Multiobjective Duality in Variational Problems with Higher Order Derivatives ()
A multiobjective variational problem involving higher order derivatives is considered and optimality conditions for this problem are derived. A Mond-Weir type dual to this problem is constructed and various duality results are validated under generalized invexity. Some special cases are mentioned and it is also pointed out that our results can be considered as a dynamic generalization of the already existing results in nonlinear programming.
1. Introduction
Calculus of variation is a powerful technique for the solution of various problems appearing in dynamics of rigid bodies, optimization of orbits, theory of variations and many other fields. The subjects whose importance is fast growing in science and engineering primarily concern with finding optimal of a definite integral involving a certain function subject to fixed point boundary conditions. In [1] Courant and Hilbert, quoting an earlier work of Friedrichs [2], gave a dual relationship for a simple type of unconstrained variational problem. Subsequently, Hanson [3] pointed out that some of the duality results of mathematical programming have analogous in variational calculus. Exploring this relationship between mathematical programming and the classical calculus of variations, Mond and Hanson [4] formulated a constrained variational problem as a mathematical programming problem and using Valentine’s [5] optimality conditions for the same, presented its Wolfe type dual variational problem for validating various duality results under convexity. Later Bector, Chandra and Husain [6] studied Mond-Weir type duality for the problem of [4] for weakening its convexity requirement. In [7] Chandra, Craven and Husain studied optimality and duality for a class of non-differentiable variational problem with non-differentiable term in the integrand of the objective functional while in [8] they derived optimality conditions and duality results for a constrained variational problem having terms with arbitrary norms in the objective as well as constrained functions.
Recently Husain and Jabeen [9] studied a wider class of variational problem in which the arc function is twice differentiable by extending the notion of invexity given in [10]. They obtained Fritz John as well as KarushKuhn-Tucker necessary optimality conditions as an application of Karush-Kuhn-Tucker optimality conditions studied various duality results for Wolfe and Mond and Weir type models.
In single objective programming we must settle on a single objective such as minimizing cost or maximizing profit. However, generally any real world problems can be identified with multiple conflicting criteria e.g., the problems of oil refinery scheduling, production planning, portfolio selection and many others can be modelled as multiobjective programming problems.
Duality results are very useful in the development of numerical algorithms for solving certain classes of optimization problems. Duality for multiobjective variational problem has been studied by a number of authors, notably Bector and Husain [11], Chen [12] and many others cited in these references. Applications of duality theory are prominent in physics, economics, management sciences, etc.
Since mathematical programming and classical calculus of variations have undergone independent development, it is felt that mutual adaptation of ideas and techniques may prove useful. Motivated with this idea in this exposition, we propose to study optimality criteria and duality for a wider class of multiobjective variational problems involving higher order derivative. These results not only generalize the results of Husain and Jabeen [9] and Bector and Husain [11] but also present a dynamic generalization of some of the results in multiobjective nonlinear programming already existing.
2. Invexity and Generalized Invexity
Invexity was introduced for functions in variational problems by Mond, Chandra and Husain [10] while Mond and Smart [13] defined invexity for functionals instead of functions. Here we introduce extended forms of definitions of invexity and various generalized invexity for functional in variational problems involving higher order derivatives.
Consider the real interval 
and the continuously differentiable function
, where x is twice differentiable with its first and second order derivatives 
and 
respectively. If
, the gradient vectors of f with respect to
, 
and 
respectively denoted by
DEFINITION 1. (Invexity): If there exists vector function 
with 
and 
and 
for 
such that for a scalar function
, the functional 
satisfies
Φ is said to be invex in 
on I with respect to η.
Here D is a differentiation operator defined later.
DEFINITION 2. (Pseudoinvexity): Φ is said to be pseudoinvex in 
with respect to 
if
implies
.
DEFINITION 3. (Quasi-invex): The functional Φ is said to quasi-invex in 
with respect to η if
implies
3. Variational Problem and Optimality Conditions
Before stating our variational problem and deriving its necessary optimality condition, we mention the following conventions for vectors x and y in n-dimensional Euclidian space Rn will be used throughout the analysis of this research.
x
is the negation of 
For 
and 
have the usual meaning.
In this section, we present the following variational problem whose optimality conditions will be derived and duality will be investigated in the subsequent sections:
(VPE) Minimize
Subject to
(1)
(2)
(3)
(4)
where
,
and 
are continuously differentiable functions, and X designates the space of piecewise functions 
possessing derivatives 
and
with the norm 
, where the differentiation operator D is given by
where α is given boundary value; thus 
except at discontinuities.
In the results to follow, we use 
to denote the space of continuous functions 
with the uniform norm
; the partial derivatives of g and h are 
and 
matrices respectively; superscript T denotes matrix transpose.
We require the following definition of efficient solution for our further analysis.
DEFINITION 4. (Efficient Solution): A feasible solution 
is efficient for (VPE) if there exist no other feasible 
for (VPE) such that for some
,
and 
for all
,
.
In relation to (VPE), we introduce the following set of problems 
for each 
in the spirit of [14], with a single objective,
Minimize 
Subject to
,
,
,
,
The following lemma can be proved on the lines of Chankong and Haimes [14].
LEMMA 1: 
is an efficient solution of (VPE) if and only if 
is an optimal solution of 
for each
.
(P0) Minimize 
Subject to
,
,
,
where
.
PROPOSITION 1. [9]: (Fritz John Optimality Conditions) If 
is an optimal solution of (P0) and 
maps X into the subspace of
, then there exists Lagrange multiplier
, the piecewise smooth 
and
, such that
,
,
.
If
, then the above optimality conditions will reduce to the Karush-Kuhn-Tucker type optimality conditions and the solution 
is referred to as a normal solution.
We now establish the following theorem that gives the necessary optimality conditions for (VPE).
THEOREM 1: (Fritz-John Conditions): Let 
be an efficient solution of (VPE) and 
maps X into the subspace of
, then there exist 
and the piecewise smooth 
and
, such that
, (5)
, (6)
. (7)
. (8)
PROOF: Since 
is an efficient solution of (VPE) by Lemma 1, 
is an optimal solution of (
), for each
. From Proposition1, it follows that, there exist scalars 
and piecewise smooth function 
and 
, such that
,
.
Summing over r, we have
,
.
Setting
,
and
, we have
,
,
.
4. Mond-Weir Type Duality
In this section, we consider the following variational problem involving higher order derivatives, by suppressing the equality constraint in (VPE).
(VP) Minimize
Subject to
We formulate the following Mond-Weir type dual to the problem (VP) and establish various duality results under invexity defined in the preceding section.
(M-WD) Maximize
Subject to
(9)
(10)
(11)
(12)
(13)
. (14)
THEOREM 2. (Weak Duality): Let 
be feasible for (VP) and 
be feasible for (M-WD) if for allfeasible 
is pseudoinvex and 
is quasi-invex with respect to the same η.
Then,
.
PROOF: The relations 
imply
This, because of the quasi-invexity of
, implies that
(By integration by parts)
Using the boundary conditions which gives
at 
(By integration by parts)
Using the boundary conditions which give 
at 
From Equation (11) this yields,
This by integration by parts and then using boundary conditions gives,
This, in view of psedoinvexity of 
implies that
.
For this, it follows
.
THEOREM 3. (Strong Duality): If 
is a feasible solution for (VP) and assume that 
is an efficient solution and for at least one 
satisfies a regularity condition for [7] for
.
Then there exists one 
such that 
is efficient for (VD). Further if the assumptions of Theorem 2 are satisfied, then 
is an efficient solution of (VD).
PROOF: Since 
is efficient solution by Lemma 1, it is an optimal solution of
. By Proposition 1, this implies that there exists 
and piecewise smooth 
such that,
(15)
(16)
(17)
(18)
(19)
From (17), we have
(20)
Equations (16), (17) and (18) imply that 
is feasible for (M-WD). The equality of objective functional of primal and dual problems is obvious from their formulations. Efficiency of 
is immediate from the application of Theorem 2.
As in [4], by employing chain rule in calculus, it can be easily seen that the expression 
,may be regarded as a function 
of variables 
and
, where 
and
. That is, we can write
In order to prove converse duality between (VP) and (M-WD), the space X is now replaced by a smaller space 
of piecewise smooth thrice differentiable function 
with the norm 
The problem (M-WD) may now be briefly written as, Minimize
Subject to
Consider 
as defining a mapping 
where Y is a space of piecewise twice differentiable function and B is the Banach Space. In order to apply Theorem 1 to the problem (M-WD), the infinite dimensional inequality must be restricted. In the following theorem, we use 
to represent the Frèchèt derivative
THEOREM 4. (Converse Duality): Let D be an efficient solution with 
and 
and 
have a (weak*) closed range hypothesis. Let f and g be twice continuously differentiable. Assume that
(H1) 
be pseudoinvex and 
be quasi-invex with respect to same
.
(H2)
.
(H3) 
are linearly independent.
Then 
is an efficient solution of (VP).
Proof: Since 
where 
and 
having a closed range, is an efficient solution of (M-WD), by Theorem 2, it implies that there exist
, 
and piecewise smooth 
and 
satisfying the following conditions .
(21)
(22)
(23)
(24)
(25)
and
(26)
Since 
which implies 
This yields from (23)
(27)
Using the equality constraint (11) in (21), we have
(28)
Postmultiplying Equation (21) by 
and using (27) in (28) we get,
This by hypothesis (H2) implies 
Also from (28) we have
This, because of linear independence of 
, gives
(29)
Now suppose
, then, from (22) and (29) we have 
and 
respectively.
This implies
, which is the contradiction to
,
.
Hence 
and by (29) we have,
.
The relation (22) in conjunction with
, and 
gives
This implies the feasibility of 
for (VP) and its efficiency is evident from and application of Theorem 2.
5. Natural Boundary Values
The duality results obtained in the preceding sections can easily be extended to the multiobjective variational problems with natural boundary values rather than fixed end points.
Primal (P1) Minimize
Subject to
Dual (D1) Maximize
Subject to
and
,
and
,
.
6. Nonlinear Programming
If the problems (P1) and (D1) are independent of t, then they will reduce to the following multiobjective nonlinear programming problems studied in [15]
(NP): Minimize 
Subject to
(ND): Maximize 
Subject to
,
.
[1] R. Courant and D. Hilbert, “Methods of Mathematical Physics,” Wiley, New York, Vol. 1, 1943.
[2] K. O. Friedrichs, “Ein Verfrahren der Variations-Rechung das Minimum eines Integrals Maximum eines Anderen Ausdruckes Dazustellan,” Göttingen Nachrichten, 1929.
[3] M. A. Hanson, “Bonds for Functionally Convex Optimal Control Problems,” Journal of Mathematical Analysis and Applications, Vol. 8, No. 1, February 1964, pp. 84-89.
[4] B. Mond and M. A. Hanson, “Duality for Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 18, No. 2, May 1967, pp. 355-364.
[5] F. A. Valentine, “The Problem of Lagrange with Differential Inequalities as Added Side Conditions,” Contributions to the Calculus of Variations, 1933-1937, University of Chicago Press, 1937, pp. 407-448.
[6] C. R. Bector, S. Chandra and I. Husain, “Generalized Concavity and Duality in Continuous Programming,” Utilitas Mathematica, Vol. 25, 1984, pp. 171-190.
[7] S. Chandra, B. D. Craven and I. Husain, “A Class of Nondifferentiable Continuous Programming Problems,” Journal of Mathematical Analysis Applications, Vol. 107, No. 1, April 1985, pp. 122-131.
[8] S. Chandra, B. D. Craven and I. Husain, “Continuous Programming Containing Arbitrary Norms,” Journal of Australian Mathematical Society (Series A), Vol. 39, No. 1, 1985, pp. 28-38.
[9] I. Husain and Z. Jabeen, “On Variational Problems Involving Higher Order Derivatives,” Journal of Applied Mathematics and Computing, Vol. 27, No. 1-2, March 2005, pp. 433-455.
[10] B. Mond and S. Chandra and I. Husain, “Duality of Variational Problems with Invexity,” Journal of Mathematical Analysis and Applications, Vol. 134, No. 2, September 1988, pp. 322-328.
[11] C. R. Bector and I. H. Husain, “Duality for Multiobjective Variational Problems,” Journal of Mathematical Analysis and Applications, Vol. 166, No. 1, 1 May 1992, pp. 214-224.
[12] X. H. Chen, “Duality for Multiobjective Variational Problems with Invexity” Journal of Mathematical Analysis and Appliactions, Vol. 203, No. 1, October 1996, pp. 236-253.
[13] B. Mond and I. Smart, “Duality with Invexity for a Class of Nondifferentiable Static and Continuous Programming Problems,” Journal of Mathematical Analysis and Applications, Vol. 136, 1988, pp. 325-333.
[14] V. Chankong and Y. Y. Haimes, “Multiobjective Decision Making: Theory and Methodology,” North Holland, New York, 1983.
[15] R. R. Egudo and M. A. Hanson, “Multiobjective Duality with Invexity,” Journal of Mathematical Analysis and Appliactions, Vol. 126, No. 2, September 1987, pp. 469- 477.