Connections between Non-Linear Optimization Problems and Associated Variational Inequalities

Abstract

In this paper, by using the invexity (or pseudoinvexity) and Fréchet differentiability of some integral functionals of curvilinear type, we state some relations between the solutions of a new non-linear optimization problem and the associated variational inequality. In order to prove the results derived in this paper, we use the new notion of invex set by considering some given functions. To justify the effectiveness and outstanding applicability of this work, some illustrative examples are provided.

1. Introduction

As is well known, by multi-objective optimization problems, we understand some optimization problems having more than one objective function, that need to be optimized simultaneously. Since there exist rarely feasible points that simultaneously extremize (maximize or minimize) all the objective functions, it is necessary to introduce some concepts of efficient solutions. In this direction, Geoffrion [1] proposed the notion of proper efficient solutions. Furthermore, Klinger [2] investigated improper solutions for the multi-objective maximum problem, and Kazmi [3] proved the existence results of a weak minimum for constrained vector optimization problems via vector variational-like inequalities. Ghaznavi-ghosoni and Khorram [4] established efficiency conditions associated with (weakly, properly) approximating efficient points for multi-objective optimization problems via the approximate solutions of scalarized problems.

On the other hand, convexity is a concept almost inevitable in control and optimization theory in order to formulate the optimality conditions. However, convexity is not sufficient in many concrete problems in applied sciences, its generalization is necessary. Therefore, Hanson [5] formulated the notion of invexity. Of course, various extensions of convexity have been defined, namely, preinvexity, univexity, approximate convexity, quasiinvexity, pseudoinvexity, and so on (Antczak [6,7], Mishra et al. [8], Arana-Jiménez et al. [9]). Moreover, these concepts were translated into the multi-dimensional context with multiple or curvilinear integrals (Treanţă [10,11], Mititelu and Treanţă [12]).

Since many problems in physics, mechanics, traffic analysis, and engineering are modelled as variational inequalities, these mathematical tools have been thoroughly studied. In this sense, the vector case of variational inequalities was developed, with remarkable results, by Giannessi [13]. As is well-known, under some convexity or generalized convexity assumptions, variational inequalities of vector type provide some existence results of solutions for the corresponding multi-objective optimization problems. Over time, several papers have studied the connections between the solution sets of vector variational inequalities and the associated multi-objective variational problems (Ruiz-Garzón et al. [14,15], Jayswal et al. [16]). Moreover, Treanţă [17] recently introduced and analysed the variational control inequalities given by integral functionals of curvilinear type.

Variational problems are structured as follows: variational problems of classic type, and (multi-objective) variational problems with continuous-time. In 2004, Kim [18] stated some relations between multiple objective programs with continuous-time and variational-type inequalities of vector type. The problems dependant on control, seen as variational problems with continuous-time, are a powerful tool in the study of various engineering problems, game theory, economics, and operations research. Following this, Jha et al. [19] and Treanţă [20,21] have contributed by formulating and proving well-posedness and existence results for classes of multi-dimensional optimization problems given by functionals of various types.

Despite all previously mentioned advances, we present this paper in which we define (weak) variational control inequalities of vector type, and multiple objective variational control problems defined by curvilinear integral-type functionals (independent of the path). More precisely, we establish some relations between the solution sets of the considered multi-dimensional variational control problems. The presence of the control function in this context is an element of novelty. Furthermore, in order to prove the theoretical results derived in the paper, we use the new notion of invex set by considering some given functions (${\displaystyle \vartheta }$ and ${\displaystyle \upsilon }$, see Definition 7) For similar studies in this area, the reader is directed to [10,11]. In [10], the authors considered the case of multiple integral functionals instead of the curvilinear ones, and in [11] the authors dealt with the well-posedness study of a class of variational inequality constrained control problems driven also by multiple integrals. To justify the effectiveness and outstanding applicability of this paper, some illustrative examples are provided.

Further, the paper continues with preliminaries and formulation of the considered problem. In Section 3, some characterization results of the solution sets for the given variational control problems are established. Section 4 contains an illustrative application. In Section 5, we conclude this study and provide a further research direction.

2. Preliminaries

In this paper, consider ${\displaystyle L}$ as a compact set in ${\displaystyle {\mathbb{R}}^m}$, and ${\displaystyle L\ni \varsigma =\left({\varsigma }^{\nu }\right),\nu =\overline{1,m}}$. Let ${\displaystyle L\supset \Gamma :\varsigma =\varsigma \left(\theta \right),\theta \in [{\theta }_0,{\theta }_1]}$ be a curve (piece-wise smooth) linking the points ${\displaystyle {\varsigma }_1=({\varsigma }_1^1,\dots ,{\varsigma }_1^m)}$ and ${\varsigma }_2=({\varsigma }_2^1,\dots ,{\varsigma }_2^m)$ in ${\displaystyle L}$. Consider ${\displaystyle \Theta }$ is the space of state functions (piece-wise smooth) ${\displaystyle u:L\to {\mathbb{R}}^n}$, and ${\displaystyle \Omega }$ is the space of control functions (piece-wise continuous) ${\displaystyle \omega :L\to {\mathbb{R}}^k}$. Define on ${\displaystyle \Theta \times \Omega }$ the following scalar product

$${\displaystyle \langle (u,\omega ),(\tau ,x)\rangle ={\int }_{\Gamma }[u\left(\varsigma \right)·\tau \left(\varsigma \right)+\omega \left(\varsigma \right)·x\left(\varsigma \right)]d{\varsigma }^{\nu }}$$

$${\displaystyle ={\int }_{\Gamma }\left[\sum _{i=1}^n{u}^i\left(\varsigma \right){\tau }^i\left(\varsigma \right)+\sum _{j=1}^k{\omega }^j\left(\varsigma \right)x^j\left(\varsigma \right)\right]d{\varsigma }^1}$$

$${\displaystyle +\cdots +\left[\sum _{i=1}^n{u}^i\left(\varsigma \right){\tau }^i\left(\varsigma \right)+\sum _{j=1}^k{\omega }^j\left(\varsigma \right)x^j\left(\varsigma \right)\right]d{\varsigma }^m,}$$

for all $(u,\omega ),(\tau ,x)\in \Theta \times \Omega$, and the associated induced norm.

Consider the $C^1$-class functions ${\displaystyle {\psi }_{\nu }=\left({\psi }_{\nu }^l\right):L\times {\mathbb{R}}^n\times {\mathbb{R}}^k\to {\mathbb{R}}^p,\nu =\overline{1,m},l=\overline{1,p}}$, in order to define the functional

$${\displaystyle \Psi :\Theta \times \Omega \to {\mathbb{R}}^p,\Psi (u,\omega )={\int }_{\Gamma }{\psi }_{\nu }\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }=}$$

$${\displaystyle =\left({\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },\cdots ,{\int }_{\Gamma }{\psi }_{\nu }^p\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }\right).}$$

Further, we assume that ${\displaystyle D_{\mu },\mu \in \{1,\dots ,m\}}$, is the derivative operator, and the 1-forms

$${\displaystyle {\psi }_{\nu }=\left({\psi }_{\nu }^1,\dots ,{\psi }_{\nu }^p\right):L\times {\mathbb{R}}^n\times {\mathbb{R}}^k\to {\mathbb{R}}^p,\nu =\overline{1,m},}$$

are closed (${\displaystyle D_{\mu }{\psi }_{\nu }^l=D_{\nu }{\psi }_{\mu }^l,\nu ,\mu =\overline{1,m},\nu \ne \mu ,l=\overline{1,p}}$. Furthermore, the following rules are used:

$${\displaystyle a=b\iff a^l=b^l,a\le b\iff a^l\le b^l,a<b\iff a^l<b^l,a⪯b\iff a\le b,a\ne b,l=\overline{1,p},}$$

for ${\displaystyle a=\left(a^1,\cdots ,a^p\right),b=\left(b^1,\cdots ,b^p\right)}$ in ${\displaystyle {\mathbb{R}}^p}$.

At this moment, we introduce the following constrained multiple objective optimization problem:

$${\displaystyle \left(P\right)\underset{(u,\omega )}{min}\left\{\Psi (u,\omega )={\int }_{\Gamma }{\psi }_{\nu }\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }\right\}\mathrm{subject}\mathrm{to}(u,\omega )\in W,}$$

where

$${\displaystyle \Psi (u,\omega )={\int }_{\Gamma }{\psi }_{\nu }\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }}$$

$${\displaystyle =\left({\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },\cdots ,{\int }_{\Gamma }{\psi }_{\nu }^p\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }\right)=\left({\Psi }^1(u,\omega ),...,{\Psi }^p(u,\omega )\right)}$$

and

$${\displaystyle W=\{(u,\omega )\in \Theta \times \Omega |u_{\mu }^i\left(\varsigma \right)={\displaystyle \frac{\partial u^i}{\partial {\varsigma }^{\mu }}}\left(\varsigma \right)=F_{\mu }^i\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right),Z\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)\le 0,}$$

$${\displaystyle {(u,\omega )|}_{\varsigma ={\varsigma }_1,{\varsigma }_2}=\mathrm{given}\}.}$$

In the above context, we consider the $C^1$-class functions ${\displaystyle F_{\mu }=\left(F_{\mu }^i\right):L\times {\mathbb{R}}^n\times {\mathbb{R}}^k\to {\mathbb{R}}^n,i=\overline{1,n},\mathrm{and}\mu =\overline{1,m}}$, to generate the PDEs

$${\displaystyle u_{\mu }^i\left(\varsigma \right)=F_{\mu }^i\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right),i=\overline{1,n},\mathrm{and}\mu =\overline{1,m},}$$

satisfying ${\displaystyle D_{\nu }{F}_{\mu }^i=D_{\mu }{F}_{\nu }^i,\mu ,\nu =\overline{1,m},\mu \ne \nu ,i=\overline{1,n}}$, and ${\displaystyle Z=\left(Z^r\right):L\times {\mathbb{R}}^n\times {\mathbb{R}}^k\to {\mathbb{R}}^q,r=\overline{1,q},}$ are functions of $C^1$-class.

A solution of $\left(P\right)$ means a feasible pair that minimizes (simultaneously) all objective functions ${\Psi }^l(u,\omega ),l=\overline{1,p}$. Thus, the following kinds of solutions of $\left(P\right)$ are needed.

Definition 1

(see Mititelu and Treanţă [12]). The pair ${\displaystyle (u^0,{\omega }^0)\in W}$ is an efficient solution to $\left(P\right)$ if there exists no other ${\displaystyle (u,\omega )\in W}$ with ${\displaystyle \Psi (u,\omega )⪯\Psi (u^0,{\omega }^0)}$, or ${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)\le 0,(\forall )l=\overline{1,p}}$, with a strict inequality for at least one l.

Definition 2

(see Geoffrion [1]). The pair ${\displaystyle (u^0,{\omega }^0)\in W}$ is a proper efficient solution to $\left(P\right)$ if ${\displaystyle (u^0,{\omega }^0)\in W}$ is an efficient solution in $\left(P\right)$ and there exists $M>0$ such that, for all $l=\overline{1,p}$, the inequality

$${\displaystyle {\Psi }^l(u^0,{\omega }^0)-{\Psi }^l(u,\omega )\le M\left({\Psi }^s(u,\omega )-{\Psi }^s(u^0,{\omega }^0)\right),}$$

is true for some $s\in \{1,\cdots ,p\}$, such that

$${\displaystyle {\Psi }^s(u,\omega )>{\Psi }^s(u^0,{\omega }^0),}$$

whenever $(u,\omega )\in W$ and

$${\displaystyle {\Psi }^l(u,\omega )<{\Psi }^l(u^0,{\omega }^0).}$$

Definition 3.

The pair ${\displaystyle (u^0,{\omega }^0)\in W}$ is a weak efficient solution to $\left(P\right)$ if there exists no other ${\displaystyle (u,\omega )\in W}$ such that ${\displaystyle \Psi (u,\omega )<\Psi (u^0,{\omega }^0)}$, or ${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)<0,(\forall )l=\overline{1,p}}$.

Taking into account Treanţă [11], in order to introduce the invexity and pseudoinvexity, we consider the following functional (which is independent of the path):

$${\displaystyle K:\Theta \times \Omega \to {\mathbb{R}}^p,K\left(u,\omega \right)={\int }_{\Gamma }{\kappa }_{\nu }\left(\varsigma ,u\left(\varsigma \right),u_{\mu }\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }.}$$

Definition 4.

The functional ${\displaystyle K}$ is called invex at ${\displaystyle \left(u^0,{\omega }^0\right)\in \Theta \times \Omega }$ with respect to ${\displaystyle \vartheta }$ and ${\displaystyle \upsilon }$ if there exist

$${\displaystyle \vartheta :L\times {({\mathbb{R}}^n\times {\mathbb{R}}^k)}^2\to {\mathbb{R}}^n,}$$

$${\displaystyle \vartheta =\vartheta \left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=\left({\vartheta }^i\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\right),i=\overline{1,n},}$$

of $C^1$-class with ${\displaystyle \vartheta \left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=0,(\forall )\varsigma \in L,\vartheta \left({\varsigma }_1\right)=\vartheta \left({\varsigma }_2\right)=0}$, and

$${\displaystyle \upsilon :L\times {({\mathbb{R}}^n\times {\mathbb{R}}^k)}^2\to {\mathbb{R}}^k,}$$

$${\displaystyle \upsilon =\upsilon \left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=\left({\upsilon }^j\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\right),j=\overline{1,k},}$$

of $C^0$-class with ${\displaystyle \upsilon \left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=0,(\forall )\varsigma \in L,\upsilon \left({\varsigma }_1\right)=\upsilon \left({\varsigma }_2\right)=0}$, satisfying

$${\displaystyle K\left(u,\omega \right)-K\left(u^0,{\omega }^0\right)}$$

$${\displaystyle \ge {\int }_{\Gamma }\left[\frac{\partial {\kappa }_{\nu }}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\kappa }_{\nu }}{\partial u_{\mu }}\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)D_{\mu }\vartheta \right]d{\varsigma }^{\nu }}$$

$${\displaystyle +{\int }_{\Gamma }\left[\frac{\partial {\kappa }_{\nu }}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },}$$

for any ${\displaystyle (u,\omega )\in \Theta \times \Omega }$.

Definition 5.

If we replace ${\displaystyle \ge }$ with ${\displaystyle >}$, ${\displaystyle \left(u,\omega \right)\ne \left(u^0,{\omega }^0\right)}$, we say that ${\displaystyle K}$ is called strictly invex at ${\displaystyle \left(u^0,{\omega }^0\right)\in \Theta \times \Omega }$ with respect to ${\displaystyle \vartheta }$ and ${\displaystyle \upsilon }$.

Definition 6.

The functional ${\displaystyle K}$ is said to be pseudoinvex at ${\displaystyle \left(u^0,{\omega }^0\right)\in \Theta \times \Omega }$ with respect to ${\displaystyle \vartheta }$ and ${\displaystyle \upsilon }$ if there exist

$${\displaystyle \vartheta :L\times {({\mathbb{R}}^n\times {\mathbb{R}}^k)}^2\to {\mathbb{R}}^n,}$$

$${\displaystyle \vartheta =\vartheta \left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=\left({\vartheta }^i\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\right),i=\overline{1,n},}$$

of $C^1$-class with ${\displaystyle \vartheta \left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=0,(\forall )\varsigma \in L,\vartheta \left({\varsigma }_1\right)=\vartheta \left({\varsigma }_2\right)=0}$, and

$${\displaystyle \upsilon :L\times {({\mathbb{R}}^n\times {\mathbb{R}}^k)}^2\to {\mathbb{R}}^k,}$$

$${\displaystyle \upsilon =\upsilon \left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=\left({\upsilon }^j\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\right),j=\overline{1,k},}$$

of $C^0$-class with ${\displaystyle \upsilon \left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right),u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)=0,(\forall )\varsigma \in L,\upsilon \left({\varsigma }_1\right)=\upsilon \left({\varsigma }_2\right)=0}$, satisfying

$${\displaystyle K\left(u,\omega \right)-K\left(u^0,{\omega }^0\right)<0}$$

$${\displaystyle \Rightarrow {\int }_{\Gamma }\left[\frac{\partial {\kappa }_{\nu }}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\kappa }_{\nu }}{\partial u_{\mu }}\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)D_{\mu }\vartheta \right]d{\varsigma }^{\nu }}$$

$${\displaystyle +{\int }_{\Gamma }\left[\frac{\partial {\kappa }_{\nu }}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }<0,}$$

or, in an equivalent manner,

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\kappa }_{\nu }}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\kappa }_{\nu }}{\partial u_{\mu }}\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)D_{\mu }\vartheta \right]d{\varsigma }^{\nu }}$$

$${\displaystyle +{\int }_{\Gamma }\left[\frac{\partial {\kappa }_{\nu }}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),u_{\mu }^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }\ge 0\Rightarrow K\left(u,\omega \right)-K\left(u^0,{\omega }^0\right)\ge 0,}$$

for any ${\displaystyle (u,\omega )\in \Theta \times \Omega }$.

For invex and pseudoinvex integral functionals of curvilinear type, the reader can consult Treanţă [11] (Examples 2.1 and 2.2).

Definition 7.

The subset $\varnothing \ne \mathsf{X}\times \mathsf{U}\subset \Theta \times \Omega$ is called invex with respect to ${\displaystyle \vartheta }$ and ${\displaystyle \upsilon }$ if

$$(u^0,{\omega }^0)+\lambda \left(\vartheta \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right),\upsilon \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right)\right)\in \mathsf{X}\times \mathsf{U},$$

for $\lambda \in [0,1]$ and $(u,\omega ),(u^0,{\omega }^0)\in \mathsf{X}\times \mathsf{U}$.

Now, for establishing the existence results of solution sets for $\left(P\right)$, we consider (weak) variational control inequalities of vector type:

• Find $(u^0,{\omega }^0)\in W$ such that there exists no $(u,\omega )\in W$ with

  $${\displaystyle \left(I\right)({\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^1}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^1}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },\cdots ,}$$

  $${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^p}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^p}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu })\le 0;}$$
• Find $(u^0,{\omega }^0)\in W$ such that there exists no $(u,\omega )\in W$ with

  $${\displaystyle \left(WI\right)({\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^1}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^1}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },\cdots ,}$$

  $${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^p}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^p}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }\left)\right)<0.}$$

The class of (weak) variational control inequalities of vector type is solvable for a given point (see Example 1).

Example 1.

For $m=p=2,n=k=1,L=[0,1]\times [0,1]$, and $\Gamma \subset L$ a curve (differentiable) linking $(0,0)$ and $(1,1)$, assuming that $u,\omega :L\to \mathbb{R}$ are functions (piece-wise differentiable), and $\vartheta ,\upsilon :L\times {\mathbb{R}}^4\to \mathbb{R}$ are defined by: $\vartheta =0$, and $\upsilon =e^{u^0\left(\varsigma \right)}-e^{u\left(\varsigma \right)},(\forall )\varsigma \in L\backslash \{{\varsigma }_1,{\varsigma }_2\}$ and $\upsilon =0$ for $\varsigma \in \{{\varsigma }_1,{\varsigma }_2\}$, we define the 1-forms

$${\displaystyle {\psi }_{\nu }=\left({\psi }_{\nu }^1,{\psi }_{\nu }^2\right):L\times {\mathbb{R}}^2\to {\mathbb{R}}^2,\nu =\overline{1,2},}$$

as

$${\displaystyle {\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)=\left(-\omega \left(\varsigma \right)-\frac{1}{2},-u\left(\varsigma \right)\right),}$$

$${\displaystyle {\psi }_{\nu }^2\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)=\left(e^{\omega \left(\varsigma \right)}+\frac{1}{2},e^{\omega \left(\varsigma \right)}\right).}$$

Further, we can easily notice that $(u^0,{\omega }^0)=(0,0)$ is a solution of $\left(I\right)$. Indeed, it results

$${\displaystyle ({\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^1}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^1}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },}$$

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^2}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^2}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu })}$$

$${\displaystyle =\left({\int }_{\Gamma }\left(e^{u\left(\varsigma \right)}-1\right)d{\varsigma }^1,{\int }_{\Gamma }{e}^{\omega \left(\varsigma \right)}\left(1-e^{u\left(\varsigma \right)}\right)\left(d{\varsigma }^1+d{\varsigma }^2\right)\right)\nleqq (0,0),}$$

for all functions $u,\omega :L\to \mathbb{R}$ (piece-wise differentiable).

3. Main Results

In this section, of the present paper, we will state some existence results and connections between the solution sets of the mentioned (weak) variational control inequalities of vector type and the associated multiple objective optimization problem $\left(P\right)$.

Theorem 1.

Consider $W\subset \Theta \times \Omega$ is an invex set, $(u^0,{\omega }^0)\in W$ is a proper efficient solution to $\left(P\right)$, and the integral functionals

$${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p},}$$

are Fréchet differentiable at $(u^0,{\omega }^0)\in W$. Then $(u^0,{\omega }^0)$ solves $\left(I\right)$.

Proof. 

Consider, in contrast, that $(u^0,{\omega }^0)\in W$ is a proper efficient solution to $\left(P\right)$ and it does not solve $\left(I\right)$. Thus, there exists $(u,\omega )\in W$, for all $l=\overline{1,p}$, such that

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }<0}$$

(1)

and

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^s}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^s}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }\le 0,}$$

(2)

for $s\ne l$. As $W\subset \Theta \times \Omega$ is an invex set, we consider the pair

$${\displaystyle (h,w)=(u^0,{\omega }^0)+{\lambda }_n\left(\vartheta \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right),\upsilon \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right)\right)\in W,(\forall )n,}$$

where $\left\{{\lambda }_n\right\}$ is a sequence of positive real numbers, with ${\lambda }_n\to 0$ as $n\to \infty$. Now, since each integral functional ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p}}$, is Fréchet differentiable at $(u^0,{\omega }^0)\in W$, it results

$${\displaystyle {\Psi }^l(h,w)-{\Psi }^l(u^0,{\omega }^0)={\int }_{\Gamma }{\lambda }_n\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }}$$

$${\displaystyle +\Vert {\lambda }_n\left(\vartheta \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right),\upsilon \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right)\right)\Vert ·G^l(h,w)}$$

(3)

where the continuous function $G^l:V_{(u^0,{\omega }^0)}\to \mathbb{R}$ is defined on a neighbourhood of $(u^0,{\omega }^0)$, and fulfils ${lim}_{n\to \infty }{G}^l(h,w)=0$. Dividing Relation (3) with ${\lambda }_n$ and by considering the limit, it follows

$${\displaystyle \underset{n\to \infty }{lim}\frac{1}{{\lambda }_n}\left[{\Psi }^l(h,w)-{\Psi }^l(u^0,{\omega }^0)\right]}$$

$${\displaystyle ={\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }.}$$

(4)

By Integrals (1) and (4), we obtain

$${\displaystyle {\Psi }^l(h,w)-{\Psi }^l(u^0,{\omega }^0)<0,}$$

for $n\ge N$ (see N as a natural number).

Next, considering the non-empty set (since $(u^0,{\omega }^0)\in W$ is a proper efficient solution to $\left(P\right)$)

$${\displaystyle \mathcal{M}=\left\{s\in \{1,\cdots ,p\}|{\Psi }^s(u^0,{\omega }^0)-{\Psi }^s(h,w)\le 0,(\forall )n\ge N\right\}.}$$

The Fréchet differentiability property of ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^s\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }}$ at $(u^0,{\omega }^0)\in W$, for $s\in \mathcal{M}$, gives

$${\displaystyle {\Psi }^s(h,w)-{\Psi }^s(u^0,{\omega }^0)={\int }_{\Gamma }{\lambda }_n\left[\frac{\partial {\psi }_{\nu }^s}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^s}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }}$$

$${\displaystyle +\Vert {\lambda }_n\left(\vartheta \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right),\upsilon \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right)\right)\Vert ·G^s(h,w)}$$

(5)

where the continuous function $G^s:V_{(u^0,{\omega }^0)}\to \mathbb{R}$ is defined on a neighbourhood of $(u^0,{\omega }^0)$, and fulfils ${lim}_{n\to \infty }{G}^s(h,w)=0$. Dividing the Relation (5) with ${\lambda }_n$ and bay considering the limit, it follows

$${\displaystyle \underset{n\to \infty }{lim}\frac{1}{{\lambda }_n}\left[{\Psi }^s(h,w)-{\Psi }^s(u^0,{\omega }^0)\right]}$$

$${\displaystyle ={\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^s}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^s}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }.}$$

In the following, for $n\ge N$, by considering the set $\mathcal{M}$, it results

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^s}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^s}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }\ge 0.}$$

(6)

By Integrals (2) and (6), we obtain that

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^s}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^s}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }=0}$$

for $n\ge N$ (see N as a natural number), and $s\ne l,s\in \mathcal{M}$.

By computing the limit, for $s\ne l,s\in \mathcal{M}$,

$${\displaystyle \frac{\frac{1}{{\lambda }_n}\left[{\Psi }^l(u^0,{\omega }^0)-{\Psi }^l(h,w)\right]}{\frac{1}{{\lambda }_n}\left[{\Psi }^s(h,w)-{\Psi }^s(u^0,{\omega }^0)\right]},}$$

we obtain it as ∞ as $n\to \infty$. This contradicts that $(u^0,{\omega }^0)$ is a proper efficient solution of $\left(P\right)$. □

Now, by using the variational control inequality $\left(I\right)$ of vector type, we formulate a characterization of efficient solutions to $\left(P\right)$.

Theorem 2.

Consider $(u^0,{\omega }^0)\in W$ is a solution to $\left(I\right)$ and the integral functionals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u(\varsigma \right),}$$\omega \left(\varsigma )\right)d{\varsigma }^{\nu },l=\overline{1,p}$, are invex and Fréchet differentiable at $(u^0,{\omega }^0)\in W$. Then the pair $(u^0,{\omega }^0)$ is an efficient solution to $\left(P\right)$.

Proof. 

Consider, in contrast, the pair $(u^0,{\omega }^0)\in W$ is a solution to $\left(I\right)$ and it is not an efficient solution to $\left(P\right)$. Thus, there exists $(u,\omega )\in W$, for all $l=\overline{1,p}$, such that

$${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)\le 0,}$$

(7)

with < for at least one l. As the integral functionals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p}}$, are invex and Fréchet differentiable at $(u^0,{\omega }^0)\in W$, we obtain

$${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)}$$

$${\displaystyle \ge {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },}$$

(8)

for $l=\overline{1,p}$ and ${\displaystyle (u,\omega )\in W}$. By Relations (7) and (8), for all $l=\overline{1,p}$, there exists ${\displaystyle (u,\omega )\in W}$ satisfying

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }\le 0,}$$

with < for at least one l. This contradicts ${\displaystyle (u^0,{\omega }^0)\in W}$ as a solution to $\left(I\right)$. □

Further, a sufficient condition of ${\displaystyle (u^0,{\omega }^0)\in W}$ in order to become a solution to $\left(WI\right)$ is presented.

Theorem 3.

Consider $W\subset \Theta \times \Omega$ is an invex set, $(u^0,{\omega }^0)\in W$ is a weak efficient solution to $\left(P\right)$, and the integral functionals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p}}$, are Fréchet differentiable at $(u^0,{\omega }^0)\in W$. Then $(u^0,{\omega }^0)$ solves $\left(WI\right)$.

Proof. 

By hypothesis, we have that there exists no other ${\displaystyle (u,\omega )\in W}$ such that ${\displaystyle \Psi (u,\omega )<\Psi (u^0,{\omega }^0)}$, equivalent with

$${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)<0,(\forall )l=\overline{1,p}.}$$

(9)

As $W\subset \Theta \times \Omega$ is an invex set, for $\lambda \in [0,1]$, we obtain

$$(h,w)=(u^0,{\omega }^0)+\lambda \left(\vartheta \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right),\upsilon \left(\varsigma ,u,\omega ,u^0,{\omega }^0\right)\right)\in W.$$

Therefore, by Relation (9), we obtain there exists no other ${\displaystyle (u,\omega )\in W}$ such that

$${\displaystyle {\Psi }^l(h,w)-{\Psi }^l(u^0,{\omega }^0)<0,(\forall )l=\overline{1,p}.}$$

(10)

Further, since the integral functionals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p}}$, are Fréchet differentiable at $(u^0,{\omega }^0)\in W$, following the same manner as in Theorem 1 and by considering Relation (10), it results that there exists no other ${\displaystyle (u,\omega )\in W}$ such that

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }<0,}$$

for $l=\overline{1,p}$. □

The next theorem uses the weak variational control inequality $\left(WI\right)$ of vector type to provide a description of weak efficient solutions to $\left(P\right)$.

Theorem 4.

Consider $(u^0,{\omega }^0)\in W$ is a solution to $\left(WI\right)$, and the integral functionals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l(\varsigma ,}$$u\left(\varsigma \right),\omega \left(\varsigma \right))d{\varsigma }^{\nu },l=\overline{1,p}$, are pseudoinvex and Fréchet differentiable at $(u^0,{\omega }^0)\in W$. Then $(u^0,{\omega }^0)$ is a weak efficient solution to $\left(P\right)$.

Proof. 

Consider, in contrast, that $(u^0,{\omega }^0)\in W$ is a solution to $\left(WI\right)$ and it is not a weak efficient solution to $\left(P\right)$. Thus, there exists $(u,\omega )\in W$, for all $l=\overline{1,p}$, such that

$${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)<0.}$$

As the integral functionals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p}}$, are pseudoinvex and Fréchet differentiable at $(u^0,{\omega }^0)\in W$, we obtain

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }<0,}$$

for $l=\overline{1,p}$ and ${\displaystyle (u,\omega )\in W}$. This is in contradiction with $(u^0,{\omega }^0)\in W$ as a solution to $\left(WI\right)$. □

A sufficient condition for a weak efficient solution to become an efficient solution of $\left(P\right)$ is provided below.

Theorem 5.

Consider $(u^0,{\omega }^0)\in W$ is a weak efficient solution of $\left(P\right)$, the integrals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l(\varsigma ,u\left(\varsigma \right),}$$\omega \left(\varsigma \right))d{\varsigma }^{\nu },l=\overline{1,p}$, are strictly invex and Fréchet differentiable at $(u^0,{\omega }^0)\in W$, and W is an invex set. Then $(u^0,{\omega }^0)$ is an efficient solution to $\left(P\right)$.

Proof. 

Assume, in contrast, that $(u^0,{\omega }^0)\in W$ is a weak efficient solution to $\left(P\right)$ and it is not an efficient solution. Therefore, there exists ${\displaystyle (u,\omega )\in W}$ such that

$${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)\le 0,(\forall )l=\overline{1,p},}$$

(11)

with < for at least one l. Since the integrals ${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,p}}$, are strictly invex and Fréchet differentiable at $(u^0,{\omega }^0)\in W$, we have

$${\displaystyle {\Psi }^l(u,\omega )-{\Psi }^l(u^0,{\omega }^0)>{\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },}$$

(12)

for $l=\overline{1,p}$ and ${\displaystyle (u,\omega )\ne (u^0,{\omega }^0)\in W}$. By Relations (11) and (12), for all $l=\overline{1,p}$, there exists ${\displaystyle (u,\omega )\in W}$ such that

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^l}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^l}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu }<0.}$$

Thus, ${\displaystyle (u^0,{\omega }^0)\in W}$ is not a solution to $\left(WI\right)$ and, by Theorem 3, we get ${\displaystyle (u^0,{\omega }^0)\in W}$ is not a weak efficient solution to $\left(P\right)$. □

4. Application

The next illustrative application uses the theoretical results established in this paper.

Example 2.

Find the extremized mechanical work provided by the following forces ${\displaystyle {\overline{V}}_1(e^{-\omega \left(\varsigma \right)}+\frac{1}{2},}$$e^{-\omega \left(\varsigma \right)})$ and ${\displaystyle {\overline{V}}_2\left(e^{u\left(\varsigma \right)}+\frac{1}{2},e^{u\left(\varsigma \right)}\right)}$ in order to move its application point along the curve ${\displaystyle \Gamma }$ (piece-wise differentiable), included in ${\displaystyle L={[0,1]}^2=[0,1]\times [0,1]}$ connecting ${\displaystyle {\varsigma }_1=\left(0,0\right)}$ and ${\displaystyle {\varsigma }_2=\left(1,1\right)}$, in a such way the next dynamic control system

$${\displaystyle {\displaystyle \frac{\partial u}{\partial {\varsigma }^1}}\left(\varsigma \right)={\displaystyle \frac{\partial u}{\partial {\varsigma }^2}}\left(\varsigma \right)=\omega \left(\varsigma \right),}$$

$${\displaystyle 1-e^{u\left(\varsigma \right)+u^2\left(\varsigma \right)}\le 0,e^{\omega \left(\varsigma \right)}-2+e^{{\omega }^2\left(\varsigma \right)}{\le 0,(u,\omega )|}_{\varsigma =(0,0),(1,1)}=0}$$

to be fulfilled related to $\vartheta =e^{{\left(u^0\right)}^2\left(\varsigma \right)}-e^{u^2\left(\varsigma \right)},(\forall )\varsigma \in L\backslash \{{\varsigma }_1,{\varsigma }_2\}$ and $\vartheta =0$ for $\varsigma \in \{{\varsigma }_1,{\varsigma }_2\}$, and $\upsilon =e^{{\left({\omega }^0\right)}^2\left(\varsigma \right)}-e^{{\omega }^2\left(\varsigma \right)},(\forall )\varsigma \in L\backslash \{{\varsigma }_1,{\varsigma }_2\}$ and $\upsilon =0$ for $\varsigma \in \{{\varsigma }_1,{\varsigma }_2\}$.

For solving this concrete problem, we consider $m=p=q=2,n=k=1,L=[0,1]\times [0,1]$, $\Gamma \subset L$ is a curve that connects $(0,0)$ and $(1,1)$, $u,\omega :L\to {\mathbb{R}}^{+}$ are functions (piece-wise differentiable) with ${\displaystyle \frac{\partial u}{\partial {\varsigma }^1}}\left(\varsigma \right)={\displaystyle \frac{\partial u}{\partial {\varsigma }^2}}\left(\varsigma \right)=\omega \left(\varsigma \right),1-e^{u\left(\varsigma \right)+u^2\left(\varsigma \right)}\le 0,e^{\omega \left(\varsigma \right)}-2+e^{{\omega }^2\left(\varsigma \right)}\le 0,{(u,\omega )}_{\varsigma =(0,0),(1,1)}=0$, and $\vartheta ,\upsilon :L\times {\mathbb{R}}^4\to \mathbb{R}$ are defined by: $\vartheta =e^{{\left(u^0\right)}^2\left(\varsigma \right)}-e^{u^2\left(\varsigma \right)},(\forall )\varsigma \in L\backslash \{{\varsigma }_1,{\varsigma }_2\}$ and $\vartheta =0$ for $\varsigma \in \{{\varsigma }_1,{\varsigma }_2\}$, and $\upsilon =e^{{\left({\omega }^0\right)}^2\left(\varsigma \right)}-e^{{\omega }^2\left(\varsigma \right)},(\forall )\varsigma \in L\backslash \{{\varsigma }_1,{\varsigma }_2\}$ and $\upsilon =0$ for $\varsigma \in \{{\varsigma }_1,{\varsigma }_2\}$. Furthermore, we introduce the next closed 1-forms

$${\displaystyle {\psi }_{\nu }=\left({\psi }_{\nu }^1,{\psi }_{\nu }^2\right):L\times {\mathbb{R}}^2\to {\mathbb{R}}^2,\nu =\overline{1,2},}$$

defined as

$${\displaystyle {\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)=\left(e^{-\omega \left(\varsigma \right)}+\frac{1}{2},e^{-\omega \left(\varsigma \right)}\right),}$$

$${\displaystyle {\psi }_{\nu }^2\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)=\left(e^{u\left(\varsigma \right)}+\frac{1}{2},e^{u\left(\varsigma \right)}\right).}$$

Now, we formulate the associated constrained multi-objective optimization problem

$${\displaystyle \left(P1\right)\underset{(u,\omega )}{min}{\int }_{\Gamma }{\psi }_{\nu }\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },}$$

with

$${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }=\left({\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },{\int }_{\Gamma }{\psi }_{\nu }^2\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }\right)}$$

$${\displaystyle =\left({\Psi }^1(u,\omega ),{\Psi }^2(u,\omega )\right)}$$

and subject it to the constraints mentioned above. We find that the integral functional

$${\displaystyle \Psi (u,\omega )={\int }_{\Gamma }{\psi }_{\nu }\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }={\int }_{\Gamma }{\psi }_1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^1+{\int }_{\Gamma }{\psi }_2\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^2}$$

is Fréchet differentiable at $(u^0,{\omega }^0)=(0,0)$ and each integral functional

$${\displaystyle {\int }_{\Gamma }{\psi }_{\nu }^l\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu },l=\overline{1,2},}$$

is invex at $(u^0,{\omega }^0)=(0,0)$ (related to ϑ and υ). Furthermore, we note that $(u^0,{\omega }^0)=(0,0)$ is a solution to $\left(I\right)$:

$${\displaystyle ({\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^1}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^1}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu },}$$

$${\displaystyle {\int }_{\Gamma }\left[\frac{\partial {\psi }_{\nu }^2}{\partial u}\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\vartheta +\frac{\partial {\psi }_{\nu }^2}{\partial \omega }\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)\upsilon \right]d{\varsigma }^{\nu })}$$

$${\displaystyle =\left({\int }_{\Gamma }{e}^{-\omega \left(\varsigma \right)}\left(e^{{\omega }^2\left(\varsigma \right)}-1\right)\left(d{\varsigma }^1+d{\varsigma }^2\right),{\int }_{\Gamma }{e}^{u\left(\varsigma \right)}\left(1-e^{u^2\left(\varsigma \right)}\right)\left(d{\varsigma }^1+d{\varsigma }^2\right)\right)\nleqq (0,0),}$$

for all $u,\omega :L\to {\mathbb{R}}^{+}$ (piece-wise differentiable functions). As a consequence, by using Theorem 2, it follows that $(u^0,{\omega }^0)$ is an efficient solution to $\left(P1\right)$. Moreover, by computation, it results in

$${\displaystyle {\Psi }^1(u,\omega )-{\Psi }^1(u^0,{\omega }^0)={\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }-{\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)d{\varsigma }^{\nu }}$$

$${\displaystyle =\left[{\int }_{\Gamma }\left(e^{-\omega \left(\varsigma \right)}+\frac{1}{2}\right)d{\varsigma }^1+e^{-\omega \left(\varsigma \right)}d{\varsigma }^2\right]-\left[{\int }_{\Gamma }\left(1+\frac{1}{2}\right)d{\varsigma }^1+1d{\varsigma }^2\right]}$$

$${\displaystyle ={\int }_{\Gamma }\left(e^{-\omega \left(\varsigma \right)}-1\right)d{\varsigma }^1+\left(e^{-\omega \left(\varsigma \right)}-1\right)d{\varsigma }^2<0,}$$

for all $\omega :L\to {\mathbb{R}}^{+}\backslash \left\{0\right\}$ (piece-wise differentiable functions). In addition,

$${\displaystyle {\Psi }^2(u,\omega )-{\Psi }^2(u^0,{\omega }^0)={\int }_{\Gamma }{\psi }_{\nu }^2\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }-{\int }_{\Gamma }{\psi }_{\nu }^2\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)d{\varsigma }^{\nu }}$$

$${\displaystyle =\left[{\int }_{\Gamma }\left(e^{u\left(\varsigma \right)}+\frac{1}{2}\right)d{\varsigma }^1+e^{u\left(\varsigma \right)}d{\varsigma }^2\right]-\left[{\int }_{\Gamma }\left(1+\frac{1}{2}\right)d{\varsigma }^1+1d{\varsigma }^2\right]}$$

$${\displaystyle ={\int }_{\Gamma }\left(e^{u\left(\varsigma \right)}-1\right)d{\varsigma }^1+\left(e^{u\left(\varsigma \right)}-1\right)d{\varsigma }^2>0,}$$

for all $u:L\to {\mathbb{R}}^{+}\backslash \left\{0\right\}$ (piece-wise differentiable functions). Since, for $M=1$ and $(u^0,{\omega }^0)=(0,0)$, the following inequality

$${\displaystyle {\Psi }^1(u^0,{\omega }^0)-{\Psi }^1(u,\omega )={\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)d{\varsigma }^{\nu }-{\int }_{\Gamma }{\psi }_{\nu }^1\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }}$$

$${\displaystyle \le M\left({\Psi }^2(u,\omega )-{\Psi }^2(u^0,{\omega }^0)={\int }_{\Gamma }{\psi }_{\nu }^2\left(\varsigma ,u\left(\varsigma \right),\omega \left(\varsigma \right)\right)d{\varsigma }^{\nu }-{\int }_{\Gamma }{\psi }_{\nu }^2\left(\varsigma ,u^0\left(\varsigma \right),{\omega }^0\left(\varsigma \right)\right)d{\varsigma }^{\nu }\right),}$$

holds, we find $(u^0,{\omega }^0)=(0,0)$ is a proper efficient solution to $\left(P1\right)$.

5. Conclusions

In this paper, we have established some connections between the solution sets of a new non-linear optimization problem and the associated variational inequality. More precisely, to establish the principal results, we have used the notions of invex set, invexity (or pseudoinvexity), and Fréchet differentiability of some integral functionals of curvilinear type. In addition, to justify the effectiveness of this work, illustrative examples have been presented. As a further research direction, we mention the reformulation of these results by taking into account the notion of variational derivatives.