1. Introduction
Chen [1] was the first to identify second-order dual formulated for a constrained variational problem and established various duality results under an involved invexitylike assumptions. Husain et al. [2] have presented MondWeir type second-order duality for the problem of [1] and by introducing continuous-time version of secondorder invexity and generalized second-order invexity, validated various duality results. Subsequently, for a class of nondifferentiable continuous programming problems, Husain and Masoodi [3] studied Wolfe type second-order duality while Husain and Srivastava [4] investigated MondWeir type second-order duality. Recently, in the spirit of Mangasarian [5], Husain and Masoodi [6] studied Wolfe type second-order duality for a continuous programming problem having support functions appearing in the integrand of the functional as well as in the constraint functions under second-order invexity and second-order pseudoinvexity conditions. They also incorporated a pair of second-order dual variational problems with natural boundary values rather than fixed end points and indicated their close relationship with those of corresponding (static) second-order duality results established for nonlinear programming problem with support functions, considered by Husain et al. [7]. The popularity of this type of nondifferentiable continuous programming problems seems to originate from the fact that, even though the objective function and/or constraint functions are non-smooth, a simple representation of the dual problem may be written. The theory of non-smooth mathematical programming deals with more general type of functions by means of generalized subdifferentials. However, square root of positive semi-definite quadratic form and support functions are amongst few cases of the nondifferentiable functions for which one can write down the subdifferentials explicitly.
In this paper, we formulate Mond-Weir type secondorder dual to the continuous programming containing support functions in order to further weaken the secondorder generalized invexity of [6]. Usual duality theorems for this pair of Mond-Weir type second-order dual continuous programming problems are validated under generalized second-order invexity assumptions. Special cases are derived. Further, a pair of Mond-Weir second-order dual variational problems with natural boundary values rather than fixed end points is presented and the proofs of the duality theorems are claimed to follow analogously. It is also pointed out that our second-order duality results can be considered as dynamic generalizations of corresponding (Static) second-order duality results established for nonlinear programming problem with support functions considered by Husain et al. [7].
2. 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 derivative of 
with respect to 
and 
respectively, that is,
Denote by 
the 
Hessian matrix of 
and 
the 
Jacobian matrix respectively, that is, with respect to
, that is,
the 
Jacobian matrix.
The symbols 
have analogous representations. Designate by X the space of piecewise smooth functions
, with the norm
where the differentiation operator 
is given by
, Thus 
except at discontinuities.
We incorporate the following definitions which needed in the subsequent analysis:
Definition1. (Second-Order Invex):
If there exists a vector function 
where 
and with 
at t = a and t = b such that for a scalar function 
the functional 
where 
satisfies
Then 
is second-order invex with respect to 
where 
the space of 
-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 Strictly Pseudoinvex):
If the functional 
satisfies
then 
is said to be second-order strictly pseudoinvex with respect to
.
Definition 4. (Second-Order Quasi-invex):
If the functional 
satisfies
Then 
is said to be second-order quasiinvex with respect to 
Consider the following nondifferentiable continuous programming problem with support functions treated by Husain and Jabeen [8]:
(CP): Minimize 
subject to
(1)
(2)
where f and g are continuously differentiable and each 
is a compact convex set in 
Husain and Jabeen [8] derived the following optimality condition for (CP):
Lemma 1. (Fritz-John Necessary Optimality Conditions):
If the problem (CP) attains a minimum at 
then there exist 
and piecewise smooth functions 
with 
and 
such that
The minimum 
of (CP) may be described as normal if 
so that the Fritz John optimality conditions reduce to Karush-Kuhn-Tucker optimality conditions. It suffices for 
that Slater’s [8] condition holds at
.
Now we review some well known facts about a support function for easy reference.
Let K be a compact set in
, then the support function of 
is defined by
A support function, being convex everywhere finite, has a subdifferential in the sense of convex analysis i.e., there exist 
such that
From [9], the subdifferential of 
is given by
For any set
, the normal cone to 
at a point 
is defined by
It can be verified that for a compact convex set C, 
if and only if
3. Mond-Weir Type Second-Order Duality
In this section, we present the following problem as the Mond-Weir type dual to (CP) and validate usual duality theorems:
(M-WCD):
subject to
(3)
(4)
(5)
(6)
(7)
where
1) 
2) 
3)
Theorem 1. (Weak Duality): Let 
be feasible solution of (CP) and
be feasible for (M-WCD). Assume that for all feasible
and with respect to vector function 
1) 
is second-order pseudoinvex and
2) 
is second-order quasi-invex.
Then,
.
Proof: Since 
is feasible for (CP) and
is feasible of (M-WCD), we have
Using 
we have,
By the second-order quasi-invexity of
for 
with respect to 
from this we have,
By integrating by parts, we have
Using 
at 
and
, this yields,
Using equality constraint (4), we have
As earlier, this becomes
This, because of second-order pseudoinvexity of
with respect to
gives
Since 
we have
implying,
.
Theorem 2. (Strong Duality):
If 
be an optimal solution of (CP) and is normal, then there exist piecewise smooth functions 
and 
such that 
is a feasible solution of (CD) and the two objective values are equal. Furthermore, if the hypothesis of Theorem1 holds, then 
is an optimal solution of (M-WCD).
Proof: From Lemma 1 there exist piecewise smooth functions 
and 
such that
The above relations imply that
is feasible for (M-WCD).
Also
This shows the equality of objective functions of the problem. Hence the optimality of
for (M-WCD) follows from weak duality theorem (Theorem1).
Theorem 3 (Strict Converse Duality): Assume that
(C1): 
is second-order strictly pseudoinvex and 
is second-order quasi-invex with respect to the same
.
(C2): 
is an optimal solution for (CP), If 
is optimal solution of (MWCD), then 
Proof: We assume that 
and show that a contradiction follows. Since 
is an optimal solution of (CP), it follows from Theorem 2, there exist 
and
such that
is optimal solution of (M-WCD).
Since
is an optimal solution of (M-WCD), it follows that
This, because of the second-order strict pseudoinvexity of 
for all 
gives
(8)
From the constraint of (CP) and (M-WCD), we have
Using 
from this, we have
This, because of (C1) we have
(9)
Combining (8) and (9), we have
Using 
at
, this implies
contradicting the equality constraint of (M-WCD), hence
.
Theorem 4. (Converse Duality): Assume that
(H1): 
is an optimal solution of (M-WCD).
(H2): The vectors 
are linear independent where 
and 
are the ith row of F and G respectively, and
(H3):
and
(H4): either 
and 
or 
and 
Then 
is feasible for (CP) and the two objective functionals have the same value. Also, if Theorem1 holds for all feasible solution of (CP) and (M-WCD), then 
is an optimal solution of (CP).
Proof: Since
is an optimal solution of (M-WCD), by results of Schester [10], there exists 
and piecewise smooth function 
and 
such that following Fritz John optimality conditions are satisfied:
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
Using the hypothesis (H2) in (12), we have
(19)
(20)
Using (4), (19) and (20) in (10), we have (see (21) below)
Let 
then (20) gives 
(19) implies
.
Consequently from (21), we have
By the hypothesis (H3), this implies 
The relation (11) implies 
Hence 
contradicting the Fritz John condition, Hence 
Pre-multiplying (11) by 
and Using (16), we have
Integrating and then using (15), we have
(21)
Putting 
we have
gives
This, in view of the hypothesis (H4) yields, 
we have 
and
These respectively imply 
and
.
Multiplying the relation (11) by 
and using (16) along with
, we have
and also 
implying the feasibility of 
for (CP).
Finally,
By Theorem 1, it implies that 
is an optimal solution of (CP).
4. Special Cases
Let for 
be positive semidefinite matrices and continuous on 
Then 
where
Replacing 
by 
and
by
We have the following problems:
subject to
(M-WCD2):
subject to
If 
are suppressed from the constraints of (CP2), we have the following problem studied for duality by Husain and Srivastava [4].
subject to
(M-WCD3):
5. Problems with Natural Boundary Values
In this section, we formulate a pair of nondifferentiable dual variational problems with natural boundary values rather than fixed end points.
subject to
,
subject to
6. Nonlinear Programming Problems
If all functions in the problems (CP0) and (M-WCD0) are independent of t, then these problems will reduce to the following nonlinear programming problems studied by Husain et al. [7].
(CP1): Minimize 
subject to
subject to
where 
and 