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,
![](https://www.scirp.org/html/10-1040149\21e7795c-ab14-4bb1-b080-4400164d0c15.jpg)
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
![](https://www.scirp.org/html/10-1040149\ac95e9f9-8114-4b3e-a7f6-f5958a742863.jpg)
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
![](https://www.scirp.org/html/10-1040149\a9a50c9e-2ed0-40b0-b3ee-9849ede74c51.jpg)
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
![](https://www.scirp.org/html/10-1040149\4ede5eaf-94f7-4d3b-93f3-aef3ccffe658.jpg)
then
is said to be second-order pseudoinvex with respect to
.
Definition 3. (Second-order strict-pseudoinvex): If the functional
satisfies
![](https://www.scirp.org/html/10-1040149\280cca31-21e3-4616-9269-ee3e53b6d912.jpg)
then
is said to be second-order pseudoinvex with respect to
.
Definition 4. (Second-order Quasi-invex): If the functional
satisfies
![](https://www.scirp.org/html/10-1040149\aeaeb921-adec-4bd6-bedd-ebadfe163110.jpg)
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
![](https://www.scirp.org/html/10-1040149\e5767af0-0596-4ae0-8fd7-3d99e63184ac.jpg)
with equality in (1) if
for some ![](https://www.scirp.org/html/10-1040149\457acef9-a5be-4337-b981-55af14541e3d.jpg)
Throughout the analysis of this research, the following conventions for the inequalities will be used:
If
with
and
, then
![](https://www.scirp.org/html/10-1040149\a6f56975-dda6-4b81-980e-ee3b8b26d019.jpg)
3. Statement of the Problem and Necessary Optimality Conditions
Consider the following nondifferentiable Multiobjective variational problem:
(VCP): Minimize
![](https://www.scirp.org/html/10-1040149\b8ce9a7d-099e-434e-81c8-cd0f029fb39c.jpg)
subject to
(1)
(2)
![](https://www.scirp.org/html/10-1040149\0f7806c3-58a1-45d8-a21e-21aa3af359f4.jpg)
where 1)
denote the space of piecewise smooth functions x with norm
, where differentiation operator D already defined.
![](https://www.scirp.org/html/10-1040149\c902947d-e558-4407-865d-3120a4edd1ad.jpg)
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
![](https://www.scirp.org/html/10-1040149\b69950d5-6d5c-4f34-a3f6-4ec70a4e5d27.jpg)
for some
and
![](https://www.scirp.org/html/10-1040149\90b8b007-5ff9-4244-a494-90c0d07a2b32.jpg)
for ![](https://www.scirp.org/html/10-1040149\653b3d18-3034-4ae7-b924-c6a95b5b6845.jpg)
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 ![](https://www.scirp.org/html/10-1040149\a2f24be7-0059-4e58-807c-43ba818cc2b0.jpg)
: Minimize
![](https://www.scirp.org/html/10-1040149\4fe2b9ed-383f-4de7-bb27-fc5a4574d58d.jpg)
subject to
![](https://www.scirp.org/html/10-1040149\d0d6254c-0cda-4f19-9df9-2613c18adf33.jpg)
![](https://www.scirp.org/html/10-1040149\f6af93e7-3d49-4910-b4c8-9af6b7c98147.jpg)
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 ![](https://www.scirp.org/html/10-1040149\54bba006-9d38-47ef-83b5-a4f4a4869626.jpg)
: Minimize
![](https://www.scirp.org/html/10-1040149\46e267c3-11d5-4fed-b183-e6e3a974335d.jpg)
subject to
![](https://www.scirp.org/html/10-1040149\c64a7e4c-5bf7-4637-9cf0-89f8dbef06a6.jpg)
![](https://www.scirp.org/html/10-1040149\f499ff48-ed82-4424-95f0-f6c6b6c2462d.jpg)
Proof: Let
be an efficient solution of (VCP). Suppose that
is not optimal solution of
, for some
Then there exists an
such that
![](https://www.scirp.org/html/10-1040149\7e1aaf10-c625-433f-b016-536fe7b46568.jpg)
(3)
and
![](https://www.scirp.org/html/10-1040149\7dc7781d-a481-46cd-8f75-05f85ba26fd1.jpg)
The inequality (3) for ![](https://www.scirp.org/html/10-1040149\4b4b744a-d896-4839-883b-0d7af059a5ff.jpg)
(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 ![](https://www.scirp.org/html/10-1040149\9f55f445-0040-4a7a-b8c2-215e848f5c61.jpg)
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 ![](https://www.scirp.org/html/10-1040149\e4d563d3-b70e-48d8-8ba6-a5b2ae2ca4d4.jpg)
for each
and hence in particular
. So by the results of [6] there exist
and piecewise smooth functions
and
such that
![](https://www.scirp.org/html/10-1040149\03d8a099-4c34-47b0-a6ab-86bc0bcbf5b4.jpg)
![](https://www.scirp.org/html/10-1040149\fb33f80f-e606-4a73-8152-265f7d801444.jpg)
![](https://www.scirp.org/html/10-1040149\0a503d22-97ab-469b-83de-75efe0e468c2.jpg)
![](https://www.scirp.org/html/10-1040149\1f99605b-c321-4c6c-b94b-1a746a6ab2e4.jpg)
![](https://www.scirp.org/html/10-1040149\ef30fe71-8e61-4f61-95ae-b37ca7a3ac80.jpg)
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 ![](https://www.scirp.org/html/10-1040149\464cc956-42c2-4b04-9260-28cf0ab4c2c6.jpg)
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
![](https://www.scirp.org/html/10-1040149\81c49ebe-6757-41eb-b038-54b64263a618.jpg)
![](https://www.scirp.org/html/10-1040149\198ca8a4-8836-47d3-9899-79b876ca1326.jpg)
![](https://www.scirp.org/html/10-1040149\bc0e80a2-335e-4072-943a-517c9fd96b0c.jpg)
![](https://www.scirp.org/html/10-1040149\fd126c73-641e-4a80-864c-cd6f66317548.jpg)
![](https://www.scirp.org/html/10-1040149\7bcc11c7-6caa-4b59-8c6d-2d5b85b29b07.jpg)
Summing over
we obtain
![](https://www.scirp.org/html/10-1040149\1f34df6e-2b13-4ad2-a6b7-dcebeda9ad30.jpg)
![](https://www.scirp.org/html/10-1040149\afff609c-7fc2-4748-ae27-8fe5314aecaf.jpg)
where
for each ![](https://www.scirp.org/html/10-1040149\86d44b4a-e90a-45a7-a960-d82da1e6dbe9.jpg)
These can be written as,
![](https://www.scirp.org/html/10-1040149\dbfee28d-538c-48b2-8cb2-92659fe3efa9.jpg)
![](https://www.scirp.org/html/10-1040149\9f2a7e0d-966e-4ef2-bfb4-f720a351c36c.jpg)
where ![](https://www.scirp.org/html/10-1040149\f12e9a81-e4f6-40ce-98b9-49992a9a51c1.jpg)
and ![](https://www.scirp.org/html/10-1040149\1d585598-eb0b-4fba-a3e2-736cbe9e1bb3.jpg)
Setting
![](https://www.scirp.org/html/10-1040149\1f25730e-38a4-4b12-9959-60c2891c36b6.jpg)
we get
![](https://www.scirp.org/html/10-1040149\cdee9617-76ed-4ea7-b08a-24e3dfdc11e9.jpg)
![](https://www.scirp.org/html/10-1040149\e8ee06f8-2ad2-46c6-a3fc-a6f1547fa504.jpg)
That is
![](https://www.scirp.org/html/10-1040149\ac3063ff-c82d-4418-b826-cffb41b16410.jpg)
![](https://www.scirp.org/html/10-1040149\fd9c30ce-d878-4637-a68a-05b85fe4bf75.jpg)
![](https://www.scirp.org/html/10-1040149\8a4fc968-398c-4788-93d0-73f81b9858fa.jpg)
![](https://www.scirp.org/html/10-1040149\f887aa6b-c36b-45f5-a58c-000dbc3a86a0.jpg)
![](https://www.scirp.org/html/10-1040149\2e2fb4d8-fb13-4be3-879a-2bdaedd8399a.jpg)
![](https://www.scirp.org/html/10-1040149\70d4048c-f969-4114-9f26-8a5e1476f919.jpg)
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
![](https://www.scirp.org/html/10-1040149\3307bcf2-1ca9-4ef6-ba64-47f05ad81aa5.jpg)
subject to
(16)
(17)
(18)
(19)
where
![](https://www.scirp.org/html/10-1040149\a1a93835-378a-4f0f-95b5-bf12fc65f32c.jpg)
and
![](https://www.scirp.org/html/10-1040149\bebf6dca-4687-44f0-972d-05ead24835db.jpg)
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
![](https://www.scirp.org/html/10-1040149\bfca2c66-218a-4726-88dc-ace9ca30ba76.jpg)
Since ![](https://www.scirp.org/html/10-1040149\80fc697b-dbdd-4b36-b73d-67588f0d06f6.jpg)
we have
![](https://www.scirp.org/html/10-1040149\a7569ed7-ce7c-4e65-9af8-a8594711be06.jpg)
Now, by the constraints (2), (18) and (19), we have
![](https://www.scirp.org/html/10-1040149\78879559-d1e4-4a77-ba82-4bc7c72ef0f1.jpg)
This by (A3), yields
![](https://www.scirp.org/html/10-1040149\e1c1b3c9-b98d-44c7-adf9-fd448644fec2.jpg)
By integration by parts, we have
![](https://www.scirp.org/html/10-1040149\84796034-c8b5-489e-ae8f-b4398aaeb8b2.jpg)
![](https://www.scirp.org/html/10-1040149\33c85c29-c6aa-4309-9a0f-b8be06222ee8.jpg)
This, by using (4) gives
(22)
![](https://www.scirp.org/html/10-1040149\a4d1a563-8110-4142-8afc-93ea3df5c11d.jpg)
![](https://www.scirp.org/html/10-1040149\563c949a-a410-4bf0-83cb-d2e86b7c8b60.jpg)
![](https://www.scirp.org/html/10-1040149\8b8334d1-f640-42d3-b960-736757d5b123.jpg)
By hypothesis (A1), it implies
![](https://www.scirp.org/html/10-1040149\be9f9d7b-3c53-4fba-a9bc-7c4a2e65d6c7.jpg)
Using
in the above, we have
![](https://www.scirp.org/html/10-1040149\c959ca0c-36ff-494d-904f-a7d74d5a44ae.jpg)
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
![](https://www.scirp.org/html/10-1040149\f58943e5-daa7-4f0a-abde-2a4398797e57.jpg)
Hence
![](https://www.scirp.org/html/10-1040149\f3a842e4-65ca-46e8-9c65-a85af595380c.jpg)
satisfies the constraints of (M-WD) and
![](https://www.scirp.org/html/10-1040149\0e3102ec-c03b-46f8-80fd-d5d5b3aee30d.jpg)
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
![](https://www.scirp.org/html/10-1040149\c5c82dab-2819-4c55-8ede-20c0cc57eb78.jpg)
As
, we have
![](https://www.scirp.org/html/10-1040149\6845382c-a9fe-4c90-8cdc-57b2870272f1.jpg)
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)![](https://www.scirp.org/html/10-1040149\2085e930-4698-4d64-9cfa-49d7f3f40a67.jpg)
are linearly independent and
(A4) for
either a) ![](https://www.scirp.org/html/10-1040149\797f4b14-628e-4837-964b-cab93ded7c5b.jpg)
and ![](https://www.scirp.org/html/10-1040149\63e4ab43-81d3-4f34-b9ef-6603c96d03ad.jpg)
or b) ![](https://www.scirp.org/html/10-1040149\3dab22dc-6ae9-404c-825e-2a0df17329f3.jpg)
and ![](https://www.scirp.org/html/10-1040149\5b7e4ab1-15b6-47b0-b251-e624595f4571.jpg)
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
, ![](https://www.scirp.org/html/10-1040149\d33aa462-f700-468a-ad49-023781886dbb.jpg)
Using
and
, (42) implies
![](https://www.scirp.org/html/10-1040149\eeea7cc1-aca7-4768-a416-fc3c3edf7050.jpg)
This, because of (A3) yields
(43)
The relation (43) with
gives ![](https://www.scirp.org/html/10-1040149\ef20dbd5-09b2-4d87-b030-5d4fca7e2112.jpg)
Since
, (36) gives
The relation (30) yields
we have
,
from (32) and
,
,
from (35). These yield
,
,
.
Consequently
![](https://www.scirp.org/html/10-1040149\99888bb9-4eb1-4192-8d46-54380741acb4.jpg)
contradicting (38).
Hence
and from (43) ![](https://www.scirp.org/html/10-1040149\05741b06-555e-4d09-a8d9-e616ecf4fb29.jpg)
Multiplying (30) by
, summing over j, and then using (34) and (41), we have
![](https://www.scirp.org/html/10-1040149\e59e1b46-63de-479e-95bb-65caad5beb57.jpg)
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
![](https://www.scirp.org/html/10-1040149\687d91b1-67ca-44e2-ad1c-535163309e61.jpg)
If
(44) gives
. So we still get
![](https://www.scirp.org/html/10-1040149\a9b6c8a6-42db-4382-910c-5844bfa04a5a.jpg)
Now suppose that
is not an efficient of (VCP). Then, there exists
such that
![](https://www.scirp.org/html/10-1040149\28151c19-1f8a-4b9b-839c-1344a5c0d888.jpg)
and
![](https://www.scirp.org/html/10-1040149\88a3b2d3-c0fa-4d16-aa0f-6b951ecc42c5.jpg)
Using
and
![](https://www.scirp.org/html/10-1040149\86ec0eed-98a1-4f72-a038-7da50ffee1fc.jpg)
We have
![](https://www.scirp.org/html/10-1040149\95ba7fac-b8e8-4a33-95b5-f05e2d1a4e32.jpg)
for some
and![](https://www.scirp.org/html/10-1040149\6fd422bf-f519-4fe4-a4f3-25d369663eac.jpg)
![](https://www.scirp.org/html/10-1040149\ef396325-77df-415c-99c0-7ad91e608c45.jpg)
This contradicts Theorem 3. Hence
is an efficient solution for (VCP).
Theorem 6 (Strict converse duality): Assume that
![](https://www.scirp.org/html/10-1040149\51ab5255-82cb-4d05-9b60-71e4368bc49a.jpg)
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 ![](https://www.scirp.org/html/10-1040149\6e938f61-c8a2-48d3-b403-9a3aa2889d26.jpg)
Proof: We assume that
and exhibit a contradiction. Since
is an efficient solution, it follows from Theorem, that there exist
,
,
,
,
,
and
such that
![](https://www.scirp.org/html/10-1040149\a38ee20b-7a80-4aa8-8fd2-730f0fc93efc.jpg)
is an efficient solution of (M-WD). Since
is an optimal solution of (M-WD), it follows that
![](https://www.scirp.org/html/10-1040149\f0e88df9-1e2d-4757-9259-10c30da0b3b7.jpg)
This, because of second-order strict-pseudoinvexity of
(46)
Also from the constraint of (VCP) and (M-WD), we have
![](https://www.scirp.org/html/10-1040149\8878eb33-58aa-48ed-ae08-dd22b7ce1d02.jpg)
Because of second-order quasi-invexity of
, this implies
(47)
Combining (46) and (47), we have
![](https://www.scirp.org/html/10-1040149\c3498201-548d-450d-848c-c483e76c3d05.jpg)
(By integrating by parts)
This, by using η = 0, at t = a and t = b, implies
![](https://www.scirp.org/html/10-1040149\191da443-badb-4211-b288-dd5e6ec7debb.jpg)
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
![](https://www.scirp.org/html/10-1040149\e1c811b7-c491-4ade-992b-b570db310f1a.jpg)
Subject to
![](https://www.scirp.org/html/10-1040149\af5abdf2-8900-44ee-aa7d-e78d5ee8a8d2.jpg)
: Maximize
![](https://www.scirp.org/html/10-1040149\155ed3d2-b028-4c54-9207-5e80690bff5c.jpg)
Subject to
![](https://www.scirp.org/html/10-1040149\ad7aa2bc-d3d6-409d-8875-aa4a44f30f31.jpg)
![](https://www.scirp.org/html/10-1040149\366357e2-0588-46ee-be07-f1f48218e7a5.jpg)
![](https://www.scirp.org/html/10-1040149\a07d7dab-ed3d-4c83-938b-b928fae51c95.jpg)
![](https://www.scirp.org/html/10-1040149\233f46aa-af4d-4e97-a24f-ffe82b42f257.jpg)
and
, at
.
![](https://www.scirp.org/html/10-1040149\7ea2809d-326e-481d-917c-eab5a90d97cf.jpg)
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
![](https://www.scirp.org/html/10-1040149\9f8b2bf0-2739-44a5-9b98-79f7162f4555.jpg)
subject to
![](https://www.scirp.org/html/10-1040149\6c23c4d1-b26f-4e6d-9b73-1c695fe715bd.jpg)
(VD1): Maximize
![](https://www.scirp.org/html/10-1040149\390b8977-8293-47bc-8902-c6c4697c25f8.jpg)
subject to
![](https://www.scirp.org/html/10-1040149\33afb300-7c30-45f5-b056-a4c76ef79703.jpg)
![](https://www.scirp.org/html/10-1040149\dfae3725-5465-4304-a72a-007205bf8264.jpg)
NOTES