( {\bar{x}}\right) \right) \right) ,$ then the vector variational E-inequality has in the considered case the following form:

$$\begin{aligned} \begin{bmatrix} 15020x_1+15010x_2+15000x_3-4503000\\ 120x_1+110x_2+100x_3-33000 \end{bmatrix} \notin - R_{+}^{2}\backslash \left\{ 0\right\} \ \forall x\in S. \;\;\; (E-VVIP2) \end{aligned}$$

Since ${\bar{x}}$ solves (E-VVIP) and the objective function f is E-convex at ${\bar{x}}$ on S, by Theorem 3, we conclude that ${\bar{x}}$ is an E-Pareto solution in (VOP2).

6 Concluding remarks

In this paper, new vector variational-like inequalities have been introduced to optimization theory. Namely, based on the effect of an operator $E:R^{n}\rightarrow R^{n}$ on the sets and the domains of functions, we have formulated the vector variational E-inequality problem (E-VVIP) and the weak vector variational E-inequality problem (E-WVVIP). The existence theorem for a solution of the weak vector variational E-inequality problem (E-WVVIP) has been established under appropriate compactness hypotheses. Further, the relationship between two aforesaid introduced variational-like inequalities and differentiable vector optimization problems has been investigated. In fact, weak E-Pareto solutions and E-Pareto solutions of differentiable vector optimization problems have been characterized by solutions of two aforesaid vector variational like inequalities, respectively. Thus, the characterization of optimality by using solutions of vector variational-like inequalities has been extended to a new class of differentiable vector optimization problems. In fact, we have shown that there are nonconvex vector optimization problems for which it is not possible to characterize their optimality by some variational-like inequalities existing in the literature. Namely, the example of such a nonconvex vector optimization problem has been given for which it is not possible, under invexity hypotheses (with respect to a vector-valued function $\eta$), to find their optimal solutions by using vectorial-like inequalities existing in the literature.

However, some interesting topics for further research remain. It would be of interest to investigate whether it is possible to prove similar results for other classes of vector optimization problems. We shall investigate these questions in subsequent papers.