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Exponentially EE-preinvex and EE-invex functions in mathematical programming

Najeeb Abdulaleem
Department of Mathematics, Mahrah University, Al-Mahrah, Yemen
Faculty of Mathematics and Computer Science, University of Łódź,
Banacha 22, 90-238 Łódź, Poland
Department of Mathematics, Hadhramout University
Al-Mahrah, Yemen
e-mail: nabbas985@gmail.com
Abstract

In this paper, we introduce a new concept of generalized convexity for EE-differentiable vector optimization problems. Namely, the notion of exponentially EE-invexity is defined. Further, some properties and results of exponentially EE-invex functions are studied. The sufficient optimality conditions are derived under appropriate (generalized) exponentially EE-invexity hypotheses. To exemplify the application of our proposed concept, we have included an appropriate example.

Key Words: Exponentially EE-invex function; EE-differentiable vector optimization; optimality conditions.

AMS Classification: 90C26, 90C29, 90C30, 90C46

1 Introduction

Multiobjective optimization serves as a valuable mathematical framework to address real-world challenges involving conflicting objectives found in engineering, economics, and decision-making. However, many studies have traditionally assumed convexity in these problems (see, for example, [23], [39], [37]). To broaden the scope beyond convexity assumptions in theorems related to optimality conditions and duality, various concepts of generalized convexity have been introduced (see, for example, [1], [2], [6], [7], [8], [9], [10], [21], and others). One particularly beneficial generalization is invexity, as introduced by Hanson [16]. This involves considering differentiable functions, denoted as f:M→R,f:M\rightarrow R, where MM is a subset of Rn.R^{n}. For these functions, Hanson proposes the existence of an n-dimensional vector function η:M×M→Rn\eta:M\times M\rightarrow R^{n} such that, for all x,u∈M,x,u\in M, the inequality

f​(x)−f​(u)≥∇f​(u)​η​(x,u)f(x)-f(u)\geq\nabla f(u)\eta(x,u)

holds. Ben Israel and Mond [13], Hanson and Mond [17], Craven and Glover [14], along with numerous others, have explored various aspects, applications, and broader concepts related to these functions.

Youness [42] initially introduced the concept of EE-convexity. Recently, there has been considerable interest in expanding the idea of EE-convexity to novel classes of generalized EE-convex functions, and researchers have investigated their characteristics (see, for example, [4], [5], [8], [9], [15], [18], [19], [20], [22], [27], [32], [36], [38], [40], [41], [44], and others). Antczak [43] discussed the applications of exponentially convex functions in the mathematical programming and optimization theory. Following the research by Hanson and Craven, various forms of differentiable functions have emerged, aiming to extend the concept of invex functions. One such function involves exponential functions (see, for example, [3], [10], [28], [29], [30], [31], [33], [34], and others).

In this paper, a new class of nonconvex EE-differentiable vector optimization problems with both inequality and equality constraints is considered in which the involved functions are exponentially (generalized) EE-invex. Therefore, the concepts of exponentially pseudo-EE-invex and exponentially quasi-EE-invex functions for EE-differentiable vector optimization problems are introduced. Furthermore, we derive the sufficiency of the so-called EE-Karush-Kuhn-Tucker optimality conditions for the considered EE-differentiable vector optimization problem under appropriate exponentially (generalized) EE-invexity hypotheses. This result is illustrated by suitable example of smooth multiobjective optimization problem in which the involved functions are exponentially (generalized) EE-invex functions.

2 Preliminaries

Throughout this paper, the following conventions vectors x=(x1,x2,…,xn)Tx=\left(x_{1},x_{2},...,x_{n}\right)^{T} and y=(y1,y2,…,yn)Ty=\left(y_{1},y_{2},...,y_{n}\right)^{T} in RnR^{n} will be followed:

(i)    x=yx=y  if and only if xi=yix_{i}=y_{i} for all i=1,2,…,ni=1,2,...,n;

(ii)   x>yx>y  if and only if xi>yix_{i}>y_{i} for all i=1,2,…,ni=1,2,...,n;

(iii)  x≧yx\geqq y  if and only if xi≧yix_{i}\geqq y_{i} for all i=1,2,…,ni=1,2,...,n;

(iv)  x≥yx\geq y  if and only if xi≧yix_{i}\geqq y_{i} for all i=1,2,…,ni=1,2,...,n but x≠yx\neq y;

(v)   x≯yx\ngtr y  is the negation of x>y.x>y.

Definition 1

[4] Let E:Rn→RnE:R^{n}\rightarrow R^{n}. A set M⊆RnM\subseteq R^{n} is said to be an EE-invex set if and only if there exists a vector-valued function η:M×M→Rn\eta:M\times M\rightarrow R^{n} such that the relation

E​(x0)+τ​η​(E​(x),E​(x0))∈ME\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\in M

holds for all x,x0∈Mx,x_{0}\in M and any τ∈[0,1]\tau\in\left[0,1\right].

Remark 2

If η\eta is a vector-valued function defined by η​(z,y)=z−y\eta(z,y)=z-y, then the definition of an EE-invex set reduces to the definition of an EE-convex set (see Youness [42]).

Definition 3

Let E:Rn→RnE:R^{n}\rightarrow R^{n}. A function f:M→Rf:M\rightarrow R is said to be EE-preinvex on MM if and only if the following inequality

f​(E​(x0)+τ​η​(E​(x),E​(x0)))≦τ​f​(E​(x))+(1−τ)​f​(E​(x0))f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)\leqq\tau f\left(E\left(x\right)\right)+\left(1-\tau\right)f\left(E\left(x_{0}\right)\right) (1)

holds for all x,x0∈Mx,x_{0}\in M and any τ∈[0,1]\tau\in\left[0,1\right].

Now, we introduce a new concept of the exponentially EE-preinvex function.

Definition 4

Let E:Rn→Rn.E:R^{n}\rightarrow R^{n}. A function f:M→Rf:M\rightarrow R is said to be exponentially EE-preinvex on MM if and only if the following inequality

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))≦τ​ef​(E​(x))+(1−τ)​ef​(E​(x0))e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}\leqq\tau e^{f\left(E\left(x\right)\right)}+\left(1-\tau\right)e^{f\left(E\left(x_{0}\right)\right)} (2)

holds for all x,x0∈Mx,x_{0}\in M and any τ∈[0,1]\tau\in\left[0,1\right].

In other words, (2) is equivalent to the fact that the following inequalities

f​(E​(x0)+τ​η​(E​(x),E​(x0)))≦log⁡[τ​ef​(E​(x))+(1−τ)​ef​(E​(x0))]f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)\leqq\log\left[\tau e^{f\left(E\left(x\right)\right)}+\left(1-\tau\right)e^{f\left(E\left(x_{0}\right)\right)}\right] (3)

holds for all x,x0∈Mx,x_{0}\in M and any τ∈[0,1]\tau\in\left[0,1\right].

Definition 5

Let E:Rn→Rn.E:R^{n}\rightarrow R^{n}. A function f:M→Rf:M\rightarrow R is said to be strictly exponentially EE-preinvex on MM if and only if the following inequality

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))<τ​ef​(E​(x))+(1−τ)​ef​(E​(x0))e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}<\tau e^{f\left(E\left(x\right)\right)}+\left(1-\tau\right)e^{f\left(E\left(x_{0}\right)\right)} (4)

holds for all x,x0∈Mx,x_{0}\in M, E​(x)≠E​(x0)E(x)\neq E(x_{0}), and any τ∈(0,1)\tau\in\left(0,1\right).

Definition 6

Let E:Rn→Rn.E:R^{n}\rightarrow R^{n}. A function f:M→Rf:M\rightarrow R is said to be exponentially quasi-EE-preinvex on MM if and only if the following inequality

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))≦max⁡{ef​(E​(x)),ef​(E​(x0))}e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}\leqq\;\max\;\{e^{f(E(x))},e^{f(E(x_{0}))}\} (5)

holds for all x,x0∈Mx,x_{0}\in M and any τ∈[0,1]\tau\in\left[0,1\right].

Definition 7

Let E:Rn→Rn.E:R^{n}\rightarrow R^{n}. A function f:M→Rf:M\rightarrow R is said to be strictly exponentially quasi-EE-preinvex on MM if and only if the following inequality

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))<max⁡{ef​(E​(x)),ef​(E​(x0))}e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}<\;\max\;\{e^{f(E(x))},e^{f(E(x_{0}))}\} (6)

holds for all x,x0∈M​(x≠x0),x,x_{0}\in M(x\neq x_{0}), and any τ∈(0,1)\tau\in\left(0,1\right).

Note that every exponentially preinvex function is exponentially quasi-EE-preinvex and every exponentially EE-preinvex function is exponentially quasi-EE-preinvex. However, the converse is not true.

Definition 8

Let E:Rn→Rn.E:R^{n}\rightarrow R^{n}. We define the EE-epigraph of an exponentially function f:M→Rf:M\rightarrow R as follows

epiE​(f)={(E​(x),Υ)∈M×R:ef​(E​(x))≤Υ}.\text{epi${}_{E}$}(f)=\{(E(x),\Upsilon)\in M\times R:e^{f(E(x))}\leq\Upsilon\}.
Theorem 9

Let E:Rn→Rn,E:R^{n}\rightarrow R^{n}, MM be a nonempty EE-invex subset of RnR^{n} and f:M→Rf:M\rightarrow R be an exponentially function. Then ff is EE-preinvex if and only if its EE-epigraph is an EE-invex set.

Proof. Let ff be an exponentially EE-preinvex function. Then for any (E​(x),Υ)(E(x),\Upsilon) and (E​(x0),Γ)∈(E(x_{0}),\Gamma)\in epiE(ff), we have ef​(E​(x))≦Υ,e^{f(E(x))}\leqq\Upsilon, ef​(E​(x0))≦Γe^{f(E(x_{0}))}\leqq\Gamma . Also, for each τ∈[0,1]\tau\in[0,1], we have

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))≦τ​ef​(E​(x))+(1−τ)​ef​(E​(x0))≦τ​Υ+(1−τ)​Γ.\begin{split}e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}&\leqq\tau e^{f\left(E\left(x\right)\right)}+\left(1-\tau\right)e^{f\left(E\left(x_{0}\right)\right)}\\ &\leqq\tau\Upsilon+\left(1-\tau\right)\Gamma.\end{split} (7)

Thus, (E​(x0)+τ​η​(E​(x),E​(x0)),τ​Υ+(1−τ)​Γ)∈(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right),\tau\Upsilon+(1-\tau)\Gamma)\in epiE(ff). Hence, epiE(ff) is EE-invex.

Conversely, let epiE(ff) be an EE-invex set, (E​(x),ef​(E​(x)))∈(E(x),e^{f(E(x))})\in epiE(ff) and (E​(x0),ef​(E​(x0)))∈(E(x_{0}),e^{f(E(x_{0}))})\in epiE(ff). Then for each E​(x),E​(x0)∈ME(x),E(x_{0})\in M and each λ∈[0,1]\lambda\in[0,1], we have

(E​(x0)+τ​η​(E​(x),E​(x0)),τ​ef​(E​(x))+(1−τ)​ef​(E​(x0)))∈epiE​(f)(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right),\tau e^{f(E\left(x\right))}+(1-\tau)e^{f(E\left(x_{0}\right))})\in\text{epi}_{E}(f) (8)

and thus,

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))≦τ​ef​(E​(x))+(1−τ)​ef​(E​(x0)).e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}\leqq\tau e^{f\left(E\left(x\right)\right)}+\left(1-\tau\right)e^{f\left(E\left(x_{0}\right)\right)}. (9)

Hence, ff is exponentially EE-preinvex on M.M.   

We now give the characterization of an exponentially quasi-EE-preinvex function in terms of EE-preinvexity of its level sets.

Theorem 10

Let E:Rn→RnE:R^{n}\rightarrow R^{n} and MM be a nonempty EE-invex subset of Rn.R^{n}. A function f:M→Rf:M\rightarrow R is exponentially quasi-EE-preinvex function if and only if the level sets LE​(f,Υ)L_{E}(f,\Upsilon) are EE-invex for all Υ∈R.\Upsilon\in R.

Proof. Let ff be an exponentially quasi-EE-preinvex function, and for Υ∈R\Upsilon\in R, let E​(x),E​(x0)∈LE​(f,Υ)E(x),E(x_{0})\in L_{E}(f,\Upsilon). Then ef​(E​(x))≦Υ,e^{f(E(x))}\leqq\Upsilon, ef​(E​(x0))≦Υe^{f(E(x_{0}))}\leqq\Upsilon. Since ff is an exponentially quasi-EE-preinvex function, for each τ∈[0,1]\tau\in[0,1], we have

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))≦max⁡{ef​(E​(x)),ef​(E​(x0))}≦Υe^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}\leqq\;\max\;\{e^{f(E(x))},e^{f(E(x_{0}))}\}\leqq\Upsilon

that is, E​(x0)+τ​η​(E​(x),E​(x0))∈LE​(f,Υ)E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\in L_{E}(f,\Upsilon) for each τ∈[0,1].\tau\in[0,1]. Hence, LE​(f,Υ)L_{E}(f,\Upsilon) is EE-invex.

Conversely, let E​(x),E​(x0)∈ME(x),E(x_{0})\in M and Υ¯=max⁡{ef​(E​(x)),ef​(E​(x0))}.\overline{\Upsilon}=\max\;\{e^{f(E(x))},e^{f(E(x_{0}))}\}. Then E​(x),E​(x0)∈LE​(f,Υ¯)E(x),E(x_{0})\in L_{E}(f,\overline{\Upsilon}), and by EE-invexity of LE​(f,Υ¯)L_{E}(f,\overline{\Upsilon}), we have E​(x0)+τ​η​(E​(x),E​(x0))∈LE​(f,Υ¯)E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\in L_{E}(f,\overline{\Upsilon}) for each τ∈[0,1].\tau\in[0,1]. Thus for each τ∈[0,1]\tau\in[0,1],

ef​(E​(x0)+τ​η​(E​(x),E​(x0)))≦Υ¯=max⁡{ef​(E​(x)),ef​(E​(x0))}.e^{f\left(E\left(x_{0}\right)+\tau\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\right)}\leqq\overline{\Upsilon}=\max\;\{e^{f(E(x))},e^{f(E(x_{0}))}\}.

This completes the proof.   

Definition 11

[26] Let E:Rn→RnE:R^{n}\rightarrow R^{n} and f:M→Rf:M\rightarrow R be a (not necessarily) differentiable function at a given point x0∈Mx_{0}\in M. It is said that ff is an EE-differentiable function at x0x_{0} if and only if f∘Ef\circ E is a differentiable function at x0x_{0} (in the usual sense) and, moreover,

(f∘E)​(x)=(f∘E)​(x0)+∇(f∘E)⁡(x0)​(x−x0)+θ​(x0,x−x0)​‖x−x0‖,\left(f\circ E\right)\left(x\right)=\left(f\circ E\right)\left(x_{0}\right)+\nabla\left(f\circ E\right)\left(x_{0}\right)\left(x-x_{0}\right)+\theta\left(x_{0},x-x_{0}\right)\left\|x-x_{0}\right\|,

where θ​(x0,x−x0)→0\theta\left(x_{0},x-x_{0}\right)\rightarrow 0 as x→x0x\rightarrow x_{0}.

Now, we introduce a new concept of generalized exponentially convexity for EE-differentiable functions.

Definition 12

Let E:Rn→Rn,E:R^{n}\rightarrow R^{n}, MM be a nonempty open EE-invex subset of RnR^{n} and f:M→Rf:M\rightarrow R be an EE-differentiable function at x0x_{0} on MM. It is said that ff is exponentially EE-invex function at x0x_{0} on MM if, there exist η:M×M→Rn\eta:M\times M\rightarrow R^{n} such that, for all x∈Mx\in M, the inequality

ef​(E​(x))−ef​(E​(x0))≧∇f​(E​(x0))​ef​(E​(x0))​η​(E​(x),E​(x0))​ e^{f\left(E\left(x\right)\right)}-e^{f\left(E\left(x_{0}\right)\right)}\geqq\nabla f\left(E\left(x_{0}\right)\right)e^{f\left(E\left(x_{0}\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\text{ \ } (10)

holds. If inequality (11) holds for any x0x_{0} on MM, then ff is exponentially EE-invex function on M.M.

Definition 13

Let E:Rn→Rn,E:R^{n}\rightarrow R^{n}, MM be a nonempty open EE-invex subset of RnR^{n} and f:M→Rf:M\rightarrow R be an EE-differentiable function at x0x_{0} on MM. It is said that ff is exponentially strictly EE-invex function at x0x_{0} on MM if, there exist η:M×M→Rn\eta:M\times M\rightarrow R^{n} such that, for all x∈Mx\in M, x≠x0x\neq x_{0} the inequality

ef​(E​(x))−ef​(E​(x0))>∇f​(E​(x0))​ef​(E​(x0))​η​(E​(x),E​(x0))​ e^{f\left(E\left(x\right)\right)}-e^{f\left(E\left(x_{0}\right)\right)}>\nabla f\left(E\left(x_{0}\right)\right)e^{f\left(E\left(x_{0}\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\text{ \ } (11)

holds. If inequality (11) holds for any x0x_{0} on M,M, x≠x0x\neq x_{0}, then ff is exponentially strictly EE-invex function on M.M.

Now, we give the necessary condition for an EE-differentiable exponentially EE-invex function.

Proposition 14

Let E:Rn→Rn,E:R^{n}\rightarrow R^{n}, f:M→Rf:M\rightarrow R be an EE-differentiable exponentially EE-invex function (an EE-differentiable exponentially strictly EE-invex function) on M.M. Then, the following inequality

(∇f​(E​(x))​ef​(E​(x))−∇f​(E​(x0))​ef​(E​(x0)))​η​(E​(x),E​(x0))≧0​ ​(>)(\nabla f\left(E\left(x\right)\right)e^{f\left(E\left(x\right)\right)}-\nabla f\left(E\left(x_{0}\right)\right)e^{f\left(E\left(x_{0}\right)\right)})\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\geqq 0\text{ \ }\left(>\right) (12)

holds for all x,x0∈Mx,x_{0}\in M.

Proof. Since ff is an EE-differentiable exponentially EE-invex function, by Definition 12, the following inequalities

ef​(E​(x))−ef​(E​(x0))≧∇f​(E​(x0))​ef​(E​(x0))​η​(E​(x),E​(x0)), ​(>)e^{f\left(E\left(x\right)\right)}-e^{f\left(E\left(x_{0}\right)\right)}\geqq\nabla f\left(E\left(x_{0}\right)\right)e^{f\left(E\left(x_{0}\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right),\text{ \ }\left(>\right) (13)
ef​(E​(x0))−ef​(E​(x))≧∇f​(E​(x))​ef​(E​(x))​η​(E​(x),E​(x0))​ ​(>)e^{f\left(E\left(x_{0}\right)\right)}-e^{f\left(E\left(x\right)\right)}\geqq\nabla f\left(E\left(x\right)\right)e^{f\left(E\left(x\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\text{ \ }\left(>\right) (14)

hold for all x,x0∈M.x,x_{0}\in M. By adding above inequalities, we get that the following inequality

(∇f​(E​(x))​ef​(E​(x))−∇f​(E​(x0))​ef​(E​(x0)))​η​(E​(x),E​(x0))≧0​ ​(>)(\nabla f\left(E\left(x\right)\right)e^{f\left(E\left(x\right)\right)}-\nabla f\left(E\left(x_{0}\right)\right)e^{f\left(E\left(x_{0}\right)\right)})\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\geqq 0\text{ \ }\left(>\right) (15)

holds for all x,x0∈Mx,x_{0}\in M.   

Now, we introduce the definitions of generalized exponentially EE-invex functions. Namely, the following result gives the characterization of an exponentially quasi-EE-invex function in terms of its gradient.

Definition 15

Let E:Rn→RnE:R^{n}\rightarrow R^{n} and f:M→Rf:M\rightarrow R be an EE-differentiable function at x0∈Mx_{0}\in M. The function ff is said to be an exponentially quasi-EE-invex at x0x_{0} on MM if the relation

ef​(E​(x))≦ef​(E​(x0))⟹∇(f∘E)⁡(x0)​ef​(E​(x0))​η​(E​(x),E​(x0))≦0e^{f\left(E\left(x\right)\right)}\leqq e^{f\left(E\left(x_{0}\right)\right)}\Longrightarrow\nabla\left(f\circ E\right)\left(x_{0}\right)e^{f\left(E\left(x_{0}\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)\leqq 0 (16)

holds for each x∈Mx\in M. If (16) is satisfied for every x0∈Mx_{0}\in M, then the function ff is said to be an exponentially quasi-EE-invex on MM.

Definition 16

Let E:Rn→RnE:R^{n}\rightarrow R^{n} and f:M→Rf:M\rightarrow R be an EE-differentiable function at x0∈Mx_{0}\in M. The function ff is said to be an exponentially pseudo-EE-invex at x0x_{0} on MM if the relation

ef​(E​(x))<ef​(E​(x0))⟹∇(f∘E)⁡(x0)​ef​(E​(x0))​η​(E​(x),E​(x0))<0e^{f\left(E\left(x\right)\right)}<e^{f\left(E\left(x_{0}\right)\right)}\Longrightarrow\nabla\left(f\circ E\right)\left(x_{0}\right)e^{f\left(E\left(x_{0}\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)<0 (17)

holds for each x∈Mx\in M. If (17) is satisfied for every x0∈Mx_{0}\in M, then the function ff is said to be an exponentially pseudo-EE-invex on MM.

Definition 17

Let E:Rn→RnE:R^{n}\rightarrow R^{n} and f:M→Rf:M\rightarrow R be an EE-differentiable function at x0∈Mx_{0}\in M. The function ff is said to be an exponentially strictly pseudo-EE-invex at x0x_{0} on MM if the relation

ef​(E​(x))≦ef​(E​(x0))⟹∇(f∘E)⁡(x0)​ef​(E​(x0))​η​(E​(x),E​(x0))<0e^{f\left(E\left(x\right)\right)}\leqq e^{f\left(E\left(x_{0}\right)\right)}\Longrightarrow\nabla\left(f\circ E\right)\left(x_{0}\right)e^{f\left(E\left(x_{0}\right)\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)<0 (18)

holds for each x,x0∈Mx,x_{0}\in M, x≠x0x\neq x_{0}. If (18) is satisfied for every x0∈Mx_{0}\in M, x≠x0x\neq x_{0}, then the function ff is said to be an exponentially strictly pseudo-EE-invex on MM.

Now, we present an example of such an exponentially pseudo-EE-invex function which is not exponentially EE-invex.

Example 18

Let E:R→RE:R\rightarrow R and f:R→Rf:R\rightarrow R be a nondifferentiable function at x=−3x=-3 defined by f​(x)=(x+3)13,f(x)=(x+3)^{\frac{1}{3}}, E​(x)=(x+3)9−3E(x)=(x+3)^{9}-3 and η​(E​(x),E​(x0))=x−x0.\eta(E(x),E(x_{0}))=x-x_{0}. The function (f∘E)​(x)=(x+3)3\left(f\circ E\right)(x)=(x+3)^{3} is a differentiable function at x=−3x=-3, thus ff is an EE-differentiable function at x=−3x=-3. Now we show that ff is exponentially pseudo-EE-invex on RR. Let x,x0∈Rx,x_{0}\in R and τ∈[0,1]\tau\in[0,1], and assume that e(f∘E)​(x)<e(f∘E)​(x0)e^{\left(f\circ E\right)\left(x\right)}<e^{\left(f\circ E\right)\left(x_{0}\right)}. We have e(f∘E)​(x)=e(x+3)3<e(x0+3)3=e(f∘E)​(x0)e^{\left(f\circ E\right)\left(x\right)}=e^{(x+3)^{3}}<e^{(x_{0}+3)^{3}}=e^{\left(f\circ E\right)\left(x_{0}\right)}. This inequality implies that x<x0x<x_{0}. Hence, we have

∇(f∘E)⁡(x0)​e(f∘E)​(x0)​η​(E​(x),E​(x0))=3​(x0+3)2​e(x0+3)3​(x−x0)<0.\nabla\left(f\circ E\right)\left(x_{0}\right)e^{\left(f\circ E\right)\left(x_{0}\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)=3(x_{0}+3)^{2}e^{(x_{0}+3)^{3}}(x-x_{0})<0.

Therefore, by Definition 16, ff is exponentially pseudo-EE-invex on RR.
Further, it can be shown that ff is also exponentially quasi-EE-invex on RR. Assume that e(f∘E)​(x)≦e(f∘E)​(x0)e^{\left(f\circ E\right)\left(x\right)}\leqq e^{\left(f\circ E\right)\left(x_{0}\right)}. We have e(f∘E)​(x)=e(f∘E)​(x0)=e(x+3)3≦e(x0+3)3e^{\left(f\circ E\right)\left(x\right)}=e^{\left(f\circ E\right)\left(x_{0}\right)}=e^{(x+3)^{3}}\leqq e^{(x_{0}+3)^{3}}. This inequality implies that x≦x0x\leqq x_{0}. Hence, we have

∇(f∘E)⁡(x0)​e(f∘E)​(x0)​η​(E​(x),E​(x0))=3​(x0+3)2​e(x0+3)3​(x−x0)≦0.\nabla\left(f\circ E\right)\left(x_{0}\right)e^{\left(f\circ E\right)\left(x_{0}\right)}\eta\left(E\left(x\right),E\left(x_{0}\right)\right)=3(x_{0}+3)^{2}e^{(x_{0}+3)^{3}}(x-x_{0})\leqq 0.

Therefore, by Definition 15, ff is exponentially quasi-EE-invex on RR.

Definition 19

Let E:Rn→RnE:R^{n}\rightarrow R^{n}. It said that x¯∈Rn\overline{x}\in R^{n} is a global EE-minimizer of f:M→Rf:M\rightarrow R if the inequality

f​(E​(x¯))≦f​(E​(x))f\left(E\left(\overline{x}\right)\right)\leqq f\left(E\left(x\right)\right)

holds for all x∈Mx\in M.

Proposition 20

Let E:Rn→RnE:R^{n}\rightarrow R^{n} and f:M→Rf:M\rightarrow R be an EE-differentiable exponentially EE-invex function on M.M. If ∇f​(E​(x¯))=0\nabla f\left(E(\overline{x})\right)=0, then x¯\overline{x} is an EE-minimizer of ff.

Proof. Let E:Rn→RnE:R^{n}\rightarrow R^{n}. Further, assume that f:M→Rf:M\rightarrow R is an EE-differentiable exponentially EE-invex function on M.M. Hence, by Definition 12, the inequality

ef​(E​(x))−ef​(E​(x¯))≧∇f​(E​(x¯))​ef​(E​(x¯))​η​(E​(x),E​(x¯))e^{f\left(E\left(x\right)\right)}-e^{f\left(E\left(\overline{x}\right)\right)}\geqq\nabla f\left(E\left(\overline{x}\right)\right)e^{f\left(E\left(\overline{x}\right)\right)}\eta\left(E\left(x\right),E\left(\overline{x}\right)\right) (19)

holds for all x∈M.x\in M. Since ∇f​(E​(x¯))=0\nabla f\left(E(\overline{x})\right)=0 and (19), therefore, we have that the relation

ef​(E​(x))−ef​(E​(x¯))≧0e^{f(E(x))}-e^{f(E(\overline{x}))}\geqq 0 (20)

implies that the inequality

f​(E​(x¯))≦f​(E​(x))f\left(E\left(\overline{x}\right)\right)\leqq f\left(E\left(x\right)\right)

holds for all x∈Mx\in M. This means, by Definition 19, that x¯\overline{x} is an EE-minimizer of ff.   

Proposition 21

Let E:Rn→RnE:R^{n}\rightarrow R^{n} be an operator and f:M→Rf:M\rightarrow R be an EE-differentiable exponentially pseudo-EE-invex function on M.M. If ∇f​(E​(x¯))=0\nabla f\left(E(\overline{x})\right)=0, then x¯\overline{x} is an EE-minimizer of ff.

Proof. The proof of this proposition follows from Definitions 16 and 19.   

3 EE-optimality conditions for multiobjective programming problems


Consider the following multiobjective programming problem (VP):

minimize ​f​(x)=(f1​(x),…,fp​(x))subject to ​gk​(x)≦0​, ​k∈K={1,…,m},hj​(x)=0​, ​j∈J={1,…,q},(VP)\begin{array}[]{c}\text{minimize }f(x)=\left(f_{1}\left(x\right),...,f_{p}\left(x\right)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{subject to }g_{k}(x)\leqq 0\text{, \ }k\in K=\left\{1,...,m\right\},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \hskip 48.42076pth_{j}(x)=0\text{, \ }j\in J=\left\{1,...,q\right\},\vskip 6.0pt plus 2.0pt minus 2.0pt\end{array}\qquad\text{(VP)}

where fi:Rn→Rf_{i}:R^{n}\rightarrow R(i∈I={1,2,…,p})(i\in I=\{1,2,...,p\}), gk:Rn→Rg_{k}:R^{n}\rightarrow R (k∈K)(k\in K) and hj:Rn→Rh_{j}:R^{n}\rightarrow R (j∈J)(j\in J), are EE-differentiable functions defined on Rn.R^{n}. Let

Ω:={x∈Rn:gk​(x)≦0​, ​k∈K​, ​hj​(x)=0​, ​j∈J}\Omega:=\left\{x\in R^{n}:g_{k}(x)\leqq 0\text{, \ }k\in K\text{, }h_{j}(x)=0\text{, \ }j\in J\right\}

be the set of all feasible solutions of (VP). Further, by K​(x),K\left(x\right), the set of inequality constraint indices that are active at a feasible solution xx, that is, K​(x)={k∈K:gk​(x)=0}.K\left(x\right)=\left\{k\in K:g_{k}(x)=0\right\}.

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

minimize ​f​(E​(x))=(f1​(E​(x)),…,fp​(E​(x)))subject to ​gk​(E​(x))≦0​, ​k∈K={1,…,m},hj​(E​(x))=0​, ​j∈J={1,…,q}.(VPE)\begin{array}[]{c}\text{minimize }f(E(x))=\left(f_{1}\left(E(x)\right),...,f_{p}\left(E(x)\right)\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{subject to }g_{k}(E(x))\leqq 0\text{, \ }k\in K=\left\{1,...,m\right\},\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \hskip 48.42076pth_{j}(E(x))=0\text{, \ }j\in J=\left\{1,...,q\right\}.\vskip 6.0pt plus 2.0pt minus 2.0pt\end{array}\qquad\text{(VP${}_{E}$)}

Let ΩE:={x∈Rn:gk​(E​(x))≦0​, ​k∈K​, ​hj​(E​(x))=0​, ​j∈J}\Omega_{E}:=\left\{x\in R^{n}:g_{k}(E(x))\leqq 0\text{, \ }k\in K\text{, }h_{j}(E(x))=0\text{, \ }j\in J\right\} be the set of all feasible solutions of (VPE).

Definition 22

A feasible point E​(y¯)E(\overline{y}) is said to be a weak EE-Pareto (weakly EE-efficient) solution for (VP) if and only if there exists no feasible point E​(x)E(x) such that

f​(E​(x))<f​(E​(y¯))​.f(E(x))<f(E(\overline{y}))\text{.}
Definition 23

A feasible point E​(y¯)E(\overline{y}) is said to be an EE-Pareto (EE-efficient) solution for (VP) if and only if there exists no feasible point E​(x)E(x) such that

f​(E​(x))≤f​(E​(y¯))​.f(E(x))\leq f(E(\overline{y}))\text{.}
Remark 24

Let E​(y¯)∈ΩE(\overline{y})\in\Omega be an EE-Pareto solution (a weak EE-Pareto solution) of the problem (VP). Then, y¯∈ΩE\overline{y}\in\Omega_{E} is a Pareto solution (a weak Pareto solution) of the problem (VPE).

Theorem 25

[44] (EE-Karush-Kuhn-Tucker necessary optimality conditions). Let E​(y¯)E\left(\overline{y}\right) be a weak EE-Pareto solution of (VP). Moreover, let fif_{i} (i∈Ii\in I), gkg_{k} (k∈Kk\in K), and hjh_{j} (j∈Jj\in J), be EE-differentiable and the Kuhn-Tucker constraint qualification be satisfied at y¯\overline{y}. Then there exist τ¯∈Rp\overline{\tau}\in R^{p}, ρ¯∈Rm\overline{\rho}\in R^{m} and ξ¯∈Rq\overline{\xi}\in R^{q} such that

∑i=1pτ¯i​∇fi​(E​(y¯))+∑k=1mρ¯k​∇gk​(E​(y¯))+∑j=1qξ¯j​∇hj​(E​(y¯))=0​,\begin{array}[]{c}\sum_{i=1}^{p}\overline{\tau}_{i}\nabla f_{i}(E(\overline{y}))+\sum_{k=1}^{m}\overline{\rho}_{k}\nabla g_{k}(E(\overline{y}))+\sum_{j=1}^{q}\overline{\xi}_{j}\nabla h_{j}(E(\overline{y}))=0\text{,}\end{array} (21)
ρ¯k​gk​(E​(y¯))=0​, ​k∈K​,\overline{\rho}_{k}g_{k}(E(\overline{y}))=0\text{, \ }k\in K\text{,} (22)
τ¯≥0​, ​ρ¯≧0​.\overline{\tau}\geq 0\text{, \ }\overline{\rho}\geqq 0\text{.} (23)
Definition 26

(E​(y¯),τ¯,ρ¯,ξ¯)∈Ω×Rp×Rm×Rq\left(E(\overline{y}),\overline{\tau},\overline{\rho},\overline{\xi}\right)\in\Omega\times R^{p}\times R^{m}\times R^{q} is said to be an EE-Karush-Kuhn-Tucker point (EE-KKT point) for (VP) if the relations (21)-(23) are satisfied at E​(y¯)E(\overline{y}) with Lagrange multipliers τ¯\overline{\tau}, ρ¯\overline{\rho}, ξ¯\overline{\xi}.

Now, we prove the sufficiency of the EE-Karush-Kuhn-Tucker necessary optimality conditions for the considered EE-differentiable vector optimization problem (VP) under exponentially EE-invexity hypotheses.

Theorem 27

Let (E​(y¯),τ¯,ρ¯,ξ¯)∈Ω×Rp×Rm×Rq\left(E(\overline{y}),\overline{\tau},\overline{\rho},\overline{\xi}\right)\in\Omega\times R^{p}\times R^{m}\times R^{q} be an EE-KKT point of (VP). Let JE+​(E​(y¯))={j∈J:ξ¯j>0}J_{E}^{+}\left(E\left(\overline{y}\right)\right)=\left\{j\in J:\overline{\xi}_{j}>0\right\} and JE−​(E​(y¯))={j∈J:ξ¯j<0}J_{E}^{-}\left(E\left(\overline{y}\right)\right)=\left\{j\in J:\overline{\xi}_{j}<0\right\}. Furthermore, assume the following hypotheses are fulfilled:

  1. a)

    fif_{i}, i∈Ii\in I, is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  2. b)

    gkg_{k}, k∈K​(E​(y¯))k\in K\left(E(\overline{y})\right), is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  3. c)

    hjh_{j}, j∈J+​(E​(y¯))j\in J^{+}\left(E\left(\overline{y}\right)\right), is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  4. d)

    −hj-h_{j}, j∈J−​(E​(y¯))j\in J^{-}\left(E\left(\overline{y}\right)\right), is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E}.

Then E​(y¯)E\left(\overline{y}\right) is a weak EE-Pareto solution of (VP).

Proof. By assumption, (E​(y¯),τ¯,ρ¯,ξ¯)∈Ω×Rp×Rm×Rq\left(E(\overline{y}),\overline{\tau},\overline{\rho},\overline{\xi}\right)\in\Omega\times R^{p}\times R^{m}\times R^{q} is an EE-KKT point of (VP). Then, by Definition 26, the relations (21)-(23) are satisfied at E​(y¯)E(\overline{y}) with Lagrange multipliers τ¯∈Rp\overline{\tau}\in R^{p}, ρ¯∈Rm\overline{\rho}\in R^{m} and ξ¯∈Rq\overline{\xi}\in R^{q}. We proceed by contradiction. Assume, contrary to the conclusion, that E​(y¯)E(\overline{y}) is not a weak EE-Pareto solution of (VP). Hence, by Definition 22, there exists another E​(x⋆)∈ΩE(x^{\star})\in\Omega such that

f​(E​(x⋆))<f​(E​(y¯))​.f(E\left(x^{\star}\right))<f\left(E\left(\overline{y}\right)\right)\text{.} (24)

Using hypotheses a)-d), by Definition 4 and Definition 12, the following inequalities

efi​(E​(x⋆))−efi​(E​(y¯))≧∇fi​(E​(y¯))​efi​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​i∈I​,e^{f_{i}\left(E\left(x^{\star}\right)\right)}-e^{f_{i}\left(E\left(\overline{y}\right)\right)}\geqq\nabla f_{i}\left(E\left(\overline{y}\right)\right)e^{f_{i}\left(E\left(\overline{y}\right)\right)}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, }i\in I\text{,} (25)
egk​(E​(x⋆))−egk​(E​(y¯))≧∇gk​(E​(y¯))​egk​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​k∈K​(E​(y¯))​,e^{g_{k}(E\left(x^{\star}\right))}-e^{g_{k}(E\left(\overline{y}\right))}\geqq\nabla g_{k}\left(E\left(\overline{y}\right)\right)e^{g_{k}(E\left(\overline{y}\right))}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, }k\in K\left(E\left(\overline{y}\right)\right)\text{,} (26)
ehj​(E​(x⋆))−ehj​(E​(y¯))≧∇hj​(E​(y¯))​ehj​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​j∈J+​(E​(y¯))​,e^{h_{j}(E\left(x^{\star}\right))}-e^{h_{j}(E\left(\overline{y}\right))}\geqq\nabla h_{j}\left(E\left(\overline{y}\right)\right)e^{h_{j}(E\left(\overline{y}\right))}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, }j\in J^{+}\left(E\left(\overline{y}\right)\right)\text{,} (27)
−ehj​(E​(x⋆))+ehj​(E​(y¯))≧−∇hj​(E​(y¯))​ehj​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​j∈J−​(E​(y¯))-e^{h_{j}(E\left(x^{\star}\right))}+e^{h_{j}(E\left(\overline{y}\right))}\geqq-\nabla h_{j}\left(E\left(\overline{y}\right)\right)e^{h_{j}(E\left(\overline{y}\right))}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, \ }j\in J^{-}\left(E\left(\overline{y}\right)\right) (28)

hold, respectively. Combining (24) and (25) and then multiplying the resulting inequalities by the corresponding Lagrange multipliers and adding both their sides, we get

[∑i=1pτ¯i​∇(fi∘E)⁡(y¯)]​η​(E​(x⋆),E​(y¯))<0​.\left[\sum_{i=1}^{p}\overline{\tau}_{i}\nabla\left(f_{i}\circ E\right)\left(\overline{y}\right)\right]\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)<0\text{.} (29)

Multiplying inequalities (26)-(28) by the corresponding Lagrange multipliers, respectively, we obtain

ρ¯k​egk​(E​(x⋆))−ρ¯k​egk​(E​(y¯))≧ρ¯k​∇gk​(E​(y¯))​egk​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​k∈K​(E​(y¯))​,\overline{\rho}_{k}e^{g_{k}(E\left(x^{\star}\right))}-\overline{\rho}_{k}e^{g_{k}(E\left(\overline{y}\right))}\geqq\overline{\rho}_{k}\nabla g_{k}\left(E\left(\overline{y}\right)\right)e^{g_{k}(E\left(\overline{y}\right))}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, }k\in K\left(E\left(\overline{y}\right)\right)\text{,} (30)
ξ¯j​ehj​(E​(x⋆))−ξ¯j​ehj​(E​(y¯))≧ξ¯j​∇hj​(E​(y¯))​ehj​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​j∈J+​(E​(y¯))​,\overline{\xi}_{j}e^{h_{j}(E\left(x^{\star}\right))}-\overline{\xi}_{j}e^{h_{j}(E\left(\overline{y}\right))}\geqq\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)e^{h_{j}(E\left(\overline{y}\right))}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, }j\in J^{+}\left(E\left(\overline{y}\right)\right)\text{,} (31)
ξ¯j​ehj​(E​(x⋆))−ξ¯j​ehj​(E​(y¯))≧ξ¯j​∇hj​(E​(y¯))​ehj​(E​(y¯))​η​(E​(x⋆),E​(y¯))​, ​j∈J−​(E​(y¯))​.\overline{\xi}_{j}e^{h_{j}(E\left(x^{\star}\right))}-\overline{\xi}_{j}e^{h_{j}(E\left(\overline{y}\right))}\geqq\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)e^{h_{j}(E\left(\overline{y}\right))}\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\text{, \ }j\in J^{-}\left(E\left(\overline{y}\right)\right)\text{.} (32)

Using the condition (22) together with E​(x⋆)∈ΩE(x^{\star})\in\Omega and E​(y¯)∈ΩE(\overline{y})\in\Omega, we get, respectively,

ρ¯k​∇gk​(E​(y¯))​η​(E​(x⋆),E​(y¯))≦0​, ​k∈K​(E​(y¯))​,\overline{\rho}_{k}\nabla g_{k}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\leqq 0\text{, }k\in K\left(E\left(\overline{y}\right)\right)\text{,} (33)
ξ¯j​∇hj​(E​(y¯))​η​(E​(x⋆),E​(y¯))≦0​, ​j∈J+​(E​(y¯))​,\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\leqq 0\text{, }j\in J^{+}\left(E\left(\overline{y}\right)\right)\text{,} (34)
ξ¯j​∇hj​(E​(y¯))​η​(E​(x⋆),E​(y¯))≦0​, ​j∈J−​(E​(y¯))​.\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)\leqq 0\text{, \ }j\in J^{-}\left(E\left(\overline{y}\right)\right)\text{.} (35)

Combining (29) and (33)-(35), we obtain that the following inequality

[∑i=1pτ¯i​∇fi​(E​(y¯))+∑k=1mρ¯k​∇gk​(E​(y¯))+∑j=1qξ¯j​∇hj​(E​(y¯))]​η​(E​(x⋆),E​(y¯))<0\left[\sum_{i=1}^{p}\overline{\tau}_{i}\nabla f_{i}(E\left(\overline{y}\right))+\sum_{k=1}^{m}\overline{\rho}_{k}\nabla g_{k}\left(E\left(\overline{y}\right)\right)+\sum_{j=1}^{q}\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)\right]\eta\left(E\left(x^{\star}\right),E\left(\overline{y}\right)\right)<0

holds, which is a contradiction to the condition (21). Thus, the proof of this theorem is completed.   

If stronger EE-differentiable exponentially EE-invexity hypotheses are imposed on the functions constituting the considered vector optimization problems, then the sufficient optimality conditions for a feasible solution to be an EE-Pareto solution of the problem (VP) result is true.

Theorem 28

Let (E​(y¯),τ¯,ρ¯,ξ¯)∈Ω×Rp×Rm×Rq\left(E(\overline{y}),\overline{\tau},\overline{\rho},\overline{\xi}\right)\in\Omega\times R^{p}\times R^{m}\times R^{q} be an EE-KKT point of (VP). Furthermore, assume that the following hypotheses are fulfilled:

  1. a)

    fif_{i}, i∈Ii\in I, is EE-differentiable exponentially strictly EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  2. b)

    gkg_{k}, k∈K​(E​(y¯))k\in K\left(E(\overline{y})\right), is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  3. c)

    hjh_{j}, j∈J+​(E​(y¯))j\in J^{+}\left(E\left(\overline{y}\right)\right), is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  4. d)

    −hj-h_{j}, j∈J−​(E​(y¯))j\in J^{-}\left(E\left(\overline{y}\right)\right), is EE-differentiable exponentially EE-invex function at y¯\overline{y} on ΩE\Omega_{E}.

Then E​(y¯)E\left(\overline{y}\right) is an EE-Pareto solution of (VP).

Proof. The proof of this theorem is similar to the proof of Theorem 27 and is omitted.   

Remark 29

According to the proof of Theorem 27, the sufficient conditions are also satisfied if gkg_{k}, k∈KE​(y¯)k\in K_{E}\left(\overline{y}\right), hjh_{j}, j∈J+​(E​(y¯))j\in J^{+}\left(E\left(\overline{y}\right)\right), −hj-h_{j}, j∈J−​(E​(y¯))j\in J^{-}\left(E\left(\overline{y}\right)\right), are EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E}.

Now, under the concepts of generalized EE-differentiable exponentially EE-invexity, we prove the sufficient optimality conditions for a feasible solution to be a weak EE-Pareto solution of the problem (VP).

Theorem 30

Let (E​(y¯),τ¯,ρ¯,ξ¯)∈Ω×Rp×Rm×Rq\left(E(\overline{y}),\overline{\tau},\overline{\rho},\overline{\xi}\right)\in\Omega\times R^{p}\times R^{m}\times R^{q} be an EE-KKT point of (VP). Furthermore, assume that the following hypotheses are fulfilled:

  1. a)

    fif_{i}, i∈Ii\in I, is EE-differentiable exponentially pseudo-EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  2. b)

    gkg_{k}, k∈K​(E​(y¯))k\in K\left(E(\overline{y})\right), is EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  3. c)

    hjh_{j}, j∈J+​(E​(y¯))j\in J^{+}\left(E\left(\overline{y}\right)\right), is EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E},

  4. d)

    −hj-h_{j}, j∈J−​(E​(y¯))j\in J^{-}\left(E\left(\overline{y}\right)\right), is EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E}.

Then E​(y¯)E\left(\overline{y}\right) is a weak EE-Pareto solution of (VP).

Proof. By assumption, (E​(y¯),τ¯,ρ¯,ξ¯)∈Ω×Rp×Rm×Rq\left(E(\overline{y}),\overline{\tau},\overline{\rho},\overline{\xi}\right)\in\Omega\times R^{p}\times R^{m}\times R^{q} is a Karush-Kuhn-Tucker point in the considered constrained EE-optimization problem (VP). Then, by Definition 26, the Karush-Kuhn-Tucker necessary optimality conditions (21)-(23) are satisfied at y¯\overline{y} with Lagrange multipliers τ¯∈Rp\overline{\tau}\in R^{p}, ρ¯∈Rm\overline{\rho}\in R^{m} and ξ¯∈Rq\overline{\xi}\in R^{q}. We proceed by contradiction. Suppose, contrary to the result, that E​(y¯)E(\overline{y}) is not a weak Pareto solution in problem (VP). Hence, by Definition 23, there exists another E​(x⋏)∈ΩE(x^{\curlywedge})\in\Omega such that

fi​(E​(x⋏))<fi​(E​(y¯))​, ​i∈I​.f_{i}(E(x^{\curlywedge}))<f_{i}\left(E\left(\overline{y}\right)\right)\text{, }i\in I\text{.} (36)

Thus,

efi​(E​(x⋏))<efi​(E​(y¯))​, ​i∈I​.e^{f_{i}(E(x^{\curlywedge}))}<e^{f_{i}\left(E\left(\overline{y}\right)\right)}\text{, }i\in I\text{.} (37)

By hypothesis (a), the objective function ff is EE-differentiable exponentially pseudo-EE-invex at E​(y¯)E(\overline{y}) on Ω\Omega. Then, (37) gives

∇(fi∘E)⁡(y¯)​efi​(E​(y¯))​η​(E​(x⋏),E​(y¯))<0​, ​i∈I​,\nabla\left(f_{i}\circ E\right)\left(\overline{y}\right)e^{f_{i}\left(E\left(\overline{y}\right)\right)}\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)<0\text{, }i\in I\text{,} (38)

By the EE-Karush-Kuhn-Tucker necessary optimality condition (23) and ef​(E​(y¯))>0,e^{f\left(E\left(\overline{y}\right)\right)}>0, inequality (38) yields

[∑i=1pτ¯i​∇(fi∘E)⁡(y¯)]​η​(E​(x⋏),E​(y¯))<0​.\left[\sum_{i=1}^{p}\overline{\tau}_{i}\nabla\left(f_{i}\circ E\right)\left(\overline{y}\right)\right]\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)<0\text{.} (39)

Since E​(x⋏)∈ΩE(x^{\curlywedge})\in\Omega, E​(y¯)∈ΩE(\overline{y})\in\Omega , therefore, the EE-Karush-Kuhn-Tucker necessary optimality conditions (22) and (23) imply

gk​(E​(x⋏))−gk​(E​(y¯))≦0​, ​k∈K​(E​(y¯))​.g_{k}(E\left(x^{\curlywedge}\right))-g_{k}(E\left(\overline{y}\right))\leqq 0\text{, \ }k\in K\left(E\left(\overline{y}\right)\right)\text{.}

Thus,

egk​(E​(x⋏))≦egk​(E​(y¯))​, ​k∈K​(E​(y¯))​.e^{g_{k}(E\left(x^{\curlywedge}\right))}\leqq e^{g_{k}(E\left(\overline{y}\right))}\text{, \ }k\in K\left(E\left(\overline{y}\right)\right)\text{.}

From the assumption, each gkg_{k}, k∈Kk\in K, is an EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E}. Then, by Definition 15, we get

∇gk​(E​(y¯))​egk​(E​(y¯))​η​(E​(x⋏),E​(y¯))≦0​, ​k∈K​(E​(y¯))​.\nabla g_{k}\left(E\left(\overline{y}\right)\right)e^{g_{k}\left(E\left(\overline{y}\right)\right)}\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{, \ }k\in K\left(E\left(\overline{y}\right)\right)\text{.} (40)

Thus, by the EE-Karush-Kuhn-Tucker necessary optimality condition (23), eg​(E​(y¯))>0,e^{g\left(E\left(\overline{y}\right)\right)}>0, and by Definition 15, (40) gives

∑k∈K​(E​(y¯))ρ¯k​∇gk​(E​(y¯))​η​(E​(x⋏),E​(y¯))≦0​.\sum_{k\in K\left(E\left(\overline{y}\right)\right)}\overline{\rho}_{k}\nabla g_{k}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{.}

Hence, taking into account ρ¯k=0\overline{\rho}_{k}=0, k∉K​(E​(y¯))k\notin K\left(E\left(\overline{y}\right)\right), we have

∑k=1mρ¯k​∇gk​(E​(y¯))​η​(E​(x⋏),E​(y¯))≦0​.\sum_{k=1}^{m}\overline{\rho}_{k}\nabla g_{k}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{.} (41)

From E​(x⋏)∈ΩE(x^{\curlywedge})\in\Omega, E​(y¯)∈ΩE(\overline{y})\in\Omega, we get

hj​(E​(x⋏))−hj​(E​(y¯))=0​, ​j∈J+​(E​(y¯))​,h_{j}(E\left(x^{\curlywedge}\right))-h_{j}(E\left(\overline{y}\right))=0\text{, }j\in J^{+}\left(E\left(\overline{y}\right)\right)\text{,} (42)
−hj​(E​(x⋏))−(−hj​(E​(y¯)))=0​, ​j∈J−​(E​(y¯))​.-h_{j}(E\left(x^{\curlywedge}\right))-\left(-h_{j}(E\left(\overline{y}\right))\right)=0\text{, }j\in J^{-}\left(E\left(\overline{y}\right)\right)\text{.} (43)

Thus,

ehj​(E​(x⋏))−ehj​(E​(y¯))=0​, ​j∈J+​(E​(y¯))​,e^{h_{j}(E\left(x^{\curlywedge}\right))}-e^{h_{j}(E\left(\overline{y}\right))}=0\text{, }j\in J^{+}\left(E\left(\overline{y}\right)\right)\text{,} (44)
−ehj​(E​(x⋏))−(−ehj​(E​(y¯)))=0​, ​j∈J−​(E​(y¯))​.-e^{h_{j}(E\left(x^{\curlywedge}\right))}-\left(-e^{h_{j}(E\left(\overline{y}\right))}\right)=0\text{, }j\in J^{-}\left(E\left(\overline{y}\right)\right)\text{.} (45)

Since each equality constraint hjh_{j}, j∈J+​(E​(x¯))j\in J^{+}\left(E\left(\overline{x}\right)\right), and each function −hj-h_{j}, j∈J−​(E​(y¯))j\in J^{-}\left(E\left(\overline{y}\right)\right), is an EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E}, then by Definition 15, eh​(E​(y¯))>0,e^{h(E\left(\overline{y}\right))}>0, (44) and (45) give, respectively,

∇hj​(E​(y¯))​η​(E​(x⋏),E​(y¯))≦0​, ​j∈J+​(E​(y¯))​,\nabla h_{j}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{,\ }j\in J^{+}\left(E\left(\overline{y}\right)\right)\text{,} (46)
−∇hj​(E​(y¯))​η​(E​(x⋏),E​(y¯))≦0​, ​j∈J−​(E​(y¯))​.-\nabla h_{j}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{,\ }j\in J^{-}\left(E\left(\overline{y}\right)\right)\text{.} (47)

Thus, (46) and (47) yield

[∑j∈J+​(E​(y¯))ξ¯j​∇hj​(E​(y¯))+∑j∈J−​(E​(y¯))ξ¯t​∇hj​(E​(y¯))]​η​(E​(x⋏),E​(y¯))≦0​.\bigg{[}\sum_{j\in J^{+}\left(E\left(\overline{y}\right)\right)}\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)+\sum_{j\in J^{-}\left(E\left(\overline{y}\right)\right)}\overline{\xi}_{t}\nabla h_{j}\left(E\left(\overline{y}\right)\right)\bigg{]}\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{.}

Hence, taking into account ξ¯j=0\overline{\xi}_{j}=0, j∉J+​(E​(y¯))∪J−​(E​(y¯))j\notin J^{+}\left(E\left(\overline{y}\right)\right)\cup J^{-}\left(E\left(\overline{y}\right)\right), we have

∑j=1qξ¯j​∇hj​(E​(y¯))​η​(E​(x⋏),E​(y¯))≦0​.\sum_{j=1}^{q}\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)\leqq 0\text{.} (48)

Combining (39), (41) and (48), we get that the following inequality

[∑i=1pτ¯i​∇fi​(E​(y¯))+∑k=1mρ¯k​∇gk​(E​(x¯))+∑j=1qξ¯j​∇hj​(E​(y¯))]​η​(E​(x⋏),E​(y¯))<0​\bigg{[}\sum_{i=1}^{p}\overline{\tau}_{i}\nabla f_{i}(E\left(\overline{y}\right))+\sum_{k=1}^{m}\overline{\rho}_{k}\nabla g_{k}\left(E\left(\overline{x}\right)\right)+\sum_{j=1}^{q}\overline{\xi}_{j}\nabla h_{j}\left(E\left(\overline{y}\right)\right)\bigg{]}\eta\left(E\left(x^{\curlywedge}\right),E\left(\overline{y}\right)\right)<0\text{}

which is a contradiction to the EE-Karush-Kuhn-Tucker necessary optimality condition (21). Thus, the proof of this theorem is completed.   

In order to illustrate the above sufficient optimality conditions, we now present an example of an EE-differentiable problem in which the involved functions are exponentially (generalized) EE-invex.

Example 31

Consider the following nonconvex nondifferentiable vector optimization problem

​f​(x)=(log⁡(x13+x23+1)​ , ​log⁡(x13+x23+2))→V−mins.t. ​g1​(x)=−x13≦0, g2​(x)=−x23≦0​.​ (VP1)\begin{array}[]{c}\text{}f(x)=(\log(\sqrt[3]{x_{1}}+\sqrt[3]{x_{2}}+1)\text{ },\text{ }\log(\sqrt[3]{x_{1}}+\sqrt[3]{x_{2}}+2))\;\;\rightarrow V-\min\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{s.t. }g_{1}(x)=-\sqrt[3]{x_{1}}\leqq 0,\text{ \ }\vskip 6.0pt plus 2.0pt minus 2.0pt\\ g_{2}(x)=-\sqrt[3]{x_{2}}\leqq 0\text{.}\end{array}\text{ \ \ (VP1)}

Note that Ω={(x1,x2)∈R2:−x13≦0​ ∧ −x23≦0}\Omega=\left\{\left(x_{1},x_{2}\right)\in R^{2}:-\sqrt[3]{x_{1}}\leqq 0\text{ }\wedge\text{ }-\sqrt[3]{x_{2}}\leqq 0\right\}. Let E:R2→R2E:R^{2}\rightarrow R^{2} be defined as follows E​(x1,x2)=(x13,x23),E\left(x_{1},x_{2}\right)=\left(x_{1}^{3},x_{2}^{3}\right), η​(E​(x),E​(y¯))=(1−ey¯1−x1,1−ey¯2−x2).\eta(E(x),E(\overline{y}))=(1-e^{\overline{y}_{1}-x_{1}},1-e^{\overline{y}_{2}-x_{2}}). Now, for the considered nonconvex nondifferentiable multiobjective programming problem (VP1), we define its associated differentiable optimization problem (VP1E) as follows

​(f∘E)​(x)=(log⁡(x1+x2+1)​ , ​log⁡(x1+x2+2))→V−mins.t. ​g1​(E​(x))=−x1≦0, g2​(E​(x))=−x2≦0​.​ (VP1E​)\begin{array}[]{c}\text{}(f\circ E)(x)=(\log({x_{1}}+{x_{2}}+1)\text{ },\text{ }\log({x_{1}}+{x_{2}}+2))\;\;\rightarrow V-\min\vskip 6.0pt plus 2.0pt minus 2.0pt\\ \text{s.t. }g_{1}(E(x))=-x_{1}\leqq 0,\text{ \ \ }\vskip 6.0pt plus 2.0pt minus 2.0pt\\ g_{2}(E(x))=-x_{2}\leqq 0\text{.}\end{array}\text{ \ \ (VP1}_{E}\text{)}

Note that ΩE={(x1,x2)∈R2:x1≧0​ ∧ ​x2≧0}\Omega_{E}=\left\{\left(x_{1},x_{2}\right)\in R^{2}:x_{1}\geqq 0\text{ }\wedge\text{ }x_{2}\geqq 0\right\} and y¯=(0,0)\overline{y}=\left(0,0\right) is the set of all feasible solutions of the problem (VP1E). Further, note that all functions constituting the considered vector optimization problem (VP1) are EE-differentiable at y¯=(0,0)\overline{y}=\left(0,0\right). Then, it can also be shown that the EE-Karush-Kuhn-Tucker necessary optimality conditions (21)-(23) are fulfilled at y¯=(0,0)\overline{y}=\left(0,0\right) with Lagrange multipliers τ¯1=12\overline{\tau}_{1}=\frac{1}{2}, τ¯2=12\overline{\tau}_{2}=\frac{1}{2} and ρ¯1=ρ¯2=1\overline{\rho}_{1}=\overline{\rho}_{2}=1. Further, it can be proved that ff is an EE-differentiable exponentially pseudo-EE-invex function at y¯\overline{y} on ΩE\Omega_{E}, the constraint function g1g_{1} and g2g_{2} are an EE-differentiable exponentially quasi-EE-invex function at y¯\overline{y} on ΩE\Omega_{E}. Hence, by Theorem 30, E​(y¯)=(0,0)E(\overline{y})=\left(0,0\right) is an EE-Pareto solution of the optimization problem (VP).

Refer to caption
Refer to caption
Figure 1: Graphical view of (VP1E).
Remark 32

Note that we are not able to use the optimality conditions for differentiable multiobjective programming problems in order to find efficient solutions in the vector optimization problem (VP1) considered in Example 31 since some of the involved functions are not differentiable. Also, the sufficient optimality conditions with convexity hypotheses are not applicable for (VP1) since (VP1) is not a convex vector optimization problem.

4 Concluding remarks

In this paper, a new class of nonconvex EE-differentiable vector optimization problems with both inequality and equality constraints have been considered. We have introduced the notion of exponentially EE-invex functions, delved into their key characteristics, and expanded the framework by introducing various generalized exponentially EE-invexity concepts. Further, we have established sufficient optimality conditions for EE-differentiable vector optimization problems under (generalized) exponentially EE-invexity. To illustrate these findings, we have provided an example of nonconvex nonsmooth vector optimization problems.

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 EE-differentiable vector optimization problems. We shall investigate these questions in subsequent papers.

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