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

and Applied Analysis 3 where xγ t ∂x/∂t 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: fα ( f α ) : J2 T,M −→ R , kα ( k α ) : J2 T,M −→ R , 1, r, α 1, p, 2.2 determine the following path independent functionals, respectively, F x · ∫γt0 ,t1 f α πx t dt , K x · ∫γt0 ,t1 k α πx t dt . 2.3 We accept that the Lagrange matrix densities: g ( gb a ) : J2 T,M −→ Rmd, a 1, d, b 1, m, m < n, 2.4 of C∞-class define the partial differential inequations PDI of evolution g πx t 0, t ∈ Ωt0,t1 , 2.5 and the Lagrange matrix densities h ( ha ) : J2 T,M −→ R, a 1, d, b 1, q, q < n, 2.6 defines the partial differential equations PDE of evolution h πx t 0, t ∈ Ωt0,t1 . 2.7 Let C∞ Ωt0,t1 ,M be the space of all functions x : Ωt0,t1 −→ M of C∞-class, with the norm ‖x‖ ‖x‖∞ p ∑


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.

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 t α and x x i , respectively.Let J 2 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 R p , the hyperparallelepiped Ω t 0 ,t 1 , in R p , with the diagonal opposite points t 0 t 1 0 , . . ., t p 0 and t 1 t 1 1 , . . ., t p 1 , can be written as interval t 0 , t 1 .Suppose γ t 0 ,t 1 is a piecewise C 2 -class curve joining the points t 0 and t 1 .

Important note
To simplify the presentation, in our subsequent theory, we shall set determine the following path independent functionals, respectively, We accept that the Lagrange matrix densities: of C ∞ -class define the partial differential inequations PDI of evolution and the Lagrange matrix densities defines the partial differential equations PDE of evolution h π x t 0, t ∈ Ω t 0 ,t 1 .

2.7
Let C ∞ Ω t 0 ,t 1 , M be the space of all functions x : x θσ ∞ .

2.8
For each 1, r, suppose K x • > 0, and consider With conditions 2.5 and 2.7 , we denote by the set of all feasible solutions of problem;

2.11
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 Ω t 0 ,t 1 .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 .

Main Results
Consider that x • • ∈ F Ω t 0 ,t 1 is an efficient solution of problem MFP .In a very recent article 9 , Pitea and Postolache proved that there exist Λ 1• , Λ 2• ∈ R r and the smooth functions μ A Natural Question Arises.
Are these conditions sufficient for x • • to be an efficient solution for program MFP ?To develop our theory, we have to introduce an appropriate generalized convexity.
Let ρ be a real number, b : , the following implication holds:

3.4
The quasiinvexity is used, in appropriate forms, in recent works for studies of some multiobjective programming problems, see 15

Then the point x • • is an efficient solution of problem (MFP).
Proof.Let us suppose that the point x • • is not an efficient solution for problem MFP .Then, there is a feasible solution x • for problem MFP , such that Making the sum after 1, r, we get According to condition a , it follows that Applying property b , the inequality

3.10
Abstract and Applied Analysis 7 Taking into account condition c , the equality

3.12
Summing side by side relations 3.8 , 3.10 , 3.12 and using condition d , it follows that

3.13
This inequality implies that b x • , x • • 0, therefore, we obtain

3.14
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 between η π x t , π x • 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 Then, the point x • • is an efficient solution of problem (MFP).

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 .
at the point x • • with respect to η and θ; d one of the integrals of (a)-(c) is strictly ρ 1 , b , ρ 2 , b or ρ 3 , b -quasiinvex at the point x • • with respect to η and θ; , and a a α , α 1, p, a closed 1-form.To a, we associate the curvilinear integral.Definition 3.2.The functional A is called strictly ρ, b -quasiinvex at the point x • • if there exists a vector function η : • • 2 .Thus, the point x • • is an efficient solution for problem MFP .Replacing the integrals from hypotheses b and c , of Theorem 3.3, by the integral Corollary 3.5.Let x • • be a feasible solution of problem (MFP), μ • • , ν • • be functions, and Λ 1• , Λ 2• vectors from R r such that relations 3.2 are satisfied.Suppose that the following conditions are fulfilled: