# Duality for Multitime Multiobjective Ratio Variational Problems on First Order Jet Bundle

Authors: Mihai Postolache
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. Based on the efficiency conditions for multitime multiobjective ratio variational problems, we introduce a ratio dual of generalized Mond-Weir-Zalmai type, and under some assumptions of generalized convexity, duality theorems are stated. We prove our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained. This work further develops our studies in (Pitea and Postolache (2011)).


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## Body

## 1. Introduction

The duality for scalar variational problems involving convex functions has been formulated by Mond and Hanson in [1]. This study was further developed to various classes of convex and generalized convex functions. In [2], Hanson extended the duality results (in the sense of Wolfe) to a class of functions subsequently called invex. In order to weaken the convexity conditions, Bector et al. [3], introduced a dual to a variational problem (in the sense of Mond and Weir), different from that formulated by Mond and Hanson in [1]. Mond et al. [4], extended the concept of invexity to continuous functions and used it to generalize earlier Wolfe duality results for a class of variational problems. Duality theory for scalar optimization can be found in many other works and see Mond and Husain [5] and Preda [6].In [7], using a vector-valued Lagrangian function, Tanino and Sawaragi introduced a duality theory for multiobjective optimization. This idea is employed by Bitran [8], which associated a matrix of dual variables with the constraints in the primal problem. In [9], Mond and Weir considered a pair of symmetric dual nonlinear programs and developed a duality theory under assumptions of pseudoconvexity. Under various types of generalized convex functions, Mukherjee and Purnachandra [10], Preda [11] and Zalmai [12] established several weak efficiency conditions and developed different types of dualities for multiobjective variational problems. D. S. Kim and A. L. Kim [13] used the efficiency property of nondifferentiable multiobjective variational problems in duality theory. In a recent study [14], Pitea and Postolache considered a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on appropriate normal efficiency conditions, they studied duals of Mond-Weir type, generalized Mond-Weir-Zalmai type, and under appropriate assumptions of generalized convexity, stated duality theorems.In time, several authors have been interested in the study of (vector) ratio programs in connection with generalized convexity. This study is motivated by many practical optimization problems whose objective functions are quotients of two functions. In [15], Jagannathan introduced a duality study using some results connecting solutions of a nonlinear ratio program with those of suitably defined parametric convex program. Concerning advances on single-objective ratio programs, see [16] by Khan and Hanson and [17] by Reddy and Mukherjee, which utilized invexity assumptions in the sense of Hanson [18] to obtain optimality conditions and duality results. As concerns vector ratio problems, Singh and Hanson [19] applied invex functions to derive duality results, while Jeyakumar and Mond [20] generalized these results to the class of V-invex functions. Later, Liang et al. [21] introduced a unified formulation of the generalized convexity to derive optimality conditions and duality results for vector ratio problems.In this paper, we are motivated by previous published research articles (please refer to [22] by Antczak, [23] by Nahak and Mohapatra, and [14] by Pitea and Postolache) to consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type and, as a particular case, a vector of functionals of curvilinear integral type. We state and prove new duality results, of Mond-Weir-Zalmai type, for feasible solutions of our multitime multiobjective ratio variational problems, under assumptions of (ρ,b)-quasi-invexity. This study is encouraged by its possible application in mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. The objective vector function is of curvilinear integral type, the integrand depending on velocities; that is why we consider it adequate to introduce our results within the framework offered by the first order jet bundle, [24–26] by Pitea et al., [27] by Sounders, and [28] by Udriste et al.


## 2. Preliminaries

Before presenting our results, we need the following background, which is necessary for the completeness of the exposition. For more details, we address the reader to [27].


### 2.1. The First Order Jet Bundle

To make our presentation self-contained and reader-friendly, we recall the notion of jet bundle of the first order,J1(T,M).Let be given the smooth manifolds(M,g) and (T,h), of dimensions n, and p, respectively. The corresponding local coordinates are x=(xi), i=1,n¯, and t=(tα), α=1,p¯, respectively.In the following, the set{1,2,…,n} will be indexed by Latin characters, while the set {1,2,…,p} will be indexed by Greek characters.Denote byC∞(T,M) the set of mappings of class C∞, from T to M.Consider(t0,x0) an arbitrary point of the product manifold T×M. On the set C∞(T,M), define the equivalence relation
(2.1)f~(t0,x0)g⟺{f(t0)=g(t0)=x0dft0=dgt0.Iff and g are arbitrary mappings from C∞(T,M), we denote
(2.2)tβ(t0)=t0β,β=1,p¯,xi=xi∘f,yi=xi∘g,xi(x0)=x0i,i=1,n¯.Thus, the equivalence relationf~(t0,x0)g has the local expression
(2.3)xi(t0β)=yi(t0β)=x0i,∂xi∂tα(t0β)=∂yi∂tα(t0β),i=1,n¯,α=1,p.¯The equivalence class of a mappingf∈C∞(T,M) will be denoted by
(2.4)[f](t0,x0)={g∈C∞(T,M)∣g~(t0,x0)f}.
The quotient space obtained by the factorization of the space C∞(T,M) by the equivalence relation “~(t0,x0)”,
(2.5)J(t0,x0)1(T,M)=C∞(T,M)~(t0,x0),
is called  1-jet at the point (t0,x0).The total space of the set of 1-jets,(2.6)J1(T,M)=⋃(t0,x0)∈T×M‍J(t0,x0)1(T,M),
can be organized as a vector bundle over the base space T×M, endowed with the differentiable structure of the product space.Letς be the canonical projection, defined as
(2.7)ς:J1(T,M)→T×M,ς([f](t0,x0))=(t0,f(t0)).The mappingς is well defined, having the property to be onto.For every local chartU×V of the product manifold T×M, we define the bijection
(2.8)ϕU×V:ς-1(U×V)→U×V×ℝnp,ϕU×V([f](t0,x0))=(t0,x0,∂xi∂tα(t0β)),x0=f(t0).Therefore, the 1-jet space is endowed with a differentiable structure of dimensionn+p+np, such that the mappings ϕU×V are diffeomorphisms.The local coordinates on the spaceJ1(T,M) are (tα,xi,xαi), where
(2.9)tα([f](t0,x0))=tα(t0),xi([f](t0,x0))=xi(x0),xαi([f](t0,x0))=∂xi∂tα(t0β),α=1,p¯,i=1,n¯.Remark 2.1.
From physical viewpoint, the differentiable manifoldT should be thought as a “temporal manifold" or a “multitime", while M is a “space manifold."Remark 2.2.
From geometrical viewpoint, an element[f] of a fiber 1-jets J1(T,M) should be thought as a class of parametrized sheet. The sections of fiber 1-jets are “physical fields."
To simplify the notations, denote byxγ(t)=(∂x/∂tγ)(t), γ=1,p¯, the partial velocities. Also, in our subsequent theory, we will set πx(t)=(t,x(t),xγ(t)).


### 2.2. On Lagrange 1-Forms

A Lagrange 1-form of the first order on the jets spaceJ1(T,M) has the form
(2.10)ω=Lα(πx(t))dtα+Mi(πx(t))dxi+Niβ(πx(t))dxβi,
where Lα, Mi and Niβ are Lagrangians of the first order. The pullback
(2.11)x*ω=(Lα+Mixαi+Niβxβαi)dtα
is a Lagrange 1-form of the second order on M. The coefficients Lα+Mixαi+Niβxβαi are second order Lagrangians, which are linear in the partial accelerations.A smooth LagrangianL(πx(t)), t∈ℝ+m produces two smooth closed (completely integrable) 1-forms:(i) the differential(2.12)dL=∂L∂xidxi+∂L∂xγidxγi+∂L∂tγdtγ
of components (∂L/∂tγ,∂L/∂xi,∂L/∂xγi), with respect to the basis (dtγ,dxi,dxγi);(ii) the restriction ofdL to πx(t), that is, the pullback
(2.13)dL|πx(t)=(∂L∂xi∂xi∂tβ+∂L∂xγi∂xγi∂tβ+∂L∂tβ)dtβ
of components containing partial accelerations
(2.14)DβL=∂L∂xi(πx(t))∂xi∂tβ(t)+∂L∂xγi(πx(t))∂xγi∂tβ(t)+∂L∂tβ(πx(t)),
with respect to the basis dtβ (for other ideas, see [28]).


## 2.1. The First Order Jet Bundle

To make our presentation self-contained and reader-friendly, we recall the notion of jet bundle of the first order,J1(T,M).Let be given the smooth manifolds(M,g) and (T,h), of dimensions n, and p, respectively. The corresponding local coordinates are x=(xi), i=1,n¯, and t=(tα), α=1,p¯, respectively.In the following, the set{1,2,…,n} will be indexed by Latin characters, while the set {1,2,…,p} will be indexed by Greek characters.Denote byC∞(T,M) the set of mappings of class C∞, from T to M.Consider(t0,x0) an arbitrary point of the product manifold T×M. On the set C∞(T,M), define the equivalence relation
(2.1)f~(t0,x0)g⟺{f(t0)=g(t0)=x0dft0=dgt0.Iff and g are arbitrary mappings from C∞(T,M), we denote
(2.2)tβ(t0)=t0β,β=1,p¯,xi=xi∘f,yi=xi∘g,xi(x0)=x0i,i=1,n¯.Thus, the equivalence relationf~(t0,x0)g has the local expression
(2.3)xi(t0β)=yi(t0β)=x0i,∂xi∂tα(t0β)=∂yi∂tα(t0β),i=1,n¯,α=1,p.¯The equivalence class of a mappingf∈C∞(T,M) will be denoted by
(2.4)[f](t0,x0)={g∈C∞(T,M)∣g~(t0,x0)f}.
The quotient space obtained by the factorization of the space C∞(T,M) by the equivalence relation “~(t0,x0)”,
(2.5)J(t0,x0)1(T,M)=C∞(T,M)~(t0,x0),
is called  1-jet at the point (t0,x0).The total space of the set of 1-jets,(2.6)J1(T,M)=⋃(t0,x0)∈T×M‍J(t0,x0)1(T,M),
can be organized as a vector bundle over the base space T×M, endowed with the differentiable structure of the product space.Letς be the canonical projection, defined as
(2.7)ς:J1(T,M)→T×M,ς([f](t0,x0))=(t0,f(t0)).The mappingς is well defined, having the property to be onto.For every local chartU×V of the product manifold T×M, we define the bijection
(2.8)ϕU×V:ς-1(U×V)→U×V×ℝnp,ϕU×V([f](t0,x0))=(t0,x0,∂xi∂tα(t0β)),x0=f(t0).Therefore, the 1-jet space is endowed with a differentiable structure of dimensionn+p+np, such that the mappings ϕU×V are diffeomorphisms.The local coordinates on the spaceJ1(T,M) are (tα,xi,xαi), where
(2.9)tα([f](t0,x0))=tα(t0),xi([f](t0,x0))=xi(x0),xαi([f](t0,x0))=∂xi∂tα(t0β),α=1,p¯,i=1,n¯.Remark 2.1.
From physical viewpoint, the differentiable manifoldT should be thought as a “temporal manifold" or a “multitime", while M is a “space manifold."Remark 2.2.
From geometrical viewpoint, an element[f] of a fiber 1-jets J1(T,M) should be thought as a class of parametrized sheet. The sections of fiber 1-jets are “physical fields."
To simplify the notations, denote byxγ(t)=(∂x/∂tγ)(t), γ=1,p¯, the partial velocities. Also, in our subsequent theory, we will set πx(t)=(t,x(t),xγ(t)).


## 2.2. On Lagrange 1-Forms

A Lagrange 1-form of the first order on the jets spaceJ1(T,M) has the form
(2.10)ω=Lα(πx(t))dtα+Mi(πx(t))dxi+Niβ(πx(t))dxβi,
where Lα, Mi and Niβ are Lagrangians of the first order. The pullback
(2.11)x*ω=(Lα+Mixαi+Niβxβαi)dtα
is a Lagrange 1-form of the second order on M. The coefficients Lα+Mixαi+Niβxβαi are second order Lagrangians, which are linear in the partial accelerations.A smooth LagrangianL(πx(t)), t∈ℝ+m produces two smooth closed (completely integrable) 1-forms:(i) the differential(2.12)dL=∂L∂xidxi+∂L∂xγidxγi+∂L∂tγdtγ
of components (∂L/∂tγ,∂L/∂xi,∂L/∂xγi), with respect to the basis (dtγ,dxi,dxγi);(ii) the restriction ofdL to πx(t), that is, the pullback
(2.13)dL|πx(t)=(∂L∂xi∂xi∂tβ+∂L∂xγi∂xγi∂tβ+∂L∂tβ)dtβ
of components containing partial accelerations
(2.14)DβL=∂L∂xi(πx(t))∂xi∂tβ(t)+∂L∂xγi(πx(t))∂xγi∂tβ(t)+∂L∂tβ(πx(t)),
with respect to the basis dtβ (for other ideas, see [28]).


## 3.   Problem Description

Let(T,h) and (M,g) be Riemannian manifolds of dimensions p and n, respectively. Denote by t=(tα), α=1,p¯, and x=(xi), i=1,n¯, the local coordinates on T and M, respectively. Consider J1(T,M) the first order jet bundle associated to T and M.To develop our theory, we recall the following relations between two vectorsv=(vj) and w=(wj), j=1,δ¯:
(3.1)v=w⟺vj=wj,j=1,δ¯,v<w⟺vj<wj,j=1,δ¯,v≦w⟺vj≤wj,j=1,δ¯(productorderrelation),v≤w⟺v≦w,v≠w.Using the product order relation onℝp, the hyperparallelepiped Ωt0,t1, in ℝp, with diagonal opposite points t0=(t01,…,t0p) and t1=(t11,…,t1p), can be written as being the interval [t0,t1]. Suppose γt0,t1 is a piecewise C1-class curve joining the points t0 and t1.The closed Lagrange 1-forms densities ofC∞-class
(3.2)fα=(fαℓ):J1(T,M)→ℝr,kα=(kαℓ):J1(T,M)→ℝr,ℓ=1,r¯,α=1,p¯,
determine the following path independent curvilinear functionals (actions, mechanical work):
(3.3)Fℓ(x(·))=∫γt0,t1‍fαℓ(πx(t))dtα,Kℓ(x(·))=∫γt0,t1‍kαℓ(πx(t))dtα.The closeness conditions (complete integrability conditions) are(3.4)Dβfαℓ=Dαfβℓ,Dβkαℓ=Dαkβℓ,α,β=1,p¯,α≠β,ℓ=1,r¯,
where Dβ is the total derivative.SupposeKℓ(x(·))>0, for all ℓ=1,r¯, and accept that the Lagrange matrix densities
(3.5)g=(gab):J1(T,M)→ℝq¯s,a=1,s¯,b=1,q¯¯,q¯<n,h=(hab):J1(T,M)→ℝq~s,a=1,s¯,b=1,q~¯,q~<n,
of C∞-class define the partial differential inequations (PDIs) (of evolution)
(3.6)g(πx(t))≦0,t∈Ωt0,t1,
and the partial differential equations (PDE) (of evolution)
(3.7)h(πx(t))=0,t∈Ωt0,t1.On the setC∞(Ωt0,t1,M) of all functions x:Ωt0,t1→M of C∞-class, we set the norm ∥x∥=∥x∥∞+∑α=1p‍∥xα∥∞.For eachℓ=1,r¯, suppose Kℓ(x(·))>0, and consider
(3.8)F(x(·))K(x(·))=(F1(x(·))K1(x(·)),…,Fr(x(·))Kr(x(·))).The aim of this work is to introduce and study the variational problem of minimizing a vector of quotients of functionals of curvilinear integral type:(MFP)minx(·)(F(x(·))K(x(·)))subjecttox(·)∈ℱ(Ωt0,t1),
where ℱ(Ωt0,t1) denotes the set of all feasible solutions of problem (MFP), defined as
(3.9)ℱ(Ωt0,t1)={x∈C∞(Ωt0,t1,M)|x(t0)=x0,x(t1)=x1,orx(t)|∂Ωt0,t1=χ=given,g(πx(t))≦0,h(πx(t))=0,t∈Ωt0,t1|x(t0)=x0,x(t1)=x1,orx(t)|∂Ωt0,t1}.This kind of problems, of considerable interest, arises in various branches of mathematical, engineering, and economical sciences. We especially have in mind mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. These objectives play an essential role in mathematical modeling of certain processes in relation with robotics, tribology, engines, and much more.


## 4.   Main Results

The following two definitions are crucial in developing our results. For more details, see [26] by Pitea et al., and [14], by Pitea and Postolache.Definition 4.1.
A feasible solutionx∘(·)∈ℱ(Ωt0,t1) is called an efficient solution of  (MFP) if there is no x(·)∈ℱ(Ωt0,t1), x(·)≠x∘(·), such that F(x(·))/K(x(·))≤F(x∘(·))/K(x∘(·)).Definition 4.2.
Letρ be a real number and let b:C∞(Ωt0,t1,M)×C∞(Ωt0,t1,M)→[0,∞) be a functional. To any closed 1-form a=(aα) we associate the path independent curvilinear functional
(4.1)A(x(·))=∫γt0,t1‍aα(πx(t))dtα.
The functionalA is called [strictly] (ρ,b)-quasi-invex at the point x∘(·) if there is a vector function η:J1(Ωt0,t1,M)×J1(Ωt0,t1,M)→ℝn, vanishing at the point (πx∘(t),πx∘(t)), and θ:C∞(Ωt0,t1,M)×C∞(Ωt0,t1,M)→ℝn, such that for any x(·) in C∞(Ωt0,t1,M), [x(·)≠x∘(·)], the following implication holds:
(4.2)(A(x(·))≤A(x∘(·)))⇒(b(x(·),x∘(·))∫γt0,t1‍[〈η(πx(t),πx∘(t)),∂aα∂x(πx∘(t))〉x((·),x∘(·))∫γt0,t1‍+〈Dγη(πx(t),πx∘(t)),∂aα∂xγ(πx∘(t))〉]dtα[<]≤-ρb(x(·),x∘(·))∥θ(x(·),x∘(·))∥2∫γt0,t1‍).
Several examples which illustrate our concept could be found in [14]. However, the following example is interesting. It is a generalization of Example 1 in [14].Example 4.3.
Leta:[0,1]×C∞([0,1])×C∞([0,1])→ℝ, x(·)=(x1(·),x2(·),…,xn(·)). With x∘(·)∈C∞([0,1]), denote
(4.3)G(t)=〈∂a∂x(t,x∘(t),x˙∘(t))-D∂a∂x˙(t,x∘(t),x˙∘(t)),∂a∂x˙(t,x∘(t),x˙∘(t))〉,t∈[0,1],
where by D we denoted the total derivative operator.
Suppose thatG(1)-G(0)=0. Then, the functional
(4.4)A(x(·))=∫01‍a(t,x(t),x˙(t))dt
is (ρ,1)-quasi-invex, for ρ≤0 and any θ, at the point x∘(·), with respect to
(4.5)η(πx(t),πx∘(t))=(A(x(·))-A(x∘(·)))(∂a∂x(t,x∘(t),x˙∘(t))-D∂a∂x˙(t,x∘(t),x˙∘(t))).
In their recent work [26], Pitea et al. established necessary efficiency conditions for problem (MFP). More accurately, they proved that if x∘(·) is an efficient solution of problem (MFP), then there are two vectors Λ10, Λ20 in ℝr and the smooth functions μ∘ and ν∘, the first from Ωt0,t1 to ℝq¯sp, and the second from Ωt0,t1 to ℝq~sp, such that
(4.6)〈Λ10,∂fα∂x(πx∘(t))〉-〈Λ20,∂kα∂x(πx∘(t))〉+〈μα∘(t),∂g∂x(πx∘(t))〉+〈να∘(t),∂h∂x(πx∘(t))〉-Dγ(〈Λ10,∂fα∂xγ(πx∘(t))〉-〈Λ20,∂kα∂xγ(πx∘(t))〉DγDγ+〈μα∘(πx∘(t)),∂g∂xγ(πx∘(t))〉+〈να∘(t),∂h∂xγ(πx∘(t))〉)Dγ=0,t∈Ωt0,t1,α=1,p¯(Euler-LagrangePDEs)(4.7)〈μα∘(t),g(πx∘(t))〉=0,μα∘(t)≧0,t∈Ωt0,t1,α=1,p¯.
IfΛ10≥0 and Λ20≥0, then x∘(·), from conditions (4.6), is called normal efficient solution.
Letx∘(·) be an efficient solution of primal (MFP), the scalars Λ10, Λ20 in ℝr, and the smooth functions μ∘:Ωt0,t1→ℝq¯sp, ν∘:Ωt0,t1→ℝq~sp, given previously.
In order to use the idea of "grouping the resources," consider{P0,P1,…,Pq} and {Q0,Q1,…,Qq} partitions of the sets {1,…,q¯} and {1,…,q~}, respectively.
For eachℓ=1,r¯ and α=1,p¯, we denote
(4.8)f¯αℓ(πy(t))=fαℓ(πy(t))+〈μαP0(t),gP0(πy(t))〉+〈ναQ0(t),hQ0(πy(t))〉,F¯ℓ(y(·))=∫γt0,t1‍f¯αℓ(πy(t))dtα.
Consider a functiony(·)∈C∞(Ωt0,t1,M) and associate to (MFP) the multiobjective ratio variational problem
(MFZD)maxy(·)(F¯1(y(·))K1(y(·)),…,F¯r(y(·))Kr(y(·)))subjectto〈Λ10,∂fα∂y(πy(t))〉-〈Λ20,∂kα∂y(πy(t))〉+〈μα(t),∂g∂y(πy(t))〉+〈να(t),∂h∂y(πy(t))〉-Dγ(〈Λ10,∂fα∂yγ(πy(t))〉-〈Λ20∂kα∂yγ(πy(t))〉+〈μα(t),∂g∂yγ(πy(t))〉+〈να(t),∂h∂yγ(πy(t))〉)=0,α=1,p¯,t∈Ωt0,t1,(4.9)〈μαPκ(t),gPκ(πy(t))〉+〈ναQκ(t),hQκ(πy(t))〉≥0,κ=1,q¯,α=1,p¯,t∈Ωt0,t1Λ10≥0,
taking into account that the function y(t) has to satisfy the boundary conditions y(t0)=x0, y(t1)=x1, or y(t)|∂Ωt0,t1=χ=given.
ϖ
(
x
∘
(
·
)
) is the value of the objective function of problem (MFP) at x∘(·)∈ℱ(Ωt0,t1), and δ¯(y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) is the maximizing functional vector of dual problem (MFZD) at the point (y(·),yγ(·),Λ10,Λ20,μ(·),ν(·))∈Δ, where Δ is the domain of problem (MFZD).
We prove three duality results in a generalized sense of Mond-Weir-Zalmai.Theorem 4.4 (weak duality).
Letx∘(·) be a feasible solution of problem  (MFP) and let (y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) be a feasible point of problem  (MFZD). Assume that the following conditions are satisfied: (a)
〈μP0(t),gP0(πy(t))〉+〈νQ0(t),hQ0(πy(t))〉≤0, for all t∈Ωt0,t1;(b)
Λℓ20>0 and Λℓ10Fℓ(y(·))-Λℓ20Kℓ(y(·))=0, for each ℓ=1,r¯;(c)
for eachℓ=1,r¯, the functional Λℓ10Fℓ(x(·))-Λℓ20Kℓ(x(·)) is (ρ'ℓ,b)-quasi-invex at the point y(·) with respect to η and θ, on ℱ(Ωt0,t1);(d)
∫γt0,t1‍[〈μαPκ(t),gPκ(πx(t))〉+〈ναQκ(t),hQκ(πx(t))〉]dtα   is (ρκ′′,b)-quasi-invex at y(·) with respect to η and θ, for each κ=1,q¯, on F(Ωt0,t1);(e)
at least one of the functionals of   (c), (d) is strictly quasi-invex;(f)
∑ℓ=1rρ'ℓ+∑κ=1qρκ′′≥0.
Then, the inequalityϖ(x∘(·))≤δ¯(y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) is false.Proof.
By reductio ad absurdum, suppose we have(4.10)Fℓ(x∘(·))Kℓ(x∘(·))≤F¯ℓ(y(·))Kℓ(y(·)),ℓ=1,r¯.
From these inequalities, it follows
(4.11)Λℓ10Fℓ(x∘(·))Λℓ20Kℓ(x∘(·))≤1+1Fℓ(y(·))∫γt0,t1‍[〈μαP0(t),gP0(πy(t))〉+〈ναQ0(t),hQ0(πy(t))〉]dtα,
and taking into account the hypotheses (a) and (b), we get
(4.12)Λℓ10Fℓ(x∘(·))≤Λℓ20Kℓ(x∘(·)),∀ℓ=1,r¯.
We obtain the following implications:(4.13)(〈Λ10,F(x∘(·))〉-〈Λ20,K(x∘(·))〉≤0)⇒(∑ℓ=1rρ'ℓb(x∘(·),y(·))×∫γt0,t1‍{〈Λ20,∂kα∂yγ(πy(t))〉〉〈η(πx∘(t),πy(t)),〈Λ10,∂fα∂y(πy(t))〉-〈Λ20,∂kα∂y(πy(t))〉〉+〈Dγη(πx∘(t),πy(t)),〈Λ10,∂fα∂yγ(πy(t))〉〉-〈Λ20,∂kα∂yγ(πy(t))〉〉}dtα≤-b(x∘(·),y(·))∥θ(x∘(·),y(·))∥2∑ℓ=1rρ'ℓ).
Hypothesis (d) regarding the(ρκ′′,b)-quasi-invexity property of each functional implies (κ=1,q¯):(4.14)(∫γt0,t1‍[〈μαPκ(t),gPκ(πx∘(t))〉+〈ναQκ(t),hQκ(πx∘(t))〉]dtα≤∫γt0,t1‍[〈μαPκ(t),gPκ(πy(t))〉+〈ναQκ(t),hQκ(πy(t))〉]dtα)⇒(∫γt0,t1‍b(x∘(·),y(·))×∫γt0,t1‍[〈η(πx∘(t),πy(t)),〈μαPκ(t),∂gPκ∂y(πy(t))〉〉+〈ναQκ(t),∂hQκ∂y(πy(t))〉+〈Dγη(πx∘(t),πy(t)),〈μαPκ(t),∂gPκ∂yγ(πy(t))〉〉+〈ναQκ(t),∂hQκ∂yγ(πy(t))〉]dtα≤-ρκ′′b(x∘(·),y(·))∥θ(x∘(·),y(·))∥2∫γt0,t1‍).
Now, we make the sum of implications (4.13) and (4.14) side by side and from κ=1 to κ=q. It follows
(4.15)(∫γt0,t1‍〈Λ10,F(x∘(·))〉-〈Λ20,K(x∘(·))〉+∫γt0,t1‍[〈μα(t),g(πx∘(t))〉+〈να(t),h(πx∘(t))〉]dtα-∫γt0,t1‍[〈μα(t),g(πy(t))〉+〈να(t),h(πy(t))〉]dtα≤0)⇒((∑ℓ=1r‍ρ'ℓ+∑κ=1q‍ρκ′′)b(x∘(·),y(·))×∫γt0,t1‍[〈η(πx∘(t),πy(t)),〈Λ10,∂fα∂y(πy(t))〉-〈Λ20,∂kα∂y(πy(t))〉〉+〈μα(t),∂g∂y(πy(t))〉+〈να(t),∂h∂y(πy(t))〉+〈Dγη(πx∘(t),πy(t)),〈Λ10,∂fα∂yγ(πy(t))〉-〈Λ20,∂kα∂yγ(πy(t))〉+〈μα(t),∂g∂yγ(πy(t))〉+〈να(t),∂h∂yγ(πy(t))〉〉]dtα-<b(x∘(·),y(·))∥θ(x∘(·),y(·))∥2(∑ℓ=1r‍ρ'ℓ+∑κ=1q‍ρκ′′)).
Sinceb(x∘(·),y(·))>0, we obtain
(4.16)∫γt0,t1‍[〈η(πx∘(t),πy(t)),∂Vα∂y(πy(t),Λ10,Λ20,μ(·),ν(·))〉∫γt0,t1‍+〈Dγη(πx∘(t),πy(t)),∂Vα∂yγ(πy(t),Λ10,Λ20,μ(·),ν(·))〉]dtα<-∥θ(x∘(·),y(·))∥2(∑ℓ=1r‍ρ'ℓ+∑κ=1q‍ρκ′′),
where
(4.17)Vα(πy(·),Λ10,Λ20,μ(·),ν(·))=〈Λ10,fα(πy(t))〉-〈Λ20,kα(πy(t))〉Vα(πy(·),Λ10,Λ20,μ(·),ν(·))+〈μα(y(t)),g(πy(t))〉+〈να(t),h(πy(t))〉,t∈Ωt0,t1,α=1,p¯.
The following relation holds:(4.18)〈Dγη(πx∘(t),πy(t)),∂Vα∂yγ(πy(t),Λ10,Λ20,μ(t),ν(t))〉=Dγ〈η(πx∘(t),πy(t)),∂Vα∂yγ(πy(t),Λ10,Λ20,μ(t),ν(t))〉-〈η(πx∘(t),πy(t)),Dγ(∂Vα∂yγ)(πy(t),Λ10,Λ20,μ(t),ν(t))〉.
By replacing relations (4.18) and by using Euler-Lagrange PDE, relation (4.16) becomes
(4.19)∫γt0,t1‍Dγ〈η(πx∘(t),πy(t)),∂Vα∂yγ(πy(t),Λ10,Λ20,μ(t),ν(t))〉dtα<-∥θ(x∘(·),y(·))∥2(∑ℓ=1r‍ρ'ℓ+∑κ=1q‍ρκ′′).Forα,γ=1,p¯, let us denote by
(4.20)Qαγ(t)=〈η(πx∘(t),πy(t)),∂Vα∂yγ(πy(t),Λ10,Λ20,μ(t),ν(t))〉,I=∫γt0,t1‍DγQαγ(t)dtα.According to [28], § 9, a total divergence is equal to a total derivative. Consequently, there exists Q(t), with Q(t0)=0 and Q(t1)=0 such that DγQαγ(t)=DαQ(t) and
(4.21)I=∫γt0,t1‍DαQ(t)dtα=Q(t1)-Q(t0)=0.Replacing into inequality (4.19), it follows that
(4.22)0<-∥θ(x∘(·),y(·))∥2(∑ℓ=1r‍ρ'ℓ+∑a=1q‍ρa′′),
contradicting hypothesis (f).From relation (4.15), it follows
(4.23)0≤〈Λ10,F(x∘(·))〉-〈Λ20,K(x∘(·))〉+∫γt0,t1‍[〈μα(t),g(πx∘(t))〉+〈να(t),h(πx∘(t))〉]dtα-∫γt0,t1‍[〈μα(t),g(πy(t))〉+〈να(t),h(πy(t))〉]dtα.
According to the constraints of problems (MFP) and (MFZD), the previously mentioned relation becomes Λℓ10Fℓ(x∘(·))-Λℓ20Kℓ(x∘(·))>0, that is:
(4.24)Kℓ(x∘(·))Kℓ(y(·))[Fℓ(x∘(·))Kℓ(x∘(·))-Fℓ(y(·))Kℓ(y(·))]>0.BecauseKℓ(x∘(·))Kℓ(y(·))>0, ℓ=1,r¯, we conclude that
(4.25)(F1(x∘(·))K1(x∘(·))-F1(y(·))K1(y(·)),…,Fr(x∘(·))Kr(x∘(·))-Fr(y(·))Kr(y(·)))≰(0,…,0),
or
(4.26)(F1(x∘(·))K1(x∘(·)),…,Fr(x∘(·))Kr(x∘(·)))≰(F1(y(·))K1(y(·)),…,Fr(y(·))Kr(y(·))).Therefore, the inequalityϖ(x∘(·))≤δ¯(y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) contradicts relations (4.12), and this completes the proof.Theorem 4.5 (direct duality).
Letx∘(·) be an efficient solution of primal  (MFP). Suppose the hypotheses of Theorem 4.4 hold. Then there are Λ10 and Λ20 in ℝr, and the smooth functions μ∘:Ωt0,t1→ℝqsp and ν∘:Ωt0,t1→ℝqsp, such that (x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)) is an efficient solution of dual program (MFZD). Moreover
(4.27)ϖ(x∘(·))=δ¯(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)).Proof.
x
∘
(
·
) being efficient solution of primal (MFP), there are Λ10 and Λ20 in ℝr and the smooth functions μ∘:Ωt0,t1→ℝqsp and ν∘:Ωt0,t1→ℝqsp, such that relations (4.6) are verified. Therefore, 〈μα(t),g(πx∘(t))〉=0, α=1,p¯. It follows (x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)) is a feasible solution for program (MFZD). Obviously,
(4.28)ϖ(x∘(·))=δ¯(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)).
The weak duality theorem assures the efficiency of x∘(·).If we remark that the notion of efficient solution of problem(MFZD) is similar to those in Definition 4.1, we can state the result in the following.Theorem 4.6 (converse duality).
Let(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)) be an efficient solution to dual (MFZD) and suppose the following conditions are satisfied: (i)
x¯(·) is a normal efficient solution of primal (MFP);(ii)
the hypotheses of Theorem4.4 hold at (x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)).
Thenx∘(·) is an efficient solution to (MFP). Moreover, one has the equality
(4.29)ϖ(x∘(·))=δ¯(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)).Proof.
Supposex¯(·)≠x∘(·). The efficiency of x¯(·) of primal (MFP) implies the existence of Λ¯10 and Λ¯20 in ℝr and the smooth functions μ¯∘:Ωt0,t1→ℝqsp and ν¯∘:Ωt0,t1→ℝqsp, such that relations (4.6) are satisfied. It follows that (x¯(·),x¯γ(·),Λ¯10,Λ¯20,μ¯∘(·),ν¯∘(·)) is a feasible solution of (MFZD) and
(4.30)ϖ(x¯(·))=δ¯(x¯(·),x¯γ(·),Λ¯10,Λ¯20,μ¯∘(·),ν¯∘(·)).
On the other hand, from the weak duality theorem,(4.31)ϖ(x¯(·))≰δ(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·))
holds, that is:
(4.32)δ(x¯(·),x¯γ(·),Λ¯10,Λ¯20,μ¯∘(·),ν¯∘(·))≰δ(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)).
This last relation contradicts the efficiency of(x∘(·),xγ∘(·),Λ10,Λ20,μ∘(·),ν∘(·)) for program (MFZD). Therefore x¯(·)=x∘(·), and the theorem is proved.
We would like to conclude the study in this section with the following important particular case.
In this respect, suppose thatkαℓ=positiveconstant, ℓ=1,r¯, α=1,p¯. Denote F(x(·))=(F1(x(·)),…,Fr(x(·))). Then, program (MFP) becomes
(MP)minF(x)subjecttox(·)∈ℱ(Ωt0,t1),
and its dual is
(GMWD)max∫ab‍Le(πy(t))dtsubjecttoΛℓ10∂fαℓ∂y(πy(t))+〈μα(t),∂g∂y(πy(t))〉+〈να(t),∂h∂y(πy(t))〉-Dγ(Λℓ10∂fαℓ∂yγ(πy(t))+〈μα(t),∂g∂yγ(πy(t))〉+〈να(t),∂h∂yγ(πy(t))〉)=0,α=1,p¯,t∈Ωt0,t1,〈μαPκ(t),gPκ(πy(t))〉+〈ναQκ(t),hQκ(πy(t))〉≥0,κ=1,q¯,α=1,p¯,t∈Ωt0,t1,Λ10≥0,
where
(4.33)Le(πy(t))=f(πy(t))+[〈μP0(t),gP0(πx(t))〉+〈νQ0(t),hQ0(πx(t))〉]e,
with Le=(L1,…,Lr).
It can be seen that we have obtained precisely the programs studied in the work [14]. Therefore, the results in this paper are stronger than the ones mentioned before.


## 5.   Conclusion

In our recent study [14], we initiated an optimization theory on the first order jet bundle. As natural continuation, in this paper we considered a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. We derived duality results for efficient solutions of multitime multiobjective ratio variational problems under the assumptions of (ρ,b)-quasi-invexity. We proved our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained.Having in mind the physical significance of the objective function, this study is strongly motivated by its possible applications of nonlinear optimization to mechanical engineering and economics [29].



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*Source: 589694-2012-08-26.xml*