Multiobjective Optimization Problems on Jet Bundles Ariana Pitea, et al.

May 4, 2022

1 Introduction

Multiobjective optimization is a modern direction of study in science, from reasons related to their real world applications. In this regard, we mention the shortest path method, which involves the length of the paths and their costs. More than that, multiple criteria may refer to the length of a journey, its price, or the number of transfers. Also, the timetable information could be considered as a result of multiobjective optimization, if we have in view the unknown delays. Physics encounters many problems whose solutions can be found by using optimization approach, since a considerable number of them refer mainly to minimization principles. In this respect, there can be mentioned the study of interfaces and elastic manifolds, morphology evaluation of flow lines in high temperature superconductor or the analysis of X-ray data; for a detailed analysis, please see Hartman and Heiko [1], or Biswas et al. [2]. Another field which provides real world multiobjective optimization problems is material sciences, where an optimal estimation of the parameters of the materials is required. Further more such optimization problems can be found also in economics, or game theory, see Ehrgott et al. [3], Gal and Hanne [4] and the references therein.

One of the main directions of research in optimization refers to determining necessary or/and sufficient efficiency conditions for some vector optimization programs, and that of developing various duality results in connection to the primal multiobjective problem. These kinds of outcomes require the use of various types of generalized convexities, a direction of study started by Craven [5] and Hanson [6]. The pseudo-convexity and quasi-convexity provided to be appropriate tools for the development of duality results, please see Bector et al. [7]. Suneja and Srivastava [8] used generalized invexity in order to prove various duality results for multiobjective problems. Osuna-Gómez et al. [9] introduced optimality conditions and duality properties for a class of multiobjective programs under generalized convexity hypotheses. Antczak [10] used B-(p, r)-invexity functions to obtain sufficient optimality conditions for vector problems. Su and Hien [11] used Mordukhovich pseudoconvexity and quasiconvexity to prove strong Karush-Kuhn-Tucker optimality conditions for constrained multiobjective problems. The optimal power flow problem is solved by means of a characterization of the KT-invexity, by Bestuzheva and Hijazi [12]. Suzuki [13] joined quasiconvexity with necessary and sufficient optimality conditions in terms of Greenberg-Pierskalla subdifferential and Martínez-Legaz subdifferential. Jayswal et al. [14] developed duality results for semi-infinite problems in terms of (F, ρ)-V-invexity. The (F, ρ)-convexity introduced by Preda [15] allowed the study of efficiency of multiobjective programs. The same tool was used by Antczak and Pitea [16] to develop sufficient optimality conditions in a geometric setting, or by Antczak and Arana-Jiménez [17] who studied vector optimization problems by additional means of weighting.

The aim of this work is to develop sufficient optimality conditions and duality results, by the use of the generalized convexity introduced by Caristi et al. [18], and also one of the most effective tool in the study of multiobjective optimization, the parametric approach, whose basis were put by Saaty and Gass [19]. The class of problems which are to be proposed in the work refers to minimizing a vector of curvilinear integrals, where the integrand depends also on the velocities. This kind of problems are connected, for example, with Mechanical Engineering, considering that curvilinear integral objectives are frequently used because of their physical meaning as mechanical work, and there is a need to minimize simultaneously such kind of quantities, subject to some suitable constraints.

The paper is organized as follows. Section 2 presents preliminary issues on jet bundles, and the (Φ, ρ)-invexity, needed to develop our theory. Section 3 is dedicated to sufficient efficiency conditions for a multitime multiobjective minimization problem with constraints, by means of the generalized convexity. Section 4 consists of weak, strong, and converse duality results in the sense of Mond-Weir and Wolfe.

2 Preliminaries

2.1 On the First Order Jet Bundle

In order to make our work self contained, we recollect some basic facts on the first order jet bundle, J1 (T, M), formed by the 1-jets

$jt1ϕ$

of the local sections ϕ ∈ Γt (ϖ). A 1-jet at the point t is an equivalence class of the sections which have the same value and the same first order partial derivatives at the point t.

If the local sections check the equality ϕ (t) = ψ (t), let (tα, χi) and (tα′, χi′) be two adapted coordinate systems around ϕ (t). Suppose the following equalities hold

$∂ϕi∂tαt=∂ψi∂tαt.$

Then the next relations hold true

$∂ϕi′∂tα′t=∂ψi′∂tα′t.$

Definition 1. Two local sections ϕ, ψ ∈ Γt (ϖ) are called 1-equivalent at the point t if

$ϕt=ψt,∂ϕi∂tαt=∂ψi∂tαt.$

The equivalence class containing the section ϕ is precisely the 1-jet associated with the local section ϕ, at the point t, denoted by

$jt1ϕ$

.

Definition 2. The set

$J1(T,M)={jt1ϕ|t∈T,ϕ∈Γt(ϖ)}$

is called the first order jet bundle.If

$(U,u)$

, u = (tα, χi) is an adapted coordinate system on the product manifold T × M, the induced coordinate system,

$(U1,u1)$

, on J1 (T, M), is defined as

$U1=jt1ϕ|ϕt∈U,u1=tα,χi,χαi,$

where

$tα(jt1ϕ)=tα(t)$

, and

$χi(jt1ϕ)=χi(ϕ(t))$

.The pn functions

$χαi:U1→R$

form the coordinate derivatives.

Proposition 1. On the product manifold T × M, consider

$(U,u)$

the atlas of adapted charts. Then, the corresponding charts

$(U1,u1)$

form a finite dimensional atlas, of C-class, on the first order jet bundle J1(T, M).In order to make the presentation more readable, in the sequel we denote πχ (t) = (t, χ (t), χγ (t)), where χγ is the derivative of χ with respect to tγ.

2.2 Lagrange 1-Forms of the First Order

Any Lagrange 1-form of the first order, on the jet space J1 (T, M), takes the form

$ω=Lαπχtdtα+Miπχtdχi+Niβπχtdχβi,$

where Lα, Mi, and

$Niβ$

are Lagrangians of the first order, with the pullback

$χ*ω=Lα+Miχαi+Niβχβαidtα,$

a Lagrange 1-form of the second order on M. The coefficients

second order Lagrangians, are linear in the second order derivatives. The Pfaff equation ω = 0, and the partial differential equations

$Lα+Miχαi+Niβχβαi=0$

can be associated with the form ω.

Let Lβ (πχ(t)) dtβ be a closed Lagrange 1-form (completely integrable), that is DβLα = DαLβ.

A closed 1-form in a simple-connected domain is an exact one. Its primitive can be expressed as a curvilinear integral,

$ϕt=∫Γt0,tLαπχsdsα,ϕt0=0,$

or as a system of partial derivative eqations,

$∂ϕ∂tαt=Lαπχt,ϕt0=0.$

Suppose there is a Lagrangian-like antiderivative

$Lπχt=∫Γt0,tLαπχsdsα,Lπχt0=0,$

or DαL = Lα, where the foregoing pullback is the given closed 1-form,

$∂L∂tβ+∂L∂χi∂χi∂tβ+∂L∂χγi∂χγi∂tβ+∂L∂χμνi∂χμνi∂tβ=Lβ,$

which is a completely integrable system of partial derivatives equations, with the unknown function χ(⋅).

Each smooth Lagrangian

$Lπχ(t)$

,

$t∈R+m$

, leads to two smooth closed 1-forms:

– the differential

$dL=∂L∂tγdtγ+∂L∂χidxi+∂L∂χγidχγi,$

with the components

$∂L∂tγ,∂L∂χi$

, with respect to the corresponding basis

$(dtγ,dχi,dχγi)$

;

– the restriction of dL to πχ (t), namely the pullback

$dLπχt=∂L∂tβ+∂L∂χi∂χi∂tβ+∂L∂χγi∂χγi∂tβdtβ,$

of components

$DβL=∂L∂tβπχt+∂L∂χiπχt∂χi∂tβt+∂L∂χγiπχt∂χγi∂tβt,$

with respect to the basis dtβ.

2.3 Generalized (Φ, ρ)-Invexity

Our results are developed by means of a suitable generalized convexity, introduced in the following.

Further, let Π = J1 (T, M) be the first order jet bundle associated to T and M. By

$C∞Ωt0,t1,M$

we denote the space of all functions

$χ:Ωt0,t1→Rn$

of C-class.

Let

$A:C∞Ωt0,t1,M→Rr$

be a path independent curvilinear vector functional

$Aχ⋅=∫γt0,t1aαπχtdtα.$

Now, we introduce the definition of the vectorial (Φ, ρ)-convexity for the vectorial functional A, which will be useful to state the results established in the paper. Before we do this, we give the definition of a convex functional.

Definition 3. The functional

$F:Π×Π×C∞Ωt0,t1,Rn×R→R$

is convex with respect to the third component, if, for all χ (⋅),

$χ̄(⋅)$

, η1 (⋅), η2 (⋅), the following inequality holds

$Fπχt,πχ̄t;λη1t,q1+1−λη2t,q2≦λFπχt,πχ̄t;η1t,q1+1−λFπχt,πχ̄t;η2t,q2,$

for q, q1,

$q2∈Rn$

, λ ∈ (0, 1).It can be easily proved that a similar property holds, if, instead of λ ∈ (0, 1), and 1−λ, we use λ1, λ2, … , λk ∈ (0, 1), with

$∑i=1kλi=1$

.Let S be a nonempty subset of

$C∞Ωt0,t1,M$

, and

$χ̄(⋅)∈S$

be given. Following the footsteps of [18], we have the following definition.

Definition 4. Let

$ρ=(ρ1,…,ρr)∈Rr$

,

$Φ:Π×Π×Rr→R$

be convex with respect to the third component, and

$Φ(πχ(t),πχ̄(t);(0,ρi))≥0$

. The vectorial functional A is called (strictly) (Φ, ρ)-convex at the point

$χ̄(⋅)$

on S if, for each i,

$i=1,r̄$

, the following inequality

$Aiχ⋅−Aiχ̄⋅≧∫γt0,t1Φπχt,πχ̄t;∂aαi∂χπχ̄t−Dγ∂aαi∂χγπχ̄t,ρidtα$

holds for all χ (⋅) ∈ S,

$(χ(⋅)≠χ̄(⋅))$

). If these inequalities are satisfied at each

$χ̄(⋅)∈S$

, then A is called (strictly) (Φ, ρ)-convex on S.This class of functionals entails that of (F, ρ)-convexity introduced in [15].

3 Sufficient Efficiency Conditions

The following well-known conventions for equalities and inequalities in case of vector optimization will be used in the sequel.

For any χ = (χ1, χ2, … , χp),

$η=η1,η2,…,ηp$

, consider.

1) χ = η if and only if χi = ηi, for all

$i=\stackrel{̄}{1,p}$

;

2) χ > η if and only if χi > ηi, for all

$\stackrel{̄}{1,p}$

;

3) χη if and only if χiηi, for all

$\stackrel{̄}{1,p}$

;

4) χη if and only if χη, and χη.

This product order relation will be used on the hyperparallelepiped

$Ωt0,t1$

in

$Rp$

, with diagonal opposite points

$t0=(t01,…,t0p)$

, and

$t1=(t11,…,t1p)$

. Assume that

$γt0,t1$

is a piecewise C1-class curve joining the points t0 and t1, and that there exists an increasing piecewise smooth curve in

$Ωt0,t1$

which joins the points t0 and t1.

Let (T, h) and (M, g) be Riemannian manifolds of dimensions p and n, respectively, with the local coordinates t = (tα),

$α=1,p̄$

, and χ = (χi),

$i=1,n̄$

, respectively, and Π = J1 (T, M).

The closed Lagrange 1-forms densities of C-class

$uα=uαi:Π→Rr,i=1,r̄,α=1,p̄,$

produce the following path independent curvilinear functionals

$Uix⋅=∫γt0,t1uαiπχtdtα,i=1,r̄,α=1,p̄,$

where πχ(t) = (t, χ(t), χγ(t)), and

$χγ(t)=∂χ∂tγ(t)$

,

$γ=1,p̄$

, are partial velocities.

Presume that the Lagrange densities matrix

$g=gaj:Π→Rms,a=1,s̄,j=1,m̄,m

of C-class leads to the partial differential inequalities

and the Lagrange densities matrix

$h=hal:Π→Rms,a=1,s̄,l=1,z̄,z

defines the partial differential equalities

In the paper, we consider the multitime multiobjective variational problem

$(CUP)$

of minimizing a vector of path independent curvilinear functionals defined by

$minUχ⋅=U1χ⋅,…,Urχ⋅gπχ⋅≦0,hπχ⋅=0,χt0=χ0,χt1=χ1.CUP$

Let

$D=χ∈C∞Ωt0,t1,M:t∈Ωt0,t1,χt0=χ0,χt1=χ1,gπχt≦0,hπχt=0$

denote the set all feasible solutions of problem

$(CUP)$

.

Definition 5. A feasible solution

$χ̄(⋅)∈D$

is called an efficient solution to the problem

$(CUP)$

if there is no other feasible solution χ (⋅) ∈ D such that

If, in this relation, we use the strict inequality, then

$χ̄(⋅)$

is called a weakly efficient solution to the problem

$(CUP)$

.In [21] were proved necessary optimality conditions for a problem similar to

$(CUP)$

; for our case we obtain the next theorem.

Theorem 1. Let

$χ̄(⋅)∈D$

be a normal efficient solution in multitime multiobjective problem

$(CUP)$

. Then there exist the vector

$Λ∈Rr$

and the smooth functions

$M:Ωt0,t1→Rmsp$

,

$N:Ωt0,t1→Rrsp$

such that

$Λ,∂uα∂χπχ̄t+Mαt,∂g∂χπχ̄t+Nαt,∂h∂χπχ̄t−DγΛ,∂uα∂χγπχ̄t+Mαt,∂g∂χγπχ̄t+Nαt,∂h∂χγπχ̄t=0,(1)$
$Λ≧0,Λ,e=1,Mαt≧0,t∈Ωt0,t1,α=1,p̄.(3)$

The following theorem establishes sufficient conditions of efficiency for the problem

$(CUP)$

.

Theorem 2. Presume that the following conditions are fulfilled:

1)

$\stackrel{̄}{\chi }\left(\cdot \right)\in D$

, Λ, M (⋅) and N (⋅) satisfy the necessary conditions of efficiency (Eqs 13).

2) The objective functional U is (Φ, ρU)-convex with regard to its third argument at

$\stackrel{̄}{\chi }\left(\cdot \right)$

on D.

3)

${\int }_{{\gamma }_{{t}_{0},{t}_{1}}}⟨{M}_{\alpha j}\left(\cdot \right),{g}^{j}\left({\pi }_{\chi }\left(\cdot \right)\right)⟩d{t}^{\alpha }$

,

$j=\stackrel{̄}{1,m}$

, are

$\left(\mathrm{\Phi },{\rho }_{{g}_{j}}\right)$

-convex with regard to its third argument at

$\stackrel{̄}{\chi }\left(\cdot \right)$

on D;

4)

${\int }_{{\gamma }_{{t}_{0},{t}_{1}}}⟨N\alpha l\left(\cdot \right),{h}^{l}\left({\pi }_{\chi }\left(\cdot \right)\right)⟩d{t}^{\alpha }$

,

$l=\stackrel{̄}{1,z}$

, are

$\left(\mathrm{\Phi },{\rho }_{{h}_{l}}\right)$

-convex with regard to its third argument at

$\stackrel{̄}{\chi }\left(\cdot \right)$

on D;

5)

$⟨\mathrm{\Lambda },{\rho }_{U}⟩+{\sum }_{j=1}^{m}{\rho }_{{g}_{j}}+{\sum }_{l=1}^{z}{\rho }_{{h}_{l}}\geqq 0$

.

Then

$χ̄(⋅)$

is an efficient solution to the problem

$(CUP)$

.

Proof 1. Assume that

$χ̄(⋅)$

, Λ, M, and N fulfill the conditions from relations (Eqs 13), and that

$χ̄(⋅)$

is not an efficient solution to problem

$(CUP)$

. In this case, there can be found

$χ̃(⋅)∈Γ(Ωt0,t1)$

such that

more precisely

$Uiχ̃⋅≦Uiχ̄⋅,i=1,r̄,(4)$

with at least one index for which the inequality is a strict one.Taking advantage of the hypothesis 2), and the (Φ, ρ)-invexity, the previous relations compel

$Uiχ̃⋅−Uiχ̄⋅≧∫γt0,t1Φπχ̃t,πχ̄t;∂uαi∂χπχ̄t−Dγ∂uαi∂χγπχ̄t,ρUidtα,i=1,r̄,$

which, by inequalities (Eq. 4), imply that

$∫γt0,t1Φπχ̃t,πχ̄t;∂uαi∂χπχ̄t−Dγ∂uαi∂χγπχ̄t,ρUidtα≦0,i=1,r̄,$

where at least one inequality is a strict one. Multiplying the previous inequality by Λi accordingly,

$i=1,r̄$

, and dividing by

$L=∑i=1rΛi+m+z$

, we get

$∑i=1rΛiL∫γt0,t1Φπχ̃t,πχ̄t;∂uαi∂χπχ̄t−Dγ∂uαi∂χγπχ̄t,ρUidtα<0.(5)$

On the other hand,

$Mαjt,gjπχ̃t−Mαjt,gjπχ̄t≦0,$

which leads, by the (Φ, ρ)-invexity, to

$1L∫γt0,t1Φπχ̃t,πχ̄t;Mαjt,∂gj∂χπχ̄t−DγMαjt,∂gj∂χγπχ̄t,ρgj,ρgjdtα≦1L∫γt0,t1Mαjt,gjπχ̃t−Mαjt,gjπχ̄tdtα≦0,j=1,m̄.(6)$

Now, by the properties of h,

$χ̄(⋅)$

, and

$χ̃(⋅)$

, we get

$N̄αlt,hlπχ̃t−Nαlt,hlπχ̄t=0,$

$1L∫γt0,t1Φπχ̃t,πχ̄t;Nαlt,∂hl∂χπχ̄t−DγNαlt,∂hαl∂χγπχ̄t,ρhldtα≦1L∫γt0,t1Nαlt,hlπχ̃t−Nαlt,hlπχ̄tdtα≦0,l=1,z̄.(7)$

Using the convexity of the functional F in the third component, and adding inequalities (Eqs 5, 6), it follows that

$∫γt0,t1Φπχ̃t,πχ̄t;1LΛ,∂uαi∂χπχ̄t+1L∑j=1mMαjt,∂gj∂χπχ̄t+1L∑l=1zNαlt,∂hl∂χπχ̄t−1LDγ∑i=1rΛi∂uαi∂χγπχ̄t+∑j=1mMαjt,∂gj∂χγπχ̄t+∑l=1zNαlt,∂hαl∂χγπχ̄t,$
$1L∑i=1rΛiρUi+∑j=1mρhj+∑l=1zρhldtα≦∑i=1rΛiL∫γt0,t1Φπχ̃t,πχ̄t;∂uαi∂χπχ̄t−Dγ∂uαi∂χγπχ̄t,ρUi+1L∑j=1m∫γt0,t1Φπχ̃t,πχ̄t;Mαjt,∂gj∂χπχ̄t−DγMαjt,∂gj∂χγπχ̄t,ρgjdtα+1L∑l=1z∫γt0,t1Φπχ̃t,πχ̄t;Nαlt,∂hl∂χπχ̄t−DγNαlt,∂hαl∂χγπχ̄t,ρhldtα<0.$

By the equality from (Eq. 1), this inequality implies

$Φπχ̃t,πχ̄t;,0,1L∑i=1rΛiρUi+∑j=1mρhj+∑l=1zρhl<0,$

which is a contradiction with the properties of the function Φ.Therefore, our assumption was false, and

$χ̄(⋅)$

is an efficient solution to the problem

$(CUP)$

.

4 Dual Programming Theory

Consider the dual problem to

$(CUP)$

in the sense of Mond-Weir

$maxUχ⋅Λ,∂uα∂χπχ̄t+Mαt,∂g∂χπχ̄t+Nαt,∂h∂χπχ̄tDCUP−DγΛ,∂uα∂χγπχ̄t+Mαt,∂g∂χγπχ̄t+Nαt,∂h∂χγπχ̄t=0,Mαjt,gjπχ̄t+Nαjt,hjπχt≥0,Λ≧0,t∈Ωt0,t1,α=1,p̄,j=1,m̄m=z.$

Let ΔD be the set of the feasible solutions to the dual problem

$(DCUP)$

, and Δ = {η (⋅):[η (⋅), λ, M(⋅), ν(⋅)] ∈ ΔD}. By using (Φ, ρ)-convexity hypothesis, weak, strong, and converse duality results may be stated and proved, as in the sequel.

Theorem 3. Suppose that

$χ̄(⋅)$

and [η (⋅), λ, M (⋅), N (⋅)] are feasible solutions to the problems

$(CUP)$

, and

$(DCUP)$

, respectively. Additionally, presume that the next hypotheses are satisfied:

1) The objective functional U is (Φ, ρU)-convex with regard to its third argument at η(⋅).

2)

${\int }_{{\gamma }_{{t}_{0},{t}_{1}}}\left(⟨{M}_{\alpha j}\left(t\right),{g}^{j}\left({\pi }_{\stackrel{̄}{\chi }}\left(t\right)\right)⟩+⟨{N}_{\alpha j}\left(t\right),{h}^{j}\left({\pi }_{\chi }\left(t\right)\right)⟩\right)d{t}^{\alpha }$

,

$j=\stackrel{̄}{1,m}$

, are

$\left(\mathrm{\Phi },{\rho }_{g{h}_{j}}\right)$

-convex with regard to its third argument at

$\stackrel{̄}{\chi }\left(\cdot \right)$

;

3)

$⟨\mathrm{\Lambda },{\rho }_{U}⟩+{\sum }_{j=1}^{m}{\rho }_{g{h}_{j}}\geqq 0$

.

Then

$U(χ̄(⋅))≰U(η(⋅))$

.

Proof 2. Presume that

$U(χ̄(⋅))≤U(η(⋅))$

, that is

$Uiχ̄⋅≤Uiη⋅,i=1,r̄,$

where the inequality is strict for at least one of the indices.By the use of the (Φ, ρ)-invexity related to U, the previous relations imply

$∫γt0,t1Φπχ̄t,πηt;∂uαi∂ηπηt−Dγ∂uαi∂ηγπηt,ρUidtα≦Uiχ̄⋅−Uiη⋅≦0,i=1,r̄,$

We multiply each relation by Λi,

$i=1,r̄$

, and then dividing by

$L=∑i=1rΛi+m$

, it follows that

$∑i=1rΛiL∫γt0,t1Φπχ̄t,πηt;∂uαi∂ηπηt−Dγ∂uαi∂ηγπηt,ρUidtα<0.(8)$

Having in mind assumption (Eq. 2) from the theorem, we get, by the (Φ, ρ)-invexity, that

$1L∫γt0,t1Φπχ̄t,πηt;Mαjt,∂gj∂ηπηt+Nαjt,∂hj∂ηπηt−DγMαjt,∂gj∂ηγπηt+Nαlt,∂hαl∂ηγπηt,ρhgjdtα≦1L∫γt0,t1Mαjt,gjπχ̄t+Nαlt,hlπχ̄t(9)$
$−Mαjt,gjπηt+Nαlt,hlπηtdtα≦0,j=1,m̄.(10)$

The properties of F, jointly with inequalities (Eq. 8), and (Eq. 10), imply

$∫γt0,t1Φπχ̄t,πηt;1LΛ,∂uαi∂ηπηt+1L∑j=1mMαjt,∂gj∂ηπηt+∑l=1zNαlt,∂hl∂ηπηt−1LDγ∑i=1rΛi∂uαi∂ηγπηt+∑j=1mMαjt,∂gj∂ηγπηt+Nαlt,∂hαl∂ηγπηt,1L∑i=1rΛiρUi+∑j=1mρghjdtα≦∑i=1rΛiL∫γt0,t1Φπχ̄t,πηt;∂uαi∂ηπηt−Dγ∂uαi∂ηγπηt,ρUi+1L∑j=1m∫γt0,t1Φπχ̄t,πηt;Mαjt,∂gj∂ηπηt+N̄αlt,∂hl∂ηπη̄t−DγMαjt,∂gj∂ηγπηt,ρghj+Nαlt,∂hαl∂ηγπηt,ρghjdtα<0.$

By the constraints of the dual problem

$(DCUP)$

$Φπχ̄t,πηt;,0,1L∑i=1rΛiρUi+∑j=1mρghj<0,$

which is a contradiction with the properties of the function Φ.Therefore, our assumption was false, and U (χ(⋅))≰U (η(⋅)).In the following, we provide a strong duality result and also a converse duality one.

Theorem 4. Consider that χ (⋅) is an efficient solution to the primal problem

$(CUP)$

. Then there exists λ, M, N so that [χ (⋅), λ, M (⋅), N (⋅)] ∈ ΔD. More than that, if assumptions (Eqs 25) from Theorem 2 are fulfilled. then [χ (⋅), λ, M (⋅), N (⋅)] is an efficient solution to the dual problem

$(DCUP)$

.

Theorem 5. Let

$(η(⋅),λ,M(⋅)),N(⋅)$

be an efficient solution to the dual problem

$(DCUP)$

. Assume that conditions 2)-5) from Theorem 2 are satisfied. Then η (⋅) is an efficient solution to the primal problem

$(CUP)$

.In a similar manner, a dual problem in the sense of Wolfe can be associated to our vector problem

$(CUP)$

. First, we introduce the objective of this problem.

$φη⋅,M⋅,N⋅=∫γt0,t1uαπηt+Mαt,gπηt+Nαt,hπηtedtα,$

where

$e=(1,…,1)T∈Rr$

.The associated multitime multiobjective problem dual to

$(CUP)$

in the sense of Wolfe is

$(WDCUP)$

, as in the following.

$maxφη⋅,M⋅,N⋅Λ,∂uα∂ηπηt+Mαt,∂g∂ηπηt+Nαt,∂h∂ηπηt−DγΛ,∂uα∂ηγπηt+Nαt,∂g∂ηγπηt+Nαt,∂h∂ηγπηt=0,ηt0=χ0,ηt1=χ1,Mαjt,gjπχ̄t+Nαjt,hjπχt≥0,Λ≧0,t∈Ωt0,t1,α=1,p̄,j=1,m̄m=z.WDCUP$

Again, by the use of the notion of (Φ, ρ)-convexity, some weak, strong and converse duality results can be stated and proved, in a similar manner.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

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