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

ϕitαt=ψitαt.

Then the next relations hold true

ϕitαt=ψitαt.

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

ϕt=ψt,ϕitαt=ψitαt.

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|tT,ϕΓt(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|ϕtU,u1=tα,χi,χαi,

where

tα(jt1ϕ)=tα(t)

, and

χi(jt1ϕ)=χi(ϕ(t))

.The pn functions

χαi:U1R

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,

ϕtαt=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,

Ltβ+Lχiχitβ+Lχγiχγitβ+Lχμνiχμνitβ=Lβ,

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

Each smooth Lagrangian

Lπχ(t)

,

tR+m

, leads to two smooth closed 1-forms:

– the differential

dL=Ltγdtγ+Lχidxi+Lχγidχγi,

with the components

Ltγ,Lχi

, with respect to the corresponding basis

(dtγ,dχi,dχγi)

;

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

dLπχt=Ltβ+Lχiχitβ+Lχγiχγitβdtβ,

of components

DβL=Ltβπχt+Lχiπχtχitβt+Lχγiπχtχγitβt,

with respect to the basis dtβ.

For other important facts on jet bundles, we address the reader to the book of Saunders [20].

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

CΩt0,t1,M

we denote the space of all functions

χ:Ωt0,t1Rn

of C-class.

Let

A:CΩt0,t1,MRr

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×CΩt0,t1,Rn×RR

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

χ̄()

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

Fπχt,πχ̄t;λη1t,q1+1λη2t,q2λFπχt,πχ̄t;η1t,q1+1λFπχt,πχ̄t;η2t,q2,

for q, q1,

q2Rn

, 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

CΩt0,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×RrR

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

AiχAiχ̄γt0,t1Φπχt,πχ̄t;aαiχπχ̄tDγaαiχγπχ̄t,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=1,p̄

;

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

1,p̄

;

3) χη if and only if χiηi, for all

1,p̄

;

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:ΠRr,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)=χtγ(t)

,

γ=1,p̄

, are partial velocities.

Presume that the Lagrange densities matrix

g=gaj:ΠRms,a=1,s̄,j=1,m̄,m<n,

of C-class leads to the partial differential inequalities

and the Lagrange densities matrix

h=hal:ΠRms,a=1,s̄,l=1,z̄,z<n,

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,,Urχgπχ0,hπχ=0,χt0=χ0,χt1=χ1.CUP

Let

D=χCΩt0,t1,M:tΩt0,t1,χt0=χ0,χt1=χ1,gπχt0,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

ΛRr

and the smooth functions

M:Ωt0,t1Rmsp

,

N:Ωt0,t1Rrsp

such that

Λ,uαχπχ̄t+Mαt,gχπχ̄t+Nαt,hχπχ̄tDγΛ,uαχγπχ̄t+Mαt,gχγπχ̄t+Nαt,hχγπχ̄t=0,(1)
Λ0,Λ,e=1,Mαt0,tΩt0,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)

χ̄()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

χ̄()

on D.

3)

γt0,t1Mαj(),gj(πχ())dtα

,

j=1,m̄

, are

(Φ,ρgj)

-convex with regard to its third argument at

χ̄()

on D;

4)

γt0,t1Nαl(),hl(πχ())dtα

,

l=1,z̄

, are

(Φ,ρhl)

-convex with regard to its third argument at

χ̄()

on D;

5)

Λ,ρU+j=1mρgj+l=1zρhl0

.

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

Uiχ̃Uiχ̄,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

Uiχ̃Uiχ̄γt0,t1Φπχ̃t,πχ̄t;uαiχπχ̄tDγuαiχγπχ̄t,ρUidtα,i=1,r̄,

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

γt0,t1Φπχ̃t,πχ̄t;uαiχπχ̄tDγuαiχγπχ̄t,ρUidtα0,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=1rΛiLγt0,t1Φπχ̃t,πχ̄t;uαiχπχ̄tDγuαiχγπχ̄t,ρUidtα<0.(5)

On the other hand,

Mαjt,gjπχ̃tMαjt,gjπχ̄t0,

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

1Lγt0,t1Φπχ̃t,πχ̄t;Mαjt,gjχπχ̄tDγMαjt,gjχγπχ̄t,ρgj,ρgjdtα1Lγt0,t1Mαjt,gjπχ̃tMαjt,gjπχ̄tdtα0,j=1,m̄.(6)

Now, by the properties of h,

χ̄()

, and

χ̃()

, we get

N̄αlt,hlπχ̃tNαlt,hlπχ̄t=0,

which leads to

1Lγt0,t1Φπχ̃t,πχ̄t;Nαlt,hlχπχ̄tDγNαlt,hαlχγπχ̄t,ρhldtα1Lγt0,t1Nαlt,hlπχ̃tNαlt,hlπχ̄tdtα0,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,uαiχπχ̄t+1Lj=1mMαjt,gjχπχ̄t+1Ll=1zNαlt,hlχπχ̄t1LDγi=1rΛiuαiχγπχ̄t+j=1mMαjt,gjχγπχ̄t+l=1zNαlt,hαlχγπχ̄t,
1Li=1rΛiρUi+j=1mρhj+l=1zρhldtαi=1rΛiLγt0,t1Φπχ̃t,πχ̄t;uαiχπχ̄tDγuαiχγπχ̄t,ρUi+1Lj=1mγt0,t1Φπχ̃t,πχ̄t;Mαjt,gjχπχ̄tDγMαjt,gjχγπχ̄t,ρgjdtα+1Ll=1zγt0,t1Φπχ̃t,πχ̄t;Nαlt,hlχπχ̄tDγNαlt,hαlχγπχ̄t,ρhldtα<0.

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

Φπχ̃t,πχ̄t;,0,1Li=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

maxUχΛ,uαχπχ̄t+Mαt,gχπχ̄t+Nαt,hχπχ̄tDCUPDγΛ,uαχγπχ̄t+Mαt,gχγπχ̄t+Nαt,hχγπχ̄t=0,Mαjt,gjπχ̄t+Nαjt,hjπχt0,Λ0,tΩt0,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.

We start with a weak duality result, as follows.

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)

γt0,t1Mαj(t),gj(πχ̄(t))+Nαj(t),hj(πχ(t))dtα

,

j=1,m̄

, are

(Φ,ρghj)

-convex with regard to its third argument at

χ̄()

;

3)

Λ,ρU+j=1mρghj0

.

Then

U(χ̄())U(η())

.

Proof 2. Presume that

U(χ̄())U(η())

, that is

Uiχ̄Uiη,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;uαiηπηtDγuαiηγπηt,ρUidtαUiχ̄Uiη0,i=1,r̄,

We multiply each relation by Λi,

i=1,r̄

, and then dividing by

L=i=1rΛi+m

, it follows that

i=1rΛiLγt0,t1Φπχ̄t,πηt;uαiηπηtDγuαiηγπηt,ρUidtα<0.(8)

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

1Lγt0,t1Φπχ̄t,πηt;Mαjt,gjηπηt+Nαjt,hjηπηtDγMαjt,gjηγπηt+Nαlt,hαlηγπηt,ρhgjdtα1Lγt0,t1Mαjt,gjπχ̄t+Nαlt,hlπχ̄t(9)
Mαjt,gjπηt+Nαlt,hlπηtdtα0,j=1,m̄.(10)

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

γt0,t1Φπχ̄t,πηt;1LΛ,uαiηπηt+1Lj=1mMαjt,gjηπηt+l=1zNαlt,hlηπηt1LDγi=1rΛiuαiηγπηt+j=1mMαjt,gjηγπηt+Nαlt,hαlηγπηt,1Li=1rΛiρUi+j=1mρghjdtαi=1rΛiLγt0,t1Φπχ̄t,πηt;uαiηπηtDγuαiηγπηt,ρUi+1Lj=1mγt0,t1Φπχ̄t,πηt;Mαjt,gjηπηt+N̄αlt,hlηπη̄tDγMαjt,gjηγπηt,ρghj+Nαlt,hαlηγπηt,ρghjdtα<0.

By the constraints of the dual problem

(DCUP)

, this inequality leads to

Φπχ̄t,πηt;,0,1Li=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)TRr

.The associated multitime multiobjective problem dual to

(CUP)

in the sense of Wolfe is

(WDCUP)

, as in the following.

maxφη,M,NΛ,uαηπηt+Mαt,gηπηt+Nαt,hηπηtDγΛ,uαηγπηt+Nαt,gηγπηt+Nαt,hηγπηt=0,ηt0=χ0,ηt1=χ1,Mαjt,gjπχ̄t+Nαjt,hjπχt0,Λ0,tΩt0,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.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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