# Minimization of Vectors of Curvilinear Functionals on Second-Order Jet Bundle: Dual Program Theory

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

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## Abstract

A previous paper (2011), Pitea and
Postolache, considered the problem of minimization of vectors of curvilinear functionals
(well known as mechanical work), thought as multitime multiobjective variational
problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework
offered by the second-order jet bundle, and initiated an optimization theory for
this class of problems by introducing necessary conditions. As natural continuation
of these results, the present work introduces a dual program theory, the general setting, and the theory which is new as a whole, containing our
results.


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## Body

## 1. Introduction

Fractional programming problems arise in wide areas of research in pure and applied sciences and in the new technology as well. We have in mind portfolio selection, stock cutting, game theory, and various decision problems in management science. The study of duality for this class of problems is an active area of research due to its wide scope of applications, see [1] by Chinchuluun and Pardalos, for an excellent paper in multiobjective optimization.The duality theory for multiobjective optimization problems based on a vector-valued Lagrangian function were extensively investigated in [2] by Bitran, and [3] by Tanino and Sawaragi. Several authors were interested in this problem, see [4] by Egudo, [5] by Mond and Husain, [6] by Weir and Mond. Later, an interesting and rich duality theory for such problems was developed by Aghezzaf and Hachimi, [7], Preda [8, 9], and recently, duality problems were studied in [10] by Boţ and Grad, [1] by Chinchuluun and Pardalos, [11] by Pitea et al.In the problems of our study, the objective function is of curvilinear integral type and could mould the mechanical work when we have in mind design problems in engineering. Also, such kind of problems proved to be useful tools for minimizing the cost or maximizing the profit when we discuss economical problems.Motivated by the study reported above, in a very recent work [12], Pitea and Postolache considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second-order jet bundle and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory. It is organized as follows. Next, in Section 2, we introduce the general setting, while in Section 3 we prove our results. Finally, we conclude the paper.


## 2. General Setting

Let(T,h) and (M,g) be Riemannian manifolds of dimensions p and n, respectively. The local coordinates on T and M will be written t=(tα) and x=(xi), respectively. Let J2(T,M) be the second-order jet bundle associated to T and M, see [13].Throughout this work, we use the customary relations between two vectors of the same dimension, [11]. With the product-order relation on ℝp, the hyperparallelepiped Ωt0,t1, in ℝp, with the diagonal opposite points t0=(t01,…,t0p) and t1=(t11,…,t1p), can be written as interval [t0,t1]. Suppose γt0,t1 is a piecewise C2-class curve joining the points t0 and t1.To simplify the notations, denote by(2.1)xγ(t)=∂x∂tγ(t),γ=1,p¯,xθσ(t)=∂2x∂tθ∂tσ(t),θ,σ=1,p¯,the partial velocities and partial accelerations, respectively. Also, in our subsequent theory, we will set πx(t)=(t,x(t),xγ(t),xθσ(t)).OnJ2(T,M) with values on ℝr, consider the closed Lagrange 1-forms densities of C∞-class:
(2.2)fα=(fαl),kα=(kαl),l=1,r¯,α=1,p¯,
which determine the following path-independent functionals:
(2.3)Fl(x(⋅))=∫γt0,t1fαl(πx(t))dtα,Kl(x(⋅))=∫γt0,t1kαl(πx(t))dtα.
The closeness conditions (complete integrability conditions) are(2.4)Dβfαl=Dαfβl,Dβkαl=Dαkβl,α,β=1,p¯,α≠β,l=1,r¯,whereDβ is the total derivative.We accept that the Lagrange matrix densities:(2.5)g=(gab):J2(T,M)⟶Rmd,a=1,d¯,b=1,m¯,m<n,ofC∞-class define the partial differential inequations (PDI) of evolution
(2.6)g(πx(t))≦0,t∈Ωt0,t1,
and the Lagrange matrix densities,
(2.7)h=(hab):J2(T,M)⟶Rqd,a=1,s¯,b=1,q¯,q<n,
define the partial differential equations (PDE), of evolution
(2.8)h(πx(t))=0,t∈Ωt0,t1.For eachℓ=1,r¯, suppose Kℓ(x(·))>0, and consider
(2.9)F(x(⋅))K(x(⋅))=(F1(x(⋅))K1(x(⋅)),…,Fr(x(⋅))Kr(x(⋅))).With conditions (2.6) and (2.8), we denote by
(2.10)F(Ωt0,t1)={x∈C∞(Ωt0,t1,M)∣x(t0)=x0,x(t1)=x1,g(πx(t))≦0,h(πx(t))=0,t∈Ωt0,t1}
the set of all feasible solutions of problem:
(2.11)(MFP){min⁡F(x(⋅))K(x(⋅)),subjecttox(⋅)∈F(Ωt0,t1),
a PDI- and/or PDE-constrained minimum problem.In [12], we introduced necessary efficiency conditions for problem (MFP). The aim of this work is to introduce and study two dual programs:(1)
the multitime multiobjective fractional variational problem (MFP) of minimizing a vector of quotients of path-independent curvilinear functionals;(2)
the multiobjective variational dual problem:(2.12)(MFD){max⁡y(⋅)(F(y(⋅))K(y(⋅)))subjecttoΛl10∂fαl∂y(πy(t))-Λl20∂kαl∂y(πy(t))+〈μα(t),∂g∂y(πy(t))〉+〈να(t),∂h∂y(πy(t))〉-Dγ(Λl10∂fαl∂yγ(πy(t))-Λl20∂kαl∂yγ(πy(t))+〈μα(t),∂g∂yγ(πy(t))〉+〈να(t),∂h∂yγ(πy(t))〉)+Dθσ2(〈λ,∂fα∂xθσ(πy(t))〉+〈μα(t),∂g∂xθσ(πy(t))〉+〈να(t),∂h∂xθσ(πy(t))〉)=0,t∈Ωt0,t1,α=1,p¯〈μα(t),g(πy(t))〉+〈να(t),h(πy(t))〉≧0,α=1,p¯,t∈Ωt0,t1Λ10≥0,
taking into account that the functions x(t) and y(t) have to satisfy the boundary conditions x(t0)=x0, x(t1)=x1, or x(t)|∂Ωt0,t1=χ= given, respectively, y(t0)=x0, y(t1)=x1, or y(t)|∂Ωt0,t1=χ= given, the partial differential inequations of evolution (2.6), and the partial differential equations of evolution (2.8).To develop the theory in our main section, we need the notion of efficient solution.Definition 2.1.
A feasible solutionx°(·)∈ℱ(Ωt0,t1) is called efficient solution for the program (MFP) if and only if for any feasible solution x(·)∈ℱ(Ωt0,t1), one has the implication:(2.13)F(x(⋅))K(x(⋅))≧F(x°(⋅))K(x°(⋅))⟹F(x(⋅))K(x(⋅))=F(x°(⋅))K(x°(⋅)).


## 3. Main Results

To state our results, we have to introduce an appropriate generalized convexity. For this purpose, considerρ be a real number, b:C∞(Ωt0,t1,M)×C∞(Ωt0,t1,M)→[0,∞) a functional, and a=(aα), α=1,p¯, a closed 1-form. To a we associate the curvilinear integral
(3.1)A(x(⋅))=∫γt0,t1aα(πx(t))dtα.Definition 3.1.
The functionalA is called [strictly](ρ,b)-quasiinvex at the pointx°(·) if there exists a vector function η:J2(Ωt0,t1,M)×J2(Ωt0,t1,M)→ℝn, vanishing at the point (πx°(t),πx°(t)), and the function θ defined on C∞(Ωt0,t1,M)×C∞(Ωt0,t1,M) to ℝn, such that for any x(·)[x(·)≠x°(·)], the following implication holds:
(3.2)(A(x(⋅))≤A(x°(⋅)))⟹(bx((⋅),x°(⋅))∫γt0,t1[〈η(πx(t),πx°(t)),∂aα∂x(πx°(t))〉+〈Dγη(πx(t),πx°(t)),∂aα∂xγ(πx°(t))〉+〈Dθσ2η(πx(t),πx°(t)),∂aα∂xθσ(πx°(t))〉]dtα[<]≤-ρb(x(⋅),x°(⋅))‖θ(x(⋅),x°(⋅))‖2).
The notion of quasiinvexity is used, in appropriate forms, in recent works for studies of some multiobjective programming problems, for example, see [14] by Nahak and Mohapatra.
Let us denote byϖ(x°(·)) the minimizing functional vector of problem (MFP) at the point x°(·)∈ℱ(Ωt0,t1) and by (δ(y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) the maximizing functional vector of dual (MFD) at the point (y(·),yγ(·),Λ10,Λ20,μ(·),ν(·))∈Δ, where Δ is the domain of problem (MFD).Theorem 3.2 (Weak Duality).
Letx°(·) be a feasible solution of the problem  (MFP) and let  y(·) be an efficient solution of problem  (MFD). Assume that the following conditions are fulfilled: (a)
Λℓ10>0, Λℓ20>0, Λℓ10Fℓ(y(·))-Λℓ20Kℓ(y(·))=0, ℓ=1,r¯;(b)
for anyℓ=1,r¯, the functional Fℓ(x(·)) is (ρ′ℓ,b)-quasiinvex at the point y(·) and the functional -Kℓ(x(·)) is (ρ′′ℓ,b)-quasiinvex at the point y(·) with respect to η and θ;(c)
the functional,(3.3)∫γt0,t1[〈μα(t),g(πx(t))〉+〈να(t),h(πx(t))〉]dtα,
is (ρ′′′,b)-quasiinvex at the point y(·) with respect to η and θ;(d)
either the functional of (b) or the functional of (c) is strictly quasiinvex;(e)
ρ′ℓΛℓ10+ρ′′ℓΛℓ20+ρ′′′≥0.Then, the inequality ϖ(x°(·))≤δ(y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) is false.Proof.
From condition (b), it follows that(3.4)(Fl(x°(⋅))Fl(y(⋅)))⟹(b(x°(⋅),y(⋅))∫γt0,t1[〈η(πx°(t),πy(t)),∂fαl∂y(πy(t))〉+〈Dγη(πx°(t),πy(t)),∂fαl∂yγ(πy(t))〉+〈Dθσ2η(πx°(t),πy(t)),∂fαl∂yθσ(πy(t))〉]dtα≦-ρ′lb(x°(⋅),y(⋅))‖θ(x°(⋅),y(⋅))‖2),l=1,r¯,(3.5)(-Kl(x°(⋅))-Kl(y(⋅)))⟹(b(x°(⋅),y(⋅))∫γt0,t1[〈-η(πx°(t),πy(t)),∂kαl∂y(πy(t))〉-〈Dγη(πx°(t),πy(t)),∂kαl∂yγ(πy(t))〉-〈Dθσ2η(πx°(t),πy(t)),∂kαl∂yθσ(πy(t))〉]dtα≦ρ′′lb(x°(⋅),y(⋅))‖θ(x°(⋅),y(⋅))‖2),l=1,r¯.
We multiply (3.4) by Λℓ10>0 and (3.5) by Λℓ20>0. We make the sum and we obtain the following implications:
(3.6)(Λl10Fl(x°(⋅))-Λl20Kl(x°(⋅))≦0)⟹(b(x°(⋅),y(⋅))∫γt0,t1{〈η(πx°(t),πy(t)),Λl10∂fαl∂y(πy(t))-Λl20∂kαl∂y(πy(t))〉+〈Dγη(πx°(t),πy(t)),Λl10∂fαl∂yγ(πy(t))-Λl20∂kαl∂yγ(πy(t))〉+〈Dθσ2η(πx°(t),πy(t)),Λl10∂fαl∂yθσ(πy(t))-Λl20∂kαl∂yθσ(πy(t))〉}dtα≦-b(x°(⋅),y(⋅))‖θ(x°(⋅),y(⋅))‖2(ρ′lΛl10+ρ′′lΛl20)).
According to hypothesis (c), we have
(3.7)(∫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)⟹(b(x°(⋅),y(⋅))∫γt0,t1[〈η(πx°(t),πy(t)),〈μα(t),∂g∂y(πy(t))〉+〈να(t),∂h∂y(πy(t))〉〉+〈Dγη(πx°(t),πy(t))〉,〈μα(t),∂g∂yγ(πy(t))〉+〈να(t),∂h∂yγ(πy(t))〉〉+〈Dθσ2η(πx°(t),πy(t))〉,〈μα(t),∂g∂yθσ(πy(t))〉+〈να(t),∂h∂yθσ(πy(t))〉〉]dtα≦-ρ′′′b(x°(⋅),y(⋅))‖θ(x°(⋅),y(⋅))‖2)∫γt0,t1.
Making the sum of the implications (3.6) and (3.7), it follows that
(3.8)(Λl10Fl(x°(⋅))-Λl20Kl(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)⟹(b(x°(⋅),y(⋅))∫γt0,t1[〈η(πx°(t),πy(t)),Λl10∂fαl∂y(πy(t))-Λl20∂kαl∂y(πy(t))+〈μα(t),∂g∂y(πy(t))〉+〈να(t),∂h∂y(πy(t))〉〉+〈Dγη(πx°(t),πy(t)),Λl10∂fαl∂yγ(πy(t))-Λl20∂kαl∂yγ(πy(t))+〈μα(t),∂g∂yγ(πy(t))〉+〈να(t),∂h∂yγ(πy(t))〉〉+〈Dθσ2η(πx°(t),πy(t)),Λl10∂fαl∂yθσ(πy(t))-Λl20∂kαl∂yθσ(πy(t))+〈μα(t),∂g∂yθσ(πy(t))〉+〈να(t),∂h∂yθσ(πy(t))〉〉]dtα<-b(x°(⋅),y(⋅))‖θ(x°(⋅),y(⋅))‖2(ρ′lΛl10+ρ′′lΛl20+ρ′′′)).
Since b(x°(·),y(·))>0, we obtain
(3.9)∫γt0,t1[〈ηπx°(t),πy(t),∂Vα∂yπy(t),λ,μ(t),ν(t)〉+〈Dγη(πx°(t),πy(t)),∂Vα∂yγ(πy(t),λ,μ(t),ν(t))〉+〈Dθσ2η(πx°(t),πy(t)),∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉]dtα<-‖θ(x°(⋅),y(⋅))‖2(ρ′lΛl10+ρ′′lΛl20+ρ′′′),
where
(3.10)Vα(πy(⋅),λ,μ(⋅),ν(⋅))=Λl10fαl(πy(t))-Λl20kαl(πy(t))+〈μα(y(t)),g(πy(t))〉+〈να(t),h(πy(t))〉,t∈Ωt0,t1,α=1,p¯.
The following relations hold:
(3.11)〈Dγη(πx°(t),πy(t)),∂Vα∂yγ(πy(t),λ,μ(t),ν(t))〉=Dγ〈η(πx°(t),πy(t)),∂Vα∂yγ(πy(t),λ,μ(t),ν(t))〉-〈η(πx°(t),πy(t)),Dγ(∂Vα∂yγ)(πy(t),λ,μ(t),ν(t))〉,(3.12)〈Dθσ2η(πx°(t),πy(t)),∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉=DθDσ〈η(πx°(t),πy(t)),∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉-Dθ〈η(πx°(t),πy(t)),Dσ∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉-Dσ〈η(πx°(t),πy(t)),Dθ∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉+〈η(πx°(t),πy(t)),Dσθ2∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉.
By replacing the relations (3.11) and (3.12), and using Euler-Lagrange PDE, the relation (3.9) becomes
(3.13)∫γt0,t1[Dγ〈η(πx°(t),πy(t)),∂Vα∂yγ(πy(t),λ,μ(t),ν(t))〉+DθDσ〈η(πx°(t),πy(t)),∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉-Dθ〈η(πx°(t),πy(t)),∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉-Dσ〈η(πx°(t),πy(t)),Dθ∂Vα∂yθσ(πy(t),λ,μ(t),ν(t))〉]dtα<-‖θ(x°(⋅),y(⋅))‖2(ρ′lΛl10+ρ′′lΛl20+ρ′′′).
According to [15], §9, a total divergence is equal to a total derivative, therefore, the left-hand side of (3.13) becomes null, and replacing into the inequality (3.13), it follows that
(3.14)0<-‖θ(x°(⋅),y(⋅))‖2(ρ′lΛl10+ρ′′lΛl20+ρ′′′).
From hypothesis (e), the previous relation becomes 0<0, which is false. Next, from relation (3.8), it follows that
(3.15)0≦Λl10Fl(x°(⋅))-Λl20Kl(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.
Taking into account the inequality:
(3.16)∫γt0,t1[〈μα(t),g(πy(t))〉+〈να(t),h(πy(t))〉]dtα≧0,
the above-mentioned relation becomes
(3.17)Λl10Fl(x°(⋅))-Λl20Kl(x°(⋅))>0,
that is,
(3.18)Kl(x°(⋅))Kl(y(⋅))[Fl(x°(⋅))Kl(x°(⋅))-Fl(y(⋅))Kl(y(⋅))]>0.
Because Kℓ(x°(·))Kℓ(y(·))>0, ℓ=1,r¯, we conclude that
(3.19)F(x°(⋅))K(x°(⋅))-F(y(⋅))K(y(⋅))≰(0,…,0),
or
(3.20)F(x°(⋅))K(x°(⋅))≰F(y(⋅))K(y(⋅)).
Therefore, the relation ϖ(x°(·))≤δ(y(·),yγ(·),Λ10,Λ20,μ(·),ν(·)) is not satisfied, and this completes the proof.We will finish our considerations by giving the statements of two results on direct duality and converse duality, respectively. Their proofs follow directly from the weak duality.Theorem 3.3 (Direct Duality).
Letx°(·) be an efficient solution of  (MFP). Suppose that the hypotheses of Theorem are satisfied. Then, there are  Λ10 and  Λ20 in  ℝr, and the smooth functions  μ°:Ωt0,t1→ℝmsp and  ν°:Ωt0,t1→ℝqsp, such that  (x°(·),xγ°(·),Λ10,Λ20,μ°(·),ν°(·)) is an efficient solution of the dual  (MFD), and  ϖ(x°(·))=δ(x°(·),xγ°(·),Λ10,Λ20,μ°(·),ν°(·)).We will present now a theorem concerning the converse duality, by changing some of the hypotheses.Theorem 3.4 (Converse Duality).
Let(x°(·),xγ°(·),Λ10,Λ20,μ°(·),ν°(·)) be an efficient solution of the dual problem  (MFD). Suppose the following conditions hold: (a)
x̅(·) is an efficient solution of the primal problem (MFP);(b)
for anyℓ=1,r¯, one has(3.21)Fl(x°(⋅))>0,Kl(x°(⋅))>0,Λl10Fl(x°(⋅))-Λl20Kl(x°(⋅))=0;(c)
for anyℓ=1,r¯, the functional Fℓ(x(·)) is (ρ′ℓ,b)-quasiinvex at the point x°(·) and the functional -Kℓ(x(·)) is (ρ′′ℓ,b)-quasiinvex at the point x°(·), with respect to η and θ;(d)
the functional,(3.22)∫γt0,t1[〈μα(t),g(πx(t))〉+〈να(t),h(πx(t))〉]dtα,
is (ρ′′′,b)-quasiinvex at the point x°(·) with respect to η and θ;(e)
one of the functionals of (c), (d) is strictly(ρ′ℓ,b)-, (ρ′′ℓ,b)-, or (ρ′′′,b)-quasiinvex with respect to η and θ, respectively;(f)
ρ′ℓΛℓ10+ρ′′ℓΛℓ20+ρ′′′≥0.Then, x̅(·)=x°(·), and ϖ(x°(·))=δ(x°(·),xγ°(·),Λ10,Λ20,μ°(·),ν°(·)).


## 4. Conclusions

In our previous work [12], we initiated an optimization theory for the second-order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints (limited resources). Within this framework, we introduced necessary conditions. As natural continuation of our results in [12], and strongly motivated by its possible applications in mechanics, the present work introduced a dual program theory for this class of problems (for related but complementary research, see [16, 17]).



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