# Sufficient Efficiency Conditions for Vector Ratio Problem on the Second-Order Jet Bundle

Authors: Ariana Pitea
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Motivated by its possible applications in mechanics and mechanical engineering, in
our previous published work (Pitea and Postolache, 2011),
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 problems, subject to PDE
and/or PDI constraints. Within this framework, we introduced necessary optimality
conditions. As natural continuation of these results, the present work introduces a
study of sufficient efficiency conditions.


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## Body

## 1. Introduction

As it is known, most of the optimization problems arising in practice have several objectives which have to be optimized simultaneously. These problems, of considerable interest, include various branches of mathematical sciences, engineering design, portfolio selection, game theory, decision problems in management science, web access problems, query optimization in databases, and so forth. Also, such kind of optimization problems arise in wide areas of research for new technology as well. First of all, we have in mind the material sciences where many times optimal estimation of material parameters is required, either nondestructive determination of faults is needed. Next, chemistry which provides a huge class of constrained optimization problems such as the determination of contamination sources given the flow model and the variance of the source. Last, but not least, games theory in the main study is finding optimal wining strategies. For descriptions of the web access problem, the portfolio selection problem, and capital budgeting problem, see [1] by Chinchuluun and Pardalos, [2] by Chinchuluun et al., and some references therein.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. For advances on single-objective ratio programs, see [3] by Khan and Hanson, and [4] by Reddy and Mukherjee, which utilized invexity assumptions in the sense of Hanson [5] to obtain optimality conditions and duality results. As concerning vector ratio problems, Singh and Hanson [6] applied invex functions to derive duality results, while Jeyakumar and Mond [7] generalized these results to the class of V-invex functions. Later, Liang et al. [8] introduced a unified formulation of the generalized convexity to derive optimality conditions and duality results for vector ratio problems.Despite of these important advances in optimization, our multitime multiobjective problem—required by practical reasons—had not been studied so far. In the problem of our very recent study [9, 10], the objective vector function is of curvilinear integral type, the integrand depending both on velocities and accelerations, that is why we have chosen as framework the second-order jet bundle [11] (also, see [12]).This work extends, generalizes, and further develops our research in [13, 14] from the first-order jet bundle (which considers the movement as function of velocities only) to the second-order jet bundle (where the movement is a function both on velocities and accelerations). Note that passing from the first-order jet bundle to the second-order jet bundle is not a facile task, it requests specific techniques to the latter one (e.g., a new quasiinvexity, an appropriate mathematical framework).Our study is encouraged by its possible application, especially in mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. The objective functions of our type play an essential role in mathematical modeling of certain processes in relation with robotics, tribology, engines, and so forth.This paper aims to establish some new results on nonlinear optimization on the second order jet bundle. It is organized as follows. In Section2 our framework is introduced and the minimization problem is described, while in Section 3 sufficient efficiency conditions for our problem are given. Finally, we conclude the paper and suggest possible further development.


## 2. Our Framework and Problem Description

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 [11].Throughout this work, we use the customary relations between two vectors of the same dimension, [9]. 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.Important note
To simplify the presentation, in our subsequent theory, we shall set(2.1)πx(t)=(t,x(t),xγ(t),xθσ(t)),πx∘(t)=(t,x∘(t),xγ∘(t),xθσ∘(t)),
where xγ(t)=(∂x/∂tγ)(t), γ=1,p¯, and xθσ(t)=(∂2x/∂tθ∂tσ)(t), θ,σ=1,p¯, are partial velocities and partial accelerations respectively.The closed Lagrange 1-form densities of C∞-class:
(2.2)fα=(fαl):J2(T,M)⟶Rr,kα=(kαl):J2(T,M)⟶Rr,l=1,r¯,α=1,p¯,
determine the following path independent functionals, respectively,
(2.3)Fl(x(⋅))=∫γt0,t1fαl(πx(t))dtα,Kl(x(⋅))=∫γt0,t1kαl(πx(t))dtα.
We accept that the Lagrange matrix densities:
(2.4)g=(gab):J2(T,M)⟶Rmd,a=1,d¯,b=1,m¯,m<n,
of C∞-class define the partial differential inequations (PDI of evolution)
(2.5)g(πx(t))≧0,t∈Ωt0,t1,
and the Lagrange matrix densities
(2.6)h=(hab):J2(T,M)⟶Rqd,a=1,d¯,b=1,q¯,q<n,
defines the partial differential equations (PDE) (of evolution)
(2.7)h(πx(t))=0,t∈Ωt0,t1.
Let C∞(Ωt0,t1,M) be the space of all functions x:Ωt0,t1→M of C∞-class, with the norm
(2.8)‖x‖=‖x‖∞+∑α=1p∥xα∥∞+∑θ,σ=1p‖xθσ‖∞.
For each ℓ=1,r¯, suppose Kℓ(x(·))>0, and consider
(2.9)F(x(⋅))K(x(⋅))=(F1(x(⋅))K1(x(⋅)),…,Fr(x(⋅))Kr(x(⋅))).
With conditions (2.5) and (2.7), 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.Using the terminology from analytical mechanics, in MFP there are given  numbers of r sources, each of them producing mechanical work. This one have to be minimized on a set of limited resources, namely, ℱ(Ωt0,t1).In our previous work [9], we introduced necessary efficiency conditions for problem (MFP). In Section 3, we would like to further develop these results by introducing sufficient efficiency conditions for problem (MFP).


## 3. Main Results

Definition 3.1.
A feasible solutionx∘(·)∈ℱ(Ωt0,t1) is called efficient for program (MFP) if and only if for any feasible solution x(·)∈ℱ(Ωt0,t1):
(3.1)F(x(⋅))K(x(⋅))≦F(x∘(⋅))K(x∘(⋅))⟹F(x(⋅))K(x(⋅))=F(x∘(⋅))K(x∘(⋅)).Consider thatx∘(·)∈ℱ(Ωt0,t1) is an efficient solution of problem (MFP). In a very recent article [9], Pitea and Postolache proved that there exist Λ1∘,Λ2∘∈ℝr and the smooth functions μ∘:Ωt0,t1→ℝmdp, ν∘:Ωt0,t1→ℝqdp, such that
(3.2)〈Λ1∘,∂fα∂x(πx∘(t))〉-〈Λ2∘,∂kα∂x(πx∘(t))〉+〈μα∘(t),∂g∂x(πx∘(t))〉+〈να∘(t),∂h∂x(πx∘(t))〉-Dγ(〈Λ1∘,∂fα∂xγ(πx∘(t))〉-〈Λ2∘,∂kα∂xγ(πx∘(t))〉+〈μα∘(πx∘(t)),∂g∂xγ(πx∘(t))〉+〈να∘(t),∂h∂xγ(πx∘(t))〉)+Dθσ2(〈Λ1∘,∂fα∂xθσ(πx∘(t))〉-〈Λ2∘,∂kα∂xθσ(πx∘(t))〉+〈μα∘(πx∘(t)),∂g∂xθσ(πx∘(t))〉+〈να∘(t),∂h∂xθσ(πx∘(t))〉)=0,t∈Ωt0,t1,α=1,p¯A Natural Question Arises.
Are these conditions sufficient forx∘(·)to be an efficient solution for program(MFP)?To develop our theory, we have to introduce an appropriate generalized convexity.Letρ 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. 						(3.3)A(x(⋅))=∫γt0,t1aα(πx(t))dtα.Definition 3.2.
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:
(3.4)(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∫γt0,t1).
The quasiinvexity is used, in appropriate forms, in recent works for studies of some multiobjective programming problems, see [15–17] by Mititelu et al., [18] by Nahak and Mohapatra.
Now, we can establish efficiency sufficient conditions for problem (MFP).Theorem 3.3.
Consider the vectorsΛ1∘, Λ2∘ from ℝr and the functions x∘(·), μ∘(·), ν∘(·) satisfying conditions (3.2). Suppose that the following properties hold:(a)
the functional〈Λ1∘,F(x(·))〉-〈Λ2∘,K(x(·))〉 is (ρ1,b)-quasiinvex at the point x∘(·) with respect to η and θ;(b)
the functional∫γt0,t1〈μα∘(t),g(πx(t))〉dtα is (ρ2,b)-quasiinvex at the point x∘(·) with respect to η and θ;(c)
the functional∫γt0,t1〈να∘(t),h(πx(t))〉dtα is (ρ3,b)-quasiinvex at the point x∘(·) with respect to η and θ;(d)
one of the integrals of (a)—(c) is strictly(ρ1,b), (ρ2,b) or (ρ3,b)-quasiinvex at the point x∘(·) with respect to η and θ;(e)
ρ1+ρ2+ρ3≥0;(f)
Λℓ1∘Fℓ(x∘(·))-Λℓ2∘Kℓ(x∘(·))=0,  for each ℓ=1,r¯.Then the point x∘(·) is an efficient solution of problem (MFP).Proof.
Let us suppose that the pointx∘(·) is not an efficient solution for problem (MFP). Then, there is a feasible solution x(·) for problem (MFP), such that
(3.5)Fl(x(⋅))Kl(x(⋅))≤Fl(x∘(⋅))Kl(x∘(⋅)),l=1,r¯,
the case x(·)=x∘(·) being excluded. That is:
(3.6)Λl1∘Fl(x(⋅))-Λl2∘Kl(x(⋅))≤Λl1∘Fl(x∘(⋅))-Λl2∘Kl(x∘(⋅)),l=1,r¯.
Making the sum after ℓ=1,r¯, we get
(3.7)〈Λ1∘,F(x(⋅))〉-〈Λ2∘,K(x(⋅))〉≤〈Λ1∘,F(x∘(⋅))〉-〈Λ2∘,K(x∘(⋅))〉.
According to condition (a), it follows that
(3.8)b(x(⋅),x∘(⋅))∫γt0,t1[〈η(πx(t),πx∘(t)),〈Λ1∘,∂fα∂x(πx∘(t))〉-〈Λ2∘,∂kα∂x(πx∘(t))〉〉+〈Dγη(πx(t),πx∘(t)),〈Λ1∘,∂fα∂xγ(πx(t))〉-〈Λ2∘,∂kα∂xγ(πx∘(t))〉〉+〈Dθσ2η(πx(t),πx∘(t)),〈Λ1∘,∂fα∂xθσ(πx(t))〉-〈Λ2∘,∂kα∂xθσ(πx∘(t))〉〉]dtα≦-ρ1b(x(⋅),x∘(⋅))‖θ(x(⋅),x∘(⋅))‖2.
Applying property (b), the inequality
(3.9)∫γt0,t1〈μα∘(t),g(πx(t))〉dtα≦∫γt0,t1〈μα∘(t),g(πx∘(t))〉dtα
leads us to
(3.10)b(x(⋅),x∘(⋅))∫γt0,t1(〈η(πx(t),πx∘(t)),〈μα∘(t),∂g∂x(πx∘(t))〉〉+〈Dγη(πx(t),πx∘(t)),〈μα∘(t),∂g∂xγ(πx∘(t))〉〉+〈Dθσ2η(πx(t),πx∘(t)),〈μα∘(t),∂g∂xθσ(πx∘(t))〉〉)dtα≦-ρ2b(x(⋅),x∘(⋅))‖θ(x(⋅),x∘(⋅))‖2.
Taking into account condition (c), the equality
(3.11)∫γt0,t1〈να∘(t),h(πx(t))〉dtα=∫γt0,t1〈να∘(t),h(πx∘(t))〉dtα
implies
(3.12)b(x(⋅),x∘(⋅))∫γt0,t1(〈η(πx(t),πx∘(t)),〈να∘(t),∂h∂x(πx∘(t))〉〉+〈Dγη(πx(t),πx∘(t)),〈να∘(t),∂h∂xγ(πx∘(t))〉〉+〈Dθσ2η(πx(t),πx∘(t)),〈να∘(t),∂h∂xθσ(πx∘(t))〉〉)dtα≦-ρ3b(x(⋅),x∘(⋅))‖θ(x(⋅),x∘(⋅))‖2.
Summing side by side relations (3.8), (3.10), (3.12) and using condition (d), it follows that
(3.13)b(x(⋅),x∘(⋅))∫γt0,t1〈η(πx(t),πx∘(t)),〈Λ1∘,∂fα∂x(πx∘(t))〉-〈Λ2∘,∂kα∂x(πx∘(t))〉+〈μα∘(t),∂g∂x(πx∘(t))〉+〈να∘(t),∂h∂x(πx∘(t))〉〉dtα+b(x(⋅),x∘(⋅))∫γt0,t1〈Dγη(πx(t),πx∘(t)),〈Λ1∘,∂fα∂xγ(πx∘(t))〉-〈Λ2∘,∂kα∂xγ(πx∘(t))〉+〈μα∘(t),∂g∂xγ(πx∘(t))〉〉dtα+〈να∘(t),∂h∂xγ(πx∘(t))〉+b(x(⋅),x∘(⋅))∫γt0,t1〈Dθσ2η(πx(t),πx∘(t)),〈Λ1∘,∂fα∂xθσ(πx∘(t))〉-〈Λ2∘,∂kα∂xθσ(πx∘(t))〉+〈μα∘(t),∂g∂xθσ(πx∘(t))〉〉dtα+〈να∘(t),∂h∂xθσ(πx∘(t))〉<-(ρ1+ρ2+ρ3)b(x(⋅),x∘(⋅))‖θ(x(⋅),x∘(⋅))‖2.
This inequality implies that b(x(·),x∘(·))〉0, therefore, we obtain
(3.14)∫γt0,t1〈η(πx(t),πx∘(t))〈Λ1∘,∂fα∂x(πx∘(t))〉-〈Λ2∘,∂kα∂x(πx∘(t))〉+〈μα∘(t),∂g∂x(πx∘(t))〉+〈να∘(t),∂h∂x(πx∘(t))〉〉dtα+∫γt0,t1〈Dγη(πx(t),πx∘(t))〈Λ1∘,∂fα∂xγ(πx∘(t))〉-〈Λ2∘,∂kα∂xγ(πx∘(t))〉+〈μα∘(t),∂g∂xγ(πx∘(t))〉+〈να∘(t),∂h∂xγ(πx∘(t))〉〉dtα+∫γt0,t1〈Dθσ2η(πx(t),πx∘(t)),〈Λ1∘,∂fα∂xθσ(πx∘(t))〉-〈Λ2∘,∂kα∂xθσ(πx∘(t))〉+〈μα∘(t),∂g∂xθσ(πx∘(t))〉+〈να∘(t),∂h∂xθσ(πx∘(t))〉〉dtα<-(ρ1+ρ2+ρ3)‖θ(x(⋅),x∘(⋅))‖2.
According to [19], we have the following.
Lemma 3.4.
A total divergence is equal to a total derivative.
After integrating by parts the last two integrals, the left-hand side of the previous inequality can be written as the sum of two integrals. The former has as integrant the scalar product betweenη(πx(t),πx∘(t)) and the null expression from (3.2). The later is also null, being and integral from a total derivative by Lemma 3.4. Therefore, the previous inequality leads us to a contradiction, that is 0<-(ρ1+ρ2+ρ3)∥θ(x(·),x∘(·))∥2. Thus, the point x∘(·) is an efficient solution for problem (MFP).Replacing the integrals from hypotheses (b) and (c), of Theorem3.3, by the integral
(3.15)∫γt0,t1[〈μα∘(t),g(πx(t))〉+〈να∘(t),h(πx(t))〉]dtα,
the following statement is obtained.Corollary 3.5.
Letx∘(·) be a feasible solution of problem (MFP), μ∘(·), ν∘(·) be functions, and Λ1∘, Λ2∘ vectors from ℝr such that relations (3.2) are satisfied. Suppose that the following conditions are fulfilled:(a)
the functional〈Λ1∘,F(x(·))〉-〈Λ2∘,K(x(·))〉 is (ρ1,b)-quasiinvex at the point x∘(·) with respect to η and θ;(b)
the functional∫γt0,t1[〈μα∘(t),g(πx(t))〉+〈να∘(t),h(πx(t))〉]dtα is (ρ2,b)-quasiinvex at the point x∘(·) with respect to η and θ;(c)
one of the integrals from (a) or (b) is strictlyquasiinvex at the pointx∘(·);(d)
ρ1+ρ2≥0;(e)
Λℓ1∘Fℓ(x∘(·))-Λℓ2∘Kℓ(x∘(·))=0, for each ℓ=1,r¯.Then, the point x∘(·) is an efficient solution of problem (MFP).


## 4. Conclusion and Further Development

In our work [9], 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 [9], and strongly motivated by its possible applications in mechanics, the present work introduced a study of sufficient efficiency conditions for (MFP).Since ratio programming problems with objective function of our type arise from applied areas as decision problems in management, game theory, engineering studies, and design, we will orient our future research to the development of (strong) dual program theory for these problems [20].



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