Research Article | Open Access
Mihai Postolache, "Duality for Multitime Multiobjective Ratio Variational Problems on First Order Jet Bundle", Abstract and Applied Analysis, vol. 2012, Article ID 589694, 18 pages, 2012. https://doi.org/10.1155/2012/589694
Duality for Multitime Multiobjective Ratio Variational Problems on First Order Jet Bundle
We consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. Based on the efficiency conditions for multitime multiobjective ratio variational problems, we introduce a ratio dual of generalized Mond-Weir-Zalmai type, and under some assumptions of generalized convexity, duality theorems are stated. We prove our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained. This work further develops our studies in (Pitea and Postolache (2011)).
The duality for scalar variational problems involving convex functions has been formulated by Mond and Hanson in . This study was further developed to various classes of convex and generalized convex functions. In , Hanson extended the duality results (in the sense of Wolfe) to a class of functions subsequently called invex. In order to weaken the convexity conditions, Bector et al. , introduced a dual to a variational problem (in the sense of Mond and Weir), different from that formulated by Mond and Hanson in . Mond et al. , extended the concept of invexity to continuous functions and used it to generalize earlier Wolfe duality results for a class of variational problems. Duality theory for scalar optimization can be found in many other works and see Mond and Husain  and Preda .
In , using a vector-valued Lagrangian function, Tanino and Sawaragi introduced a duality theory for multiobjective optimization. This idea is employed by Bitran , which associated a matrix of dual variables with the constraints in the primal problem. In , Mond and Weir considered a pair of symmetric dual nonlinear programs and developed a duality theory under assumptions of pseudoconvexity. Under various types of generalized convex functions, Mukherjee and Purnachandra , Preda  and Zalmai  established several weak efficiency conditions and developed different types of dualities for multiobjective variational problems. D. S. Kim and A. L. Kim  used the efficiency property of nondifferentiable multiobjective variational problems in duality theory. In a recent study , Pitea and Postolache considered a new class of multitime multiobjective variational problems of minimizing a vector of functionals of curvilinear integral type. Based on appropriate normal efficiency conditions, they studied duals of Mond-Weir type, generalized Mond-Weir-Zalmai type, and under appropriate assumptions of generalized convexity, stated duality theorems.
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. In , Jagannathan introduced a duality study using some results connecting solutions of a nonlinear ratio program with those of suitably defined parametric convex program. Concerning advances on single-objective ratio programs, see  by Khan and Hanson and  by Reddy and Mukherjee, which utilized invexity assumptions in the sense of Hanson  to obtain optimality conditions and duality results. As concerns vector ratio problems, Singh and Hanson  applied invex functions to derive duality results, while Jeyakumar and Mond  generalized these results to the class of -invex functions. Later, Liang et al.  introduced a unified formulation of the generalized convexity to derive optimality conditions and duality results for vector ratio problems.
In this paper, we are motivated by previous published research articles (please refer to  by Antczak,  by Nahak and Mohapatra, and  by Pitea and Postolache) to consider a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type and, as a particular case, a vector of functionals of curvilinear integral type. We state and prove new duality results, of Mond-Weir-Zalmai type, for feasible solutions of our multitime multiobjective ratio variational problems, under assumptions of -quasi-invexity. This study is encouraged by its possible application in mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. The objective vector function is of curvilinear integral type, the integrand depending on velocities; that is why we consider it adequate to introduce our results within the framework offered by the first order jet bundle, [24–26] by Pitea et al.,  by Sounders, and  by Udriste et al.
Before presenting our results, we need the following background, which is necessary for the completeness of the exposition. For more details, we address the reader to .
2.1. The First Order Jet Bundle
To make our presentation self-contained and reader-friendly, we recall the notion of jet bundle of the first order, .
Let be given the smooth manifolds and , of dimensions , and , respectively. The corresponding local coordinates are , , and , , respectively.
In the following, the set will be indexed by Latin characters, while the set will be indexed by Greek characters.
Denote by the set of mappings of class , from to .
Consider an arbitrary point of the product manifold . On the set , define the equivalence relation
If and are arbitrary mappings from , we denote
Thus, the equivalence relation has the local expression
The equivalence class of a mapping will be denoted by The quotient space obtained by the factorization of the space by the equivalence relation “”, is called -jet at the point .
The total space of the set of 1-jets, can be organized as a vector bundle over the base space , endowed with the differentiable structure of the product space.
Let be the canonical projection, defined as
The mapping is well defined, having the property to be onto.
For every local chart of the product manifold , we define the bijection
Therefore, the 1-jet space is endowed with a differentiable structure of dimension , such that the mappings are diffeomorphisms.
The local coordinates on the space are , where
Remark 2.1. From physical viewpoint, the differentiable manifold should be thought as a “temporal manifold" or a “multitime", while is a “space manifold."
Remark 2.2. From geometrical viewpoint, an element of a fiber 1-jets should be thought as a class of parametrized sheet. The sections of fiber -jets are “physical fields."
To simplify the notations, denote by , , the partial velocities. Also, in our subsequent theory, we will set .
2.2. On Lagrange 1-Forms
A Lagrange 1-form of the first order on the jets space has the form where , and are Lagrangians of the first order. The pullback is a Lagrange 1-form of the second order on . The coefficients are second order Lagrangians, which are linear in the partial accelerations.
A smooth Lagrangian , produces two smooth closed (completely integrable) 1-forms:
(i) the differential of components , with respect to the basis ;
(ii) the restriction of to , that is, the pullback of components containing partial accelerations with respect to the basis (for other ideas, see ).
3. Problem Description
Let and be Riemannian manifolds of dimensions and , respectively. Denote by , , and , , the local coordinates on and , respectively. Consider the first order jet bundle associated to and .
To develop our theory, we recall the following relations between two vectors and , :
Using the product order relation on , the hyperparallelepiped , in , with diagonal opposite points and , can be written as being the interval . Suppose is a piecewise -class curve joining the points and .
The closed Lagrange 1-forms densities of -class determine the following path independent curvilinear functionals (actions, mechanical work):
The closeness conditions (complete integrability conditions) are where is the total derivative.
Suppose , for all , and accept that the Lagrange matrix densities of -class define the partial differential inequations (PDIs) (of evolution) and the partial differential equations (PDE) (of evolution)
On the set of all functions of -class, we set the norm .
For each , suppose , and consider
The aim of this work is to introduce and study the variational problem of minimizing a vector of quotients of functionals of curvilinear integral type: where denotes the set of all feasible solutions of problem , defined as
This kind of problems, of considerable interest, arises in various branches of mathematical, engineering, and economical sciences. We especially have in mind mechanical engineering, where curvilinear integral objectives are extensively used due to their physical meaning as mechanical work. These objectives play an essential role in mathematical modeling of certain processes in relation with robotics, tribology, engines, and much more.
4. Main Results
Definition 4.2. Let be a real number and let be a functional. To any closed 1-form we associate the path independent curvilinear functional
The functional is called [strictly] -quasi-invex at the point if there is a vector function , vanishing at the point , and , such that for any in , , the following implication holds:
Several examples which illustrate our concept could be found in . However, the following example is interesting. It is a generalization of Example 1 in .
Example 4.3. Let , . With , denote
where by we denoted the total derivative operator.
Suppose that . Then, the functional is -quasi-invex, for and any , at the point , with respect to
In their recent work , Pitea et al. established necessary efficiency conditions for problem . More accurately, they proved that if is an efficient solution of problem , then there are two vectors , in and the smooth functions and , the first from to , and the second from to , such that
If and , then , from conditions (4.6), is called normal efficient solution.
Let be an efficient solution of primal , the scalars , in , and the smooth functions , , given previously.
In order to use the idea of "grouping the resources," consider and partitions of the sets and , respectively.
For each and , we denote
Consider a function and associate to the multiobjective ratio variational problem taking into account that the function has to satisfy the boundary conditions , , or given.
is the value of the objective function of problem at , and is the maximizing functional vector of dual problem at the point , where is the domain of problem .
We prove three duality results in a generalized sense of Mond-Weir-Zalmai.
Theorem 4.4 (weak duality). Let be a feasible solution of problem and let be a feasible point of problem . Assume that the following conditions are satisfied: (a), for all ; (b) and , for each ; (c)for each , the functional is -quasi-invex at the point with respect to and , on ; (d) is -quasi-invex at with respect to and , for each , on ; (e)at least one of the functionals of (c), (d) is strictly quasi-invex; (f).
Then, the inequality is false.
Proof. By reductio ad absurdum, suppose we have
From these inequalities, it follows
and taking into account the hypotheses (a) and (b), we get
We obtain the following implications:
Hypothesis (d) regarding the -quasi-invexity property of each functional implies
Now, we make the sum of implications (4.13) and (4.14) side by side and from to . It follows
Since , we obtain where
The following relation holds:
By replacing relations (4.18) and by using Euler-Lagrange PDE, relation (4.16) becomes
For , let us denote by
According to , § 9, a total divergence is equal to a total derivative. Consequently, there exists , with and such that and
Replacing into inequality (4.19), it follows that contradicting hypothesis (f).
From relation (4.15), it follows According to the constraints of problems and , the previously mentioned relation becomes , that is:
Because , , we conclude that or
Therefore, the inequality contradicts relations (4.12), and this completes the proof.
Theorem 4.5 (direct duality). Let be an efficient solution of primal . Suppose the hypotheses of Theorem 4.4 hold. Then there are and in , and the smooth functions and , such that is an efficient solution of dual program . Moreover
Proof. being efficient solution of primal , there are and in and the smooth functions and , such that relations (4.6) are verified. Therefore, , . It follows is a feasible solution for program . Obviously, The weak duality theorem assures the efficiency of .
If we remark that the notion of efficient solution of problem is similar to those in Definition 4.1, we can state the result in the following.
Theorem 4.6 (converse duality). Let be an efficient solution to dual and suppose the following conditions are satisfied: (i) is a normal efficient solution of primal ; (ii)the hypotheses of Theorem 4.4 hold at .
Then is an efficient solution to . Moreover, one has the equality
Proof. Suppose . The efficiency of of primal implies the existence of and in and the smooth functions and , such that relations (4.6) are satisfied. It follows that is a feasible solution of and
On the other hand, from the weak duality theorem, holds, that is:
This last relation contradicts the efficiency of for program . Therefore , and the theorem is proved.
We would like to conclude the study in this section with the following important particular case.
In this respect, suppose that , , . Denote . Then, program becomes and its dual is where with .
It can be seen that we have obtained precisely the programs studied in the work . Therefore, the results in this paper are stronger than the ones mentioned before.
In our recent study , we initiated an optimization theory on the first order jet bundle. As natural continuation, in this paper we considered a new class of multitime multiobjective variational problems of minimizing a vector of quotients of functionals of curvilinear integral type. We derived duality results for efficient solutions of multitime multiobjective ratio variational problems under the assumptions of -quasi-invexity. We proved our weak duality theorem for efficient solutions, showing that the value of the objective function of the primal cannot exceed the value of the dual. Direct and converse duality theorems are stated, underlying the connections between the values of the objective functions of the primal and dual programs. As special cases, duality results of Mond-Weir-Zalmai type for a multitime multiobjective variational problem are obtained.
Having in mind the physical significance of the objective function, this study is strongly motivated by its possible applications of nonlinear optimization to mechanical engineering and economics .
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