ABSTRACT

A second-order Mond-Weir type dual problem is formulated for a class of continuous programming problems in which both objective and constraint functions contain support functions; hence it is nondifferentiable. Under second-order strict pseudoinvexity, second-order pseudoinvexity and second-order quasi-invexity assumptions on functionals, weak, strong, strict converse and converse duality theorems are established for this pair of dual continuous programming problems. Special cases are deduced and a pair of dual continuous problems with natural boundary values is constructed. A close relationship between the duality results of our problems and those of the corresponding (static) nonlinear programming problem with support functions is briefly outlined.

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 i^th 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

REFERENCES