Abstract and Applied Analysis

Volume 2012 (2012), Article ID 535416, 12 pages

http://dx.doi.org/10.1155/2012/535416

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

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

Received 6 December 2011; Accepted 27 December 2011

Academic Editor: Ngai-Ching Wong

Copyright © 2012 Mihai Postolache. 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

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.

#### 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 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 [13].

Throughout this work, we use the customary relations between two vectors of the same dimension, [11]. 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 .

To simplify the notations, denote bythe partial velocities and partial accelerations, respectively. Also, in our subsequent theory, we will set .

On with values on , consider the closed Lagrange 1-forms densities of -class: which determine the following path-independent functionals: The closeness conditions (complete integrability conditions) are

where is the total derivative.

We accept that the Lagrange matrix densities:

of -class define the partial differential inequations (PDI) of evolution and the Lagrange matrix densities, define the partial differential equations (PDE), of evolution

For each , suppose , and consider

With conditions (2.6) and (2.8), we denote by
the *set of all feasible solutions* of problem:
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: taking into account that the functions and have to satisfy the boundary conditions , , or given, respectively, , , or 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 solution is called *efficient solution* for the program (MFP) if and only if for any feasible solution , one has the implication:

#### 3. Main Results

To state our results, we have to introduce an appropriate generalized convexity. For this purpose, consider be a real number, a functional, and , , a closed 1-form. To we associate the curvilinear integral

*Definition 3.1. *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 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 by the minimizing functional vector of problem (MFP) at the point and by the maximizing functional vector of dual (MFD) at the point , where is the domain of problem (MFD).

Theorem 3.2 (Weak Duality). *Let be a feasible solution of the problem and let be an efficient solution of problem . Assume that the following conditions are fulfilled: *(a)*, , , ; *(b)*for any , the functional is -quasiinvex at the point and the functional is -quasiinvex at the point with respect to and ;*(c)*the functional,**
is -quasiinvex at the point with respect to and ;*(d)*either the functional of (b) or the functional of (c) is strictly quasiinvex;*(e)*.**Then, the inequality is false.*

*Proof. *From condition (b), it follows that
We multiply (3.4) by and (3.5) by . We make the sum and we obtain the following implications:
According to hypothesis (c), we have
Making the sum of the implications (3.6) and (3.7), it follows that
Since , we obtain
where
The following relations hold:
By replacing the relations (3.11) and (3.12), and using Euler-Lagrange PDE, the relation (3.9) becomes
According to [15], , 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
From hypothesis (e), the previous relation becomes , which is false. Next, from relation (3.8), it follows that
Taking into account the inequality:
the above-mentioned relation becomes
that is,
Because , , we conclude that
or
Therefore, the relation 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). *Let be an efficient solution of . Suppose that the hypotheses of Theorem are satisfied. Then, there are and in , and the smooth functions and , such that is an efficient solution of the dual , and .*

We will present now a theorem concerning the converse duality, by changing some of the hypotheses.

Theorem 3.4 (Converse Duality). *Let be an efficient solution of the dual problem . Suppose the following conditions hold: *(a)* is an efficient solution of the primal problem ;*(b)*for any , one has*(c)*for any , the functional is -quasiinvex at the point and the functional is -quasiinvex at the point , with respect to and ;*(d)*the functional,**
is -quasiinvex at the point with respect to and ;*(e)*one of the functionals of (c), (d) is strictly -, -, or -quasiinvex with respect to and , respectively;*(f)*. **Then, , and .*

#### 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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