10_husain.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 3, 2012, 390-400

ISSN 1307-5543 – www.ejpam.com

Second-Order Duality For a Class of Nondifferentiable

Continuous Programming Problems

Iqbal Husain1,∗, Mashoob Masoodi2

1 Department of Mathematics, Jaypee University of Engineering and Technology, Guna (M.P.), India
2 Department of Statistics, University of Kashmir, Srinagar (Kashmir), India

Abstract. A dual problem associated with a class of non-differentiable continuous programming prob-

lems is formulated. Under the second-order pseudo-invexity, various duality theorems are validated

for this pair of dual problems. A pair of dual problems with natural boundary values is constructed

and the proofs of its various duality results are merely indicated. Further, it is shown that our results

can be viewed as dynamic generalizations of corresponding (static) second-order duality theorems for

a class of nondifferentiable nonlinear programming problems existing in the literature.

2010 Mathematics Subject Classifications: 90C30, 90C11, 90C20, 90C26.

Key Words and Phrases: Continuous programming, Second-order invexity, Second-order Pseudoin-

vexity, Second-order duality, Nonlinear programming

1. Introduction

Second-order duality in mathematical programming has been extensively investigated in

the literature. A second-order dual formulation for a non-linear programming problem was

introduced by Mangasarian [5]. Later Mond [6] established various duality theorems under a

condition which is called "Second-order convexity". This condition is much simpler than that

used by Mangasarian [5]. In [9], Mond and Weir reconstructed the second-order duals and

higher order dual models to drive usual duality results. It is remarked here that second-order

dual to a mathematical programming problem presents a tighter bound and because of which

it enjoys computational advantage over a first order dual.

Duality and optimality for continuous programming have been widely investigated by

many authors in the recent past notably, Mond and Hanson [7], Bector, Chandra and Husain

[1], Mond and Husain [8] and Chen [3] and other cited references in these expositions.

Chen [3] was the first to identify second-order dual formulated for a constrained vari-

ational problem and established various duality results under an involved invexity- like as-

sumptions. Recently, Husain et al [4] have presented Mond-Weir type duality for the problem

∗Corresponding author.

Email addresses: ihusain11�yahoo.
om (I. Husain), masoodisaba�yahoo.
om (M. Masoodi)

http://www.ejpam.com 390 c© 2012 EJPAM All rights reserved.



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 391

of [3] and by introducing continuous-time version of second-order invexity and generalized

second-order invexity, validated various duality results.

In this paper we formulate a Wolfe type second order dual to a class of nondifferentiability

continuous programming problems where nondifferentiability enters due to the square root

of a certain quadratic form appearing in the integrand of the objective functional. The popu-

larity of this type of 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 found. The theory of non-smooth mathematical programming deals with

more general type of functions by means of generalized sub- differentials. However, square

root of positive semi-definite quadratic form is one of the few cases of the nondifferentiable

functions for which one can write down the sub-or quasi-differentials explicitly. Here, var-

ious duality theorems for this pair of Wolfe type dual problems are validated under second

order pseudo-invexity condition. A pair of Wolfe type dual variational problems with natural

boundary values rather than fixed end points is presented and the proofs of its duality results

are indicated. It is also shown that our second-order duality results can be considered as dy-

namic generalizations of corresponding (Static) second-order duality results established for

nondifferentiable nonlinear programming problem, considered by Zhang and Mond [10].

2. Definitions and Related Pre-requistes

Let I = [a, b] be a real interval, φ : I ×Rn×Rn −→ R and ψ : I ×Rn×Rn −→ Rm be twice

continuously differentiable functions. In order to consider φ(t, x(t), ẋ(t)) where x : I −→ Rn

is differentiable with derivative ẋ , denoted by φx and φ ẋ , the first order of φ with respect to

x(t) and ẋ(t), respectively, that is,

φx =

�

∂ φ

∂ x1
,
∂ φ

∂ x2
, ...,

∂ φ

∂ xn

�T

, φ ẋ =

�

∂ φ

∂ ẋ1
,
∂ φ

∂ ẋ2
, ...,

∂ φ

∂ ẋn

�T

Denote by φx x the Hessian matrix of φ, and ψx the m× n Jacobian matrix respectively, that

is, with respect to x(t), that is, φx x = (
∂ 2φ

∂ x i∂ x j ), i, j = 1,2, ...n, ψx the m× n Jacobian matrix

ψx =















∂ ψ1

∂ x1

∂ ψ1

∂ x2 · · ·
∂ ψ1

∂ xn

∂ ψ2

∂ x1

∂ ψ2

∂ x2 · · ·
∂ ψ2

∂ xn

...
...

...
∂ ψm

∂ x1

∂ ψm

∂ x2 · · ·
∂ ψm

∂ xn















n×n

The symbols φ ẋ ,φ ẋ x ,φx ẋ and ψ ẋ have analogous representations.

Designate by X the space of piecewise smooth functions x : I −→ Rn, with the norm

‖x‖= ‖x‖∞+ ‖Dx‖∞, where the differentiation operator D is given by

u = Dx ⇐⇒ x(t) =

∫ t

a

u(s)ds,



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 392

Thus d

d t
= D except at discontinuities.

We incorporate the following definitions which are required in the subsequent analysis.

Definition 1 (Second order Invex). If there exists a vector function η = η(t, x , x̄) ∈ Rn where

η : I × Rn×Rn −→ Rn with η = 0 at t = a and t = b, such that for a scalar function φ(t, x , ẋ),

the functional
∫

I
φ(t, x , ẋ)d t where φ : I × Rn× Rn −→ R satisfies

∫

I

φ(t, x , ẋ)d t−

∫

I

{φ(t, x̄ , ˙̄x)−
1

2
pT (t)Gp(t)}d t

≥

∫

I

{ηTφx (t, x̄ , ˙̄x) + (Dη)Tφ ẋ(t, x̄ , ˙̄x)+ηT Gp(t)}d t

then
∫

I
φ(t, x , ẋ)d t is second-order invex with respect to η where

G = φx x − 2Dφx ẋ + D2φ ẋ ẋ − D3φ ẋ ẍ and p ∈ C(I ,Rn), the space of continuous n-dimensional

continuous vector functions.

Definition 2 (Second order Pseudoinvex). If the functional
∫

I
φ(t, x , ẋ)d t satisfies

∫

I

{ηTφx+(Dη)
Tφ ẋ +η

T Gp(t)}d t ≥ 0=⇒

∫

I

φ(t, x , ẋ)d t ≥

∫

I

{φ(t, x̄ , ˙̄x)−
1

2
p(t)T Gp(t)}d t,

then
∫

I
φ(t, x , ẋ)d t is said to be second-order pseudoinvex with respect to η.

Definition 3 (Second order Quasi-invex). If the functional
∫

I
φ(t, x , ẋ)d t satisfies

∫

I

φ(t, x , ẋ)d t ≤

∫

I

{φ(t, x̄ , ˙̄x)−
1

2
p(t)T (t)Gp(t)}d t

=⇒

∫

I

{ηTφx + (Dη)
Tφ ẋ +η

T G(t)p(t)}d t ≤ 0,

then
∫

I
φ(t, x , ẋ)d t is said to be second-order quasi-invex with respect to η.

Remark 1. If φ does not depend explicitly on t, then the above definitions reduce to those given

in [6] for static cases.

Consider the following class of non-differentiable continuous programming problem stud-

ied in [2]:

(P+) Minimize

∫

I

{ f (t, x(t), ẋ(t)) + (x(t)T B(t)x(t))1/2}d t

subject to x(a) = 0= x(b),

g(t, x(t), ẋ(t)) ≤ 0, t ∈ I ,

h(t, x(t), ẋ(t)) = 0, t ∈ I

where



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 393

(i) f , g and h are twice differentiable functions from I × Rn× Rn into R, Rm and Rk respec-

tively, and

(ii) B(t) is a positive semidefinite n× n matrix with B(.) continuous on I .

The following proposition gives the Fritz John type of optimality conditions which are derived

by Chandra, Craven and Husain [2]:

Proposition 1 (Fritz-John Conditions). If (VP+) attains a local minimum at x̄ ∈ X and if

hx(·, x̄(·), ˙̄x(·)) maps X onto a closed subspace of C(I ,Rp), then there exist Lagrange multipliers

τ ∈ R+, piecewise smooth ȳ : I → Rm and λ̄ : I → Rk, not all zero, and also piecewise smooth

z̄ : I → Rn satisfying for all t ∈ I ,

τ fx (t, x̄(t), ˙̄x(t)) + z̄(t)T B(t) + ȳ(t)T gx(t, x̄(t), ˙̄x(t)) + µ̄(t)T hx(t, x̄(t), ˙̄x(t))

= D[τ fx(t, x̄(t), ˙̄x(t)) + ȳ(t)T g ẋ(t, x̄(t), ˙̄x(t)) + µ̄(t)T h ẋ(t, x̄(t), ˙̄x(t))] t ∈ T,

ȳ(t)T g(t, x̄(t), ˙̄x(t)) = 0, t ∈ I

z̄(t)T B(t) z̄(t) ≤ 1, t ∈ I

x̄(t)T B(t) z̄(t) = ( x̄(t)T B(t) x̄(t))1/2, t ∈ I

If hx(·, x̄(·), ˙̄x(·)) is subjective, then τ and ȳ are not both zero.

Lemma 1 (Schwartz inequality). It states that

x(t)T B(t)z(t) ≤ (x(t)T B(t) x(t))1/2(z(t)T B(t) z(t))1/2 , t ∈ I (1)

with equality in (1) if (and only if)

B(t)(x(t)− q(t)z(t)) = 0 for some q(t) ∈ R.

Remark 2. The Fritz John necessary optimality conditions in Proposition 1 for (P+), become the

Karush-Kuhn-Tucker type optimality conditions if τ = 1. It suffices for τ = 1, that the following

Slater’s condition holds:

g(t, x̄(t), ˙̄x(t)) + gx(t, x̄(t), ˙̄x(t))ν(t) + g ẋ(t, x̄(t), ˙̄x(t))ν̇(t) < 0, ν(t) ∈ X , t ∈ I .

3. Second-Order Duality

Consider the following continuous programming problem (CP) by ignoring the equality

constraint h(t, x̄(t), ˙̄x(t)) = 0, t ∈ I in the problem (P+):

(CP) Minimize

∫

I

{ f (t, x(t), ẋ(t)) + (x(t)T B(t)x(t))1/2}d t

subject to x(a) = 0= x(b), (2)

g(t, x(t), ẋ(t))≤ 0, t ∈ I (3)



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 394

Analogously to the second-order dual problem introduced by Mangasarian [5] for a nonlinear

programming problem, we consider the following second order dual continuous programming

problem (CD) for (CP).

(CD) Maximize

∫

I

{ f (t,u(t), u̇(t)) + u(t)T B(t)z(t) + y(t)T g(t,u(t), u̇(t))−
1

2
p(t)T H p(t)}d t

subject to u(a) = 0= u(b), (4)

fu(t,u(t), u̇(t)) + B(t)z(t) + y(t)T gu(t,u(t), u̇(t))

−D( fu̇(t,u(t), u̇(t)) + y(t)T (5)

z(t)T B(t)z(t) ≤ 1, t ∈ I (6)

y(t) ≥ 0, t ∈ I (7)

where

H = fuu(t,u, u̇) + (y(t)T gu(t,u, u̇))u− 2D[ fuu̇(t,u, u̇) + (y(t)T gu(t,u, u̇))u̇]

+ D2[ fu̇u̇(t,u, u̇) + (y(t)T gu̇(t,u, u̇))u̇]− D3[ fu̇ü(t,u, u̇) + (y(t)T gu̇(t,u, u̇))ü]

Theorem 1 (Weak duality). Let x(t) ∈ X be a feasible solution of (CP) and u(t), y(t), z(t) be a

feasible solution of (CD). If

∫

I

{ f (t, ., .) + (·)T B(t)z(t) + y(t)T g(t, ., .)}d t

is second-order pseudo-invex with respect to η = η(t, x ,u), then

inf(C P)≥ sup(C D).

Proof. From (5), we have

∫

I

ηT { fu(t,u(t), u̇(t)) + B(t)z(t) + y(t)T gu(t,u(t), u̇(t))

− D( fu̇(t,u(t), u̇(t)) + y(t)T gu̇(t,u(t), u̇(t)))}d t +

∫

I

ηT H p(t)d t

=

∫

I

[ηT { fu(t,u(t), u̇(t)) + B(t)z(t) + y(t)T gu(t,u(t), u̇(t))

+ (Dη)T ( fu̇(t,u(t), u̇(t)) + y(t)T gu̇(t,u(t), u̇(t))) +η
T H p(t)}]d t

−ηT ( fu̇(t,u(t), u̇(t)) + y(t)T gu̇(t,u(t), u̇(t)))|
t=b
t=a,

by integration by parts. Using the boundary conditions (2) and (4), we have

∫

I

[ηT { fu(t,u(t), u̇(t)) + B(t)z(t) + y(t)T gu(t,u(t), u̇(t))

+ (Dη)T ( fu̇(t,u(t), u̇(t)) + y(t)T gu̇(t,u(t), u̇(t))) +η
T H p(t)}]d t = 0.



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 395

This, in view of second-order pseudo-invexity of
∫

I
{ f (t, ., .)+(·)T B(t)z(t)+ y(t)T g(t, ., .)}d t

yields

∫

I

{ f (t, x , ẋ) + x(t)T B(t)z(t) + y(t)T g(t, x , ẋ)}d t

≥

∫

I

{ f (t,u, u̇) + u(t)T B(t)z(t) + y(t)T g(t,u, u̇)−
1

2
p(t)T H p(t)}d t

Because of Schwartz’s inequality (1) along with (5), (6) and (2), this implies

∫

I

{ f (t, x(t), ẋ(t)) + (x(t)T B(t)x(t))1/2}d t

≥

∫

I

{ f (t,u, u̇) + u(t)T B(t)z(t) + y(t)T g(t,u, u̇)−
1

2
p(t)T H p(t)}d t

yielding,

inf(C P)≥ sup(C D).

Theorem 2 (Strong duality). If x̄(t) ∈ X is an optimal solution of (CP) and is also normal, then

there exist piecewise smooth functions y → Rm and z→ Rn such that

( x̄(t), ȳ(t), z̄(t), p(t) = 0) is a feasible solution of (CD) and the two objective values are equal.

Furthermore, if the hypotheses of Theorem 1 hold, then ( x̄(t), ȳ(t), z̄(t), p(t)) is an optimal of

(CD).

Proof. From Proposition 1, there exist piecewise smooth functions ȳ : I → Rm and

z̄ : I → Rn such that (for t ∈ I)

( fx(t, x̄ , ˙̄x)+ B(t)z̄(t) + ȳ(t)T g(t, x̄ , ˙̄x))− D( f ẋ (t, x̄ , ˙̄x) + ȳ(t)T g ẋ(t, x̄ , ˙̄x)) = 0, (8)

ȳ(t)T g(t, x̄ , ˙̄x) = 0, (9)

x̄(t)T B(t)z̄(t) = ( x̄(t)T B(t)x(t))1/2, (10)

z̄(t)T B(t)z̄(t) ≤ 1, (11)

y(t) ≥ 1. (12)

Hence ( x̄(t), ȳ(t), z̄(t), p̄(t) = 0) satisfies the constraints of (CD) and the objective values are

equal. Furthermore, for every feasible solution (u(t), y(t), z(t), p(t) = 0), from the above

conditions and using (9), (10) and p̄(t) = 0 we have

∫

I

{ f (t, x̄ , ˙̄x)+ x̄(t)T B(t)z̄(t)+ ȳ(t)T g(t, x̄(t), ˙̄x(t))−
1

2
p̄(t)T H p̄(t)}d t

=

∫

I

{ f (t, x̄ , ˙̄x)+ ( x̄(t)T B(t) x̄(t))1/2}d t



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 396

≥

∫

I

{ f (t,u, u̇) + u(t)T B(t)z(t) + y(t)T g(t,u, u̇)−
1

2
p(t)T H p(t)}d t

So, ( x̄(t), ȳ(t), z̄(t), p̄(t)) is an optimal solution of (CD).

Theorem 3 (Converse duality). Assume that f and g are thrice continuously differentiable and

( x̄(t), ȳ(t), z̄(t), p̄(t)) be an optimal solution of (CD). Let the following conditions hold:

(i) The Hessian matrix H is non-singular, and

(ii)

(ψ(t)T Hψ(t))x + 2(D(ψ(t)T Hψ(t)) ẋ −ψ(t)
T D(Hψ(t)) ẋ)

− (D2(ψ(t)T Hψ(t)) ẍ −ψ(t)
T D2(Hψ(t)) ẍ)

+ (D3(ψ(t)T Hψ(t))...x −ψ(t)
T D3(Hψ(t))...x )

− (D4(ψ(t)T Hψ(t))....x −ψ(t)
T D4(Hψ(t))....x ) = 0, t ∈ I

=⇒ψ(t) = 0, t ∈ I

Then x(t) is feasible for (CP), ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I . In addition, if the hypotheses in

Theorem 1 hold, then x̄(t) is an optimal solution.

Proof. Since ( x̄(t), ȳ(t), z̄(t), p̄(t)) is an optimal solution for (CD), by Proposition 1, there

exist Lagrange multiplier τ ∈ R, and piecewise smooth θ : I → Rn, µ : I → Rm and α : I → Rn

such that following conditions hold at the feasible point of (CD).

τ[( fx + B(t)z(t) + y(t)T gx)− D( f ẋ + y(t)T g ẋ)−
1

2
(p(t)T H p(t))x + D(p(t)T H p(t)) ẋ

−
1

2
D2(p(t)T H p(t)) ẍ +

1

2
D3(p(t)T H p(t))...x −

1

2
D4(p(t)T H p(t))....x ]

+θ(t)T [ fx x − (y(t)
T gx)x − 2D( f ẋ x + (y(t)

T gx) ẋ)− D2( f ẋ ẋ + y(t)T g ẋ ẋ)

+ D3( f ẋ ẍ + (y(t)
T g ẋ) ẍ + (H p(t))x − D(H p(t)) ẋ + D2(H p(t)) ẍ

− D3(H p(t))...x + D4(H p(t))....x ]

(13)

τ ( f j −
1

2
p(t)T g j

x x p(t)) + θ(t)T (g j
x x − 2Dg

j

x ẋ
+ D2 g

j

ẋ ẋ
)p(t) +µ j(t) = 0,

j = 1,2, . . . , m. (14)

τ x̄(t)T B(t) + θ(t)T B(t)− 2α(t)B(t)z(t) = 0, (15)

(θ(t)−τp(t))H = 0, (16)

fx + B(t)z(t) + ȳ(t)T gx − D( f ẋ + ȳ(t)g ẋ) +H p(t) = 0, (17)

α(t)(1− z̄(t)T B(t)z̄(t)) = 0, (18)

µ̄(t)T ȳ(t) = 0, (19)



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 397

(τ,α(t),µ(t)) ≥ 0, (20)

(τ,α(t),µ(t),θ(t)) 6= 0, (21)

where t ∈ I . By the nonsingularity of H, eqref11 yields,

θ(t) +τp̄(t) = 0, t ∈ I (22)

If τ = 0, (22) implies θ(t) = 0 t ∈ I . From (14), we have µ(t) = 0, t ∈ I . The relation (15)

together with (18) gives α(t) = 0. Hence (τ,α(t),θ(t),µ(t)) = 0, t ∈ I , contradicting (21).

Consequently τ > 0. From (22) and τ > 0, (13) becomes,

[( fx + B(t)z(t) + y(t)T gx)− D( f ẋ+y(t)T g ẋ
)−

1

2
(p(t)T H p(t))x + D(p(t)T H p(t)) ẋ

−
1

2
D2(p(t)T H p(t)) ẍ +

1

2
D3(p(t)T H p(t))...x −

1

2
D4(p(t)T H p(t))....x ]

+ p(t)T [( fx x − (y(t)
T gx)x)− D( f ẋ x + (y(t)

T gx) ẋ)− D( f ẋ x + (y(t)
T g ẋ)x)

− D (D( f ẋ ẋ + y(t)T g ẋ ẋ))+ D2 (D( f ẋ ẍ + (y(t)
T g ẋ)x))

+ (H p(t))x − D(H p(t)) ẋ + D2(H p(t)) ẍ − D3(H p(t))...x + D4(H p(t))....x ] = 0.

Using the expression for H, this gives

[( fx + B(t)z(t) + y(t)T gx)− D( f ẋ+y(t)T g ẋ
) +H p(t)−

1

2
(p(t)T H p(t))x + D(p(t)T H p(t)) ẋ

−
1

2
D2(p(t)T H p(t)) ẍ +

1

2
D3(p(t)T H p(t))...x −

1

2
D4(p(t)T H p(t))....x ]

+ p(t)T [(H p(t))x − D(H p(t)) ẋ + D2(H p(t)) ẍ − D3(H p(t))...x + D4(H p(t))....x ] = 0.

This, by using (17), reduces to

(p(t)T H p(t))x + 2(D(p(t)T H p(t)) ẋ − p(t)T D(H p(t)) ẋ )− (D
2(p(t)T H p(t)) ẍ

− 2p(t)T D2(H p(t)) ẍ ) + (D
3(p(t)T H p(t))...x − 2p(t)T D3(H p(t))...x )

− (D4(p(t)T H p(t))....x − 2p(t)T D4(H p(t))....x ) = 0, t ∈ I ,

which, because of the hypothesis (ii) implies p̄(t) = 0 t ∈ I . From (14), we have

τg j +µ j(t) = 0, t ∈ I , j = 1,2, . . . , m (23)

This because of τ > 0, yields

g j(t, x̄ , ˙̄x)≤ 0, t ∈ I

The relation (23) along with (19) and τ > 0 gives

ȳ(t)T g(t, x̄ , ˙̄x) = 0, t ∈ I (24)

Using θ(t) = 0, t ∈ I and τ > 0, (15) yields

B(t) x̄(t)T = 2(
α(t)

τ
)B(t)z̄(t), t ∈ I (25)



I. Husain, M. Masoodi / Eur. J. Pure Appl. Math, 5 (2012), 390-400 398

which is the required condition for the equality in Schwartz inequality, i.e.,

x̄(t)T B(t)z̄(t) = ( x̄(t)T B(t) x̄(t))1/2(z̄(t)T B(t)z̄(t))1/2, t ∈ I (26)

If α(t) > 0, t ∈ I (18) gives z̄(t)T B(t)z̄(t) = 1, and so (25) implies

x̄(t)T B(t)z̄(t) = ( x̄(t)T B(t) x̄(t))1/2, t ∈ I

If α(t) = 0, t ∈ I , (25) implies B(t) x̄(t), t ∈ I . So we still get

x̄(t)T B(t)z̄(t) = ( x̄(t)T B(t) x̄(t))1/2, t ∈ I (27)

Therefore from (24), (27) and p̄(t) = 0, t ∈ I , we have

∫

T

{ f (t, x̄ , ˙̄x) + ( x̄(t)T B(t) x̄(t))1/2}d t =

∫

T

{ f (t, x̄ , ˙̄x)+ x̄(t)T B(t)z̄(t) + ȳ(t)T g(t, x̄ , ˙̄x)−
1

2
p̄(t)T H p̄(t)}d t

Thus, by the application of Theorem 1 the optimality of x̄(t) for (CP) follows.

4. Natural Boundary Values

In this section, we formulate a pair of non differentiable dual variational problems with

natural boundary values rather than fixed end points.

(CP0) Minimize

∫

I

{ f (t, x(t), ẋ(t)) + ( x̄(t)T B(t) x̄(t))1/2}d t

subject to g(t, x , ẋ)≤ 0, t ∈ I

(CD0) Maximize

∫

I

{ f (t, x(t), ẋ(t)) + x(t)T B(t)z(t) + y(t)T g(t, x , ẋ)−
1

2
p(t)T H p(t)}d t

subject to ( fx (t, x , ẋ) + B(t)z̄(t) + ȳ(t)T gx(t, x , ẋ))

−D( f ẋ (t, x , ẋ) + ȳ(t)T g ẋ(t, x , ẋ)) +H p(t) = 0, t ∈ I

z(t)T B(t)z(t) ≤ 1, t ∈ I

y(t) ≥ 0, t ∈ I

f ẋ (t, x , ẋ) + ȳ(t)T g ẋ(t, x , ẋ)|t=a = 0

f ẋ (t, x , ẋ) + ȳ(t)T g ẋ(t, x , ẋ)|t=b = 0

We shall not repeat the proofs of Theorems 1-3, as these follow on the lines of the analysis of

the preceding section with slight modifications.



REFERENCES 399

5. Non-Differentiable Nonlinear Programming Problems

If all functions in the problems (CP0) and (CD0) are independent of t and b− a = 1, then

these problems will reduce to following nondifferentiable dual variational problems, treated

by Zhang and Mond [10].

(NP) Minimize f (x)+ (x T Bx)1/2

subject to g(x)≤ 0,

(ND) Maximize f (x)+ x T Bz + yT g(x)−
1

2
pT∇2( f (x)+ yT g(x))p

subject to ∇( f (x)+ x T Bz + yT g(x))+∇2( f (x)+ yT g(x))p = 0,

zT Bz ≤ 1, y ≥ 0

where

∇( f (x)+ x T Bz + yT g(x)) = fx(x)+ Bz + yT gx(x)

and

∇2( f (x)+ yT g(x)) = fx x (x)+ (y
T gx(x)) ẋ

6. Conclusion

In this exposition, we have discussed a class of nondifferentiable continuous program-

ming problems treated in [2] and formulated Wolfe type Second-order dual variation problem

which is analogous to the second-order dual problem constructed by Zhang and Mond [10]

for a nondifferentiable nonlinear programming problem. Under second-order pseudoinvexity

we established weak, strong and converse duality theorems. When functions, occurring in the

formulations of the problems, do not depend explicitly on t, our results reduce to those of

[10]. Thus our results become dynamic generalizations of the results in [10]. The problems

of this research can be revisited in multiobjective setting.

References

[1] C.R. Bector, S. Chandra, and I. Husain. Generalized concavity and duality in continuous

programming. Utilitas, Mathematica, 25:171–190, 1984.

[2] S. Chandra, B.D. Craven, and I. Husain. A class of nondifferentiable continuous pro-

gramming problems. J. Math. Anal. Appl., 107:122–131, 1985.

[3] X. Chen. Second order duality for the variational problems. J. Math. Anal. Appl.,

286:261–270, 2003.

[4] I. Husain, A. Ahmed, and M. Masoodi. Second order duality for variational problems.

European Journal of Pure and Applied Mathematics, 2(2):278– 295, 2009.



REFERENCES 400

[5] O.L. Mangasarian. Second and higher order duality in non linear programming. J. Math.

Anal. Appl., 51:605–620, 1979.

[6] B. Mond. Second order duality in non-linear programming. Opsearch, 11:90–99, 1974.

[7] B. Mond and M.A. Hanson. Duality for variational problems. J. Math. Anal. Appl.,

18:355–364, 1965.

[8] B. Mond and I. Husain. Sufficient optimality criteria and duality for variational prob-

lem with generalized invexity. Journal of the Australian Mathematical Society (Ser B),

31:108–121, 1989.

[9] B. Mond and T. Weir. Generalized convexity and higher order duality. J. Math. Anal.

App., 46:169–174, 1974.

[10] J. Zhang and B. Mond. Duality for a nondifferentiable programming problem. Bulletin

Australian Mathematical society, 55:20–44, 1997.