Abstract and Applied Analysis

Volume 2012 (2012), Article ID 713765, 9 pages

http://dx.doi.org/10.1155/2012/713765

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

Faculty of Applied Sciences, University “Politehnica” of Bucharest, Splaiul Independentei 313, Bucharest 060042, Romania

Received 30 January 2012; Accepted 13 February 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Ariana Pitea. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 -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 Section 2 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 and be Riemannian manifolds of dimensions and , respectively. The local coordinates on and will be written and , respectively. Let be the second-order jet bundle associated to and , see [11].

Throughout this work, we use the customary relations between two vectors of the same dimension, [9]. With the product-order relation on , the hyperparallelepiped , in , with the diagonal opposite points and , can be written as interval . Suppose is a piecewise -class curve joining the points and .

* Important note*

To simplify the presentation, in our subsequent theory, we shall set
where , , and , , are partial velocities and partial accelerations respectively.The closed Lagrange 1-form densities of -class:
determine the following path independent functionals, respectively,
We accept that the Lagrange matrix densities:
of -class define the partial differential inequations (PDI of evolution)
and the Lagrange matrix densities
defines the partial differential equations (PDE) (of evolution)
Let be the space of all functions of -class, with the norm
For each , suppose , and consider
With conditions (2.5) and (2.7), we denote by
the *set of all feasible solutions* of problem;
a PDI and/or PDE-constrained minimum problem.Using the terminology from analytical mechanics, in MFP there are given numbers of sources, each of them producing mechanical work. This one have to be minimized on a set of limited resources, namely, .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 solution is called *efficient* for program (MFP) if and only if for any feasible solution :

Consider that is an efficient solution of problem (MFP). In a very recent article [9], Pitea and Postolache proved that there exist and the smooth functions , , such that

* A Natural Question Arises.**Are these conditions sufficient for ** to be an efficient solution for program *(MFP)?

To develop our theory, we have to introduce an appropriate generalized convexity.

Let be a real number, a functional, and , , a closed 1-form. To , we associate the curvilinear integral.

*Definition 3.2. * The functional is called [*strictly*]*-quasiinvex at the point * if there exists a vector function , vanishing at the point , and the function defined on to , such that for any , the following implication holds:
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 vectors , from and the functions , , satisfying conditions (3.2). Suppose that the following properties hold:*(a)*the functional is -quasiinvex at the point with respect to and ;*(b)*the functional is -quasiinvex at the point with respect to and ;*(c)*the functional is -quasiinvex at the point with respect to and ;*(d)*one of the integrals of (a)—(c) is strictly , or -quasiinvex at the point with respect to and ;*(e)*;*(f)*, for each .**Then the point is an efficient solution of problem (MFP).*

*Proof. *Let us suppose that the point is not an efficient solution for problem (MFP). Then, there is a feasible solution for problem (MFP), such that
the case being excluded. That is:
Making the sum after , we get
According to condition (a), it follows that
Applying property (b), the inequality
leads us to
Taking into account condition (c), the equality
implies
Summing side by side relations (3.8), (3.10), (3.12) and using condition (d), it follows that
This inequality implies that , therefore, we obtain
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 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 . Thus, the point is an efficient solution for problem (MFP).

Replacing the integrals from hypotheses (b) and (c), of Theorem 3.3, by the integral the following statement is obtained.

Corollary 3.5. *Let be a feasible solution of problem (MFP), , be functions, and , vectors from such that relations (3.2) are satisfied. Suppose that the following conditions are fulfilled:*(a)*the functional is -quasiinvex at the point with respect to and ;*(b)*the functional is -quasiinvex at the point with respect to and ;*(c)*one of the integrals from (a) or (b) is strictlyquasiinvex at the point ;*(d)*;*(e)*, for each .**Then, the point 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].

#### References

- A. Chinchuluun and P. M. Pardalos, “A survey of recent developments in multiobjective optimization,”
*Annals of Operations Research*, vol. 154, pp. 29–50, 2007. View at Publisher · View at Google Scholar - A. Chinchuluun, P. M. Pardalos, A. Migdalas, and L. Pitsoulis,
*Pareto Optimality, Game Theory and Equilibria*, vol. 17 of*Springer Optimization and Its Applications*, Springer, New York, NY, USA, 2008, Edited by Altannar Chinchuluun, Panos M. Pardalos, Athanasios Migdalas and Leonidas Pitsoulis. View at Publisher · View at Google Scholar - Z. A. Khan and M. A. Hanson, “On ratio invexity in mathematical programming,”
*Journal of Mathematical Analysis and Applications*, vol. 205, no. 2, pp. 330–336, 1997. View at Publisher · View at Google Scholar - L. V. Reddy and R. N. Mukherjee, “Some results on mathematical programming with generalized ratio invexity,”
*Journal of Mathematical Analysis and Applications*, vol. 240, no. 2, pp. 299–310, 1999. View at Publisher · View at Google Scholar - M. A. Hanson, “A duality theorem in non-linear programming with non-linear constraints,”
*The Australian Journal of Statistics*, vol. 3, pp. 64–72, 1961. View at Google Scholar - C. Singh and M. A. Hanson, “Multiobjective fractional programming duality theory,”
*Naval Research Logistics*, vol. 38, no. 6, pp. 925–933, 1991. View at Google Scholar - V. Jeyakumar and B. Mond, “On generalised convex mathematical programming,”
*Australian Mathematical Society Journal Series B*, vol. 34, no. 1, pp. 43–53, 1992. View at Publisher · View at Google Scholar - Z.-A. Liang, H.-X. Huang, and P. M. Pardalos, “Efficiency conditions and duality for a class of multiobjective fractional programming problems,”
*Journal of Global Optimization*, vol. 27, no. 4, pp. 447–471, 2003. View at Publisher · View at Google Scholar - A. Pitea and M. Postolache, “Minimization of vectors of curvilinear functionals
on the second order jet bundle. Necessary conditions,”
*Optimization Letters*, vol. 6, no. 3, pp. 459–470, 2011. View at Google Scholar - A. Pitea,
*Integral geometry and PDE constrained optimization problems*, Ph.D. thesis, University “Politehnica” of Bucharest, 2008. - C. Udrişte and M. Postolache,
*Atlas of Magnetic Geometric Dynamics*, vol. 3, Geometry Balkan Press, Bucharest, Romania, 2001. - C. Udrişte, O. Dogaru, and I.
*Ţ*evy, “Null Lagrangian forms and Euler-Lagrange PDEs,”*Journal of Advanced Mathematical Studies*, vol. 1, no. 1-2, pp. 143–156, 2008. View at Google Scholar - A. Pitea and M. Postolache, “Duality theorems for a new class of multitime multiobjective variational problems,”
*Journal of Global Optimization*. In press. View at Publisher · View at Google Scholar - A. Pitea and C. Udrişte, “Sufficient efficiency conditions for a minimizing fractional program,”
*“Politehnica” University of Bucharest. Scientific Bulletin*, vol. 72, no. 2, pp. 13–20, 2010. View at Google Scholar - Ş. Mititelu, “Extensions in invexity theory,”
*Journal of Advanced Mathematical Studies*, vol. 1, no. 1-2, pp. 63–70, 2008. View at Google Scholar - Ş. Mititelu and M. Postolache, “Mond-Weir dualities with Lagrangians for multiobjective fractional and non-fractional variational problems,”
*Journal of Advanced Mathematical Studies*, vol. 3, no. 1, pp. 41–58, 2010. View at Google Scholar - Ş. Mititelu, V. Preda, and M. Postolache, “Duality of multitime vector integral programming with quasiinvexity,”
*Journal of Advanced Mathematical Studies*, vol. 4, no. 2, pp. 59–72, 2011. View at Google Scholar - C. Nahak and R. N. Mohapatra, “Nonsmooth $\rho \text{-}(\eta ,\theta )$-invexity in multiobjective programming problems,”
*Optimization Letters*, vol. 6, no. 2, pp. 253–260, 2012. View at Publisher · View at Google Scholar - A. Pitea, “Null Lagrangian forms on 2nd order jet bundles,”
*Journal of Advanced Mathematical Studies*, vol. 3, no. 1, pp. 73–82, 2010. View at Google Scholar - M. Postolache, “Minimization of vectors of curvilinear functionals on second order jet bundle: dual program theory,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 535416, 12 pages, 2012. View at Publisher · View at Google Scholar