Optimality Conditions and Second-Order Duality for Nondifferentiable Multiobjective Continuous Programming Problems ()
1. Introduction
Second-order duality in mathematical programming has been extensively investigated in the literature. In [1], Chen formulated second order dual for a constrained variational problem and established various duality results under an involved invexity-like assumptions. Subsequently, Husain et al. [2], have presented Mond-Weir type second order duality for the problem of [1], by introducing continuous-time version of second-order invexity and generalized second-order invexity. Husain and Masoodi [3] formulated a Wolfe type dual for a nondifferentiable variational problem and proved usual duality theorems under second-order pseudoinvexity condition while Husain and Srivastav [4] presented a MondWeir type dual to the problem of [2] to study duality under second-order pseudo-invexity and second-order quasiinvexity.
The purpose of this research is to present multiobjective version of the nondifferentiable variational problems considered in [2,4] and study various duality in terms of efficient solutions. The relationship between these multiobjective variational problems and their static counterparts is established through problems with natural boundary values.
2. Definitions and Related Pre-Requisites
Let 
be a real interval, 
and 
be twice continuously differentiable functions. In order to consider 
where 
is differentiable with derivative
, denoted by 
and 
the first order derivatives of 
with respect to
and
, respectively, that is,
Further denote by
, and 
the 
Hessian and 
Jacobian matrices respectively.
The symbols 
and 
have analogous representations.
Designate by X the space of piecewise smooth functions 
with the norm
, where the differentiation operator D is given by
Thus 
except at discontinuities.
We incorporate the following definitions which are required for the derivation of the duality results.
Definition 1. (Second-order Invex): If there exists a vector function 
where 
and with 
at 
and 
such that for a scalar function
, the functional 
where 
satisfies
then 
is second-order invex with respect to
where 
and 
the space of n-dimensional continuous vector functions.
Definition 2. (Second-order Pseudoinvex): If the functional 
satisfies
then 
is said to be second-order pseudoinvex with respect to
.
Definition 3. (Second-order strict-pseudoinvex): If the functional 
satisfies
then 
is said to be second-order pseudoinvex with respect to
.
Definition 4. (Second-order Quasi-invex): If the functional 
satisfies
then 
is said to be second-order quasi-invex with respect to
.
Remark 1. If 
does not depend explicitly on t, then the above definitions reduce to those for static cases.
The following inequality will also be required in the forthcoming analysis of the research:
Lemma: 1 (Schwartz inequality): It states that
with equality in (1) if 
for some 
Throughout the analysis of this research, the following conventions for the inequalities will be used:
If 
with 
and
, then
3. Statement of the Problem and Necessary Optimality Conditions
Consider the following nondifferentiable Multiobjective variational problem:
(VCP): Minimize
subject to
(1)
(2)
where 1) 
denote the space of piecewise smooth functions x with norm
, where differentiation operator D already defined.
are assumed to be continuously differentiable functions, and 3) for each 
is an 
positive semi definite (symmetric) matrix, with 
continuous on I.
In this section we will derive Fritz John and Karush-Kuhn-Tucker type necessary optimality conditions for (VCP).
Definition: A point 
is said to be efficient solution of (VCP) if there exist 
such that
for some 
and
for 
The following result which is a recast of a result of Chankong and Haimes [5] giving a linkage between an efficient solution of (VCP) and an optimal solution of p-single objective variational problem:
Proposition 1. (Chankong and Haimes [5]): A point 
is an efficient solution of (VCP) if and only if 
is an optimal solution of 
for each 
: Minimize
subject to
for obtaining the optimal conditions for (VCP) we will use the optimal conditions obtained by Chandra et al. [6] for a single-objective variational problem which does not contain integral inequality constraints of
.
The validity of the following proposition is quite essential in obtaining the optimality conditions for (VCP)Proposition 2. If 
is an efficient solution of (VCP), then 
is an optimal solution of the following problem 
for each 
: Minimize
subject to
Proof: Let 
be an efficient solution of (VCP). Suppose that 
is not optimal solution of
, for some 
Then there exists an 
such that
(3)
and
The inequality (3) for 
(4)
The inequalities (3) along with (4) contradicts the fact that 
is an efficient solution of (VCP).
Hence 
is an optimal solution of
, for some 
Theorem 1. (Fritz John Type necessary optimality condition): Let 
be an efficient solution of (VCP). Then there exist 
and piecewise smooth functions 
and 
such that
(5)
(6)
(7)
(8)
(9)
Proof: Since 
is an efficient solution of (VCP), by Proposition 2, 
is an efficient solution of 
for each 
and hence in particular
. So by the results of [6] there exist 
and piecewise smooth functions 
and 
such that
The above conditions yield the relations (5) to (9).
Theorem 2 (Kuhn-Tucker type necessary optimality condition):
Let 
be an efficient solution of (VCP) and let for each
, the conditions of 
satisfy Slaters or Robinson condition [6] at
. Then there exist 
and piecewise smooth functions 
and 
such that
(10)
(11)
(12)
(13)
(14)
(15)
Proof: Since 
is an efficient solution of (VCP) by Proposition 2, 
is an optimal solution of 
for each
. Since for each
, the contradicts of
, satisfy Slaters or Robinson conditions [6] at
, by Kuhn-Tucker necessary condition of [6], for each
, there exist 
and piecewise smooth function 
such that
Summing over 
we obtain
where 
for each 
These can be written as,
where 
and 
Setting
we get
That is
4. Mond-Weir Type Second Order Duality
In this section, we present the following Mond-Weir type second-order dual to (VCP) and validate duality results:
(M-WD): Maximize
subject to
(16)
(17)
(18)
(19)
where
and
We denote by CP and CD the sets of feasible solutions to (VCP) and (M-WD) respectively.
Theorem 3. (Weak Duality): Assume that
(A1) 
and
.
(A2) 
is second-order pseudoinvex.
(A3) 
is second-order quasi-invex.
Then
(20)
and
(21)
cannot hold.
Proof. Suppose to the contrary, that (20) and (21) hold.
Since 
we have
Since 
we have
Now, by the constraints (2), (18) and (19), we have
This by (A3), yields
By integration by parts, we have
This, by using (4) gives
(22)
By hypothesis (A1), it implies
Using 
in the above, we have
This contradicts (20) and (21). Hence the result.
Theorem 4 (Strong duality): Let 
be normal and is an efficient solution of (VP). Then there exist
, a piecewise smooth function 
such that 
is feasible for (M-WD) and the two objective functions are equal. Furthermore, if the hypotheses of Theorem 3 hold for all feasible solutions of (VCP) and (M-WD) ,then 
is an efficient solution of (M-WD).
Proof: Since 
is normal and an efficient solution of (VP), by Proposition 2, there exist 
and piecewise smooth 
and 
satisfying
(23)
(24)
(25)
(26)
(27)
From (24) along with
, we have
Hence
satisfies the constraints of (M-WD) and
That is, the two objective functionals have the same value.
Suppose that 
is not the efficient solution of (M-WD). Then there exists 
such that
As
, we have
This contradicts the conclusion of Theorem 3. Hence 
is an efficient solution of (M-WD).
Theorem 5 (Converse duality):
(A1): Assume that 
is an efficient solution of (M-WD)
(A2): The vectors 
are linearly independent where 
the 
row of is 
and 
is the 
row of G,
(A3)
are linearly independent and
(A4) for 
either a) 
and 
or b) 
and 
Then 
is feasible for (VCP) and the two objective functionals have the same value. Also, if Theorem 3 holds for all feasible solutions of (CP) and (M-WD), the 
is an efficient solution of (VCP).
Proof: Since 
is an efficient solution of (M-WD), there exist
, 
and
, and piecewise smooth
, 
, 
and 
such that the following Fritz John optimality conditions (Theorem 1)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
From (31), we have
(39)
This, by the hypothesis (A2) gives
(40)
and 
(41)
Using (40), (41) and (17), we have
(42)
Let 
Then (41) gives 
and (40) gives
, 
Using 
and
, (42) implies
This, because of (A3) yields
(43)
The relation (43) with 
gives 
Since
, (36) gives 
The relation (30) yields 
we have
, 
from (32) and
, 
, 
from (35). These yield
, 
,
.
Consequently
contradicting (38).
Hence 
and from (43) 
Multiplying (30) by
, summing over j, and then using (34) and (41), we have
In view of the hypothesis (A4), this gives
,
, The relation (30) implies
,
yielding the feasibility of 
for (VCP).
The relation (32) with 
and 
gives
(44)
This by Schwartz inequality gives
(45)
If 
then (35) give
,
. and so (45 ) implies
If 
(44) gives
. So we still get
Now suppose that 
is not an efficient of (VCP). Then, there exists 
such that
and
Using 
and
We have
for some 
and
This contradicts Theorem 3. Hence 
is an efficient solution for (VCP).
Theorem 6 (Strict converse duality): Assume that
is second-order strictly pseudoinvex, and
is second-order quasi-invex with respect to same
. Assume also that (VCP) has an optimal solution 
which is normal [6]. If 
is an optimal solution of (M-WD), then 
is an efficient solution of (VCP) with 
Proof: We assume that 
and exhibit a contradiction. Since 
is an efficient solution, it follows from Theorem, that there exist
, 
, 
, 
, 
, 
and 
such that
is an efficient solution of (M-WD). Since 
is an optimal solution of (M-WD), it follows that
This, because of second-order strict-pseudoinvexity of
(46)
Also from the constraint of (VCP) and (M-WD), we have
Because of second-order quasi-invexity of
, this implies
(47)
Combining (46) and (47), we have
(By integrating by parts)
This, by using η = 0, at t = a and t = b, implies
contradicting the feasibility of
for (M-WD).
5. Problems with Natural Boundary Values
In this section, we formulate a pair of nondifferentiable Mond-Weir type dual variational problems with natural boundary values rather than fixed end points given bellow
: Minimize
Subject to
: Maximize
Subject to
and
, at
.
We shall not repeat the proofs of Theorems 3-6 for the above problems, as these follow on the lines of the analysis of the preceding section with slight modifications.
6. Non-Linear Multiobjective Programming Problem
If the time dependency of 
and 
is ignored, then these problems reduce to the following nondifferentiable second-order nonlinear problems already studied in the literature:
(VP1): Minimize
subject to
(VD1): Maximize
subject to
NOTES