Journal of Applied Mathematics & Bioinformatics, vol.3, no.2, 2013, 45-64
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2013
On Nondifferentiable Nonlinear Programming
I. Husain 1 and Santosh K. Srivastava 2
Abstract
In this paper, we obtain optimality conditions for a class of nondifferentiable
nonlinear programming problems with equality and inequality constraints in
which
the objective contains the square root of a positive semidefinite quadratic
function and is ,therefore, not differentiable. Using Karush-Kuhn-Tucker
optimality, Mond-Weir dual to this problem is constructed and various duality
results are validated under suitable generalized invexity hypotheses. A mixed type
dual to the problem is also formulated and duality results are similarly derived.
Mathematics Subject Classification: 90C30, 90C11, 90C20, 90C26
Keywords: Nondifferentiable Nonlinear Programming, Generalized Invexity,
Mixed type duality, Square root function
1
2
Department of Mathematics, Jaypee University of Engineering and Technology,
Guna (M.P), India, e-mail:ihusain11@yahoo.com
Department of Mathematics, Jaypee University of Engineering and Technology,
Guna (M.P), India
Article Info: Received : December 14, 2012. Revised : February 12, 2013
Published online : June 30, 2013
46
On Nondifferentiable Nonlinear Programming
1 Introduction
In mathematical programming, there are a large number of researchers who
have discussed duality for a problem involving the square root of a positive
semidefinite. Mond studied Wolfe type duality while Chandra et. al. [1]
investigated Mond-Weir type duality for a class of nondifferentiable mathematical
programming problem containing square root term. Subsequently several research
papers have appeared in the literature, namely Chandra and Husain [2] and
references exist there. The popularity of this kind of problem seems to originate
from the fact that, even though the objective functions and or constraint functions
are non-smooth, a simple representation for the dual problem may be found. The
nonsmooth mathematical programming deals with much more general kind of
functions by means of generalized subdifferential, or quasidifferentials. However,
the square root of a positive semidefinite quadratic form is one of the four cases
of a nondifferentiable function for which subdifferential can explicitly be written.
In this paper, we obtain optimality conditions for a class of nondifferentiable
nonlinear programming problem having square root term in the objective function
with equality and inequality constraints. It is to be remarked here, a constrained
optimization problem with equality and inequality constraints
represents a
majority of real life problems, and hence it is important. For this class of problem,
Mond-Weir duality is investigated using generalized invexity assumptions. A
mixed type dual problem to this problem is also constructed to obtain various
duality results.
2 Statement of the Problem and Related Pre-requisites
We consider the following nondifferentiable nonlinear programming
problems:
I. Husain and Santosh K. Srivastava
(EP):
47
Minimize f ( x ) + ( xT B x )
1/ 2
subject to
g ( x ) ≤ 0,
(1)
h ( x ) = 0,
(2)
Where
(i) f : R n → R, g : R n → R m and h : R n → R p are continuously differentiable
functions.
(ii) B is an n × n symmetric positive semi-definite matrix.
We recall the following definitions of generalized invexity which will be used to
derive various duality results.
Definitions 2.1: (i) A function φ : R n → R is said to be quasi-invex with respect
to a vector function η = η ( x,u ) , if
φ ( x ) ≤ φ ( u ) ⇒ η T ( x,u ) ∇ φ ( u ) ≤ 0 .
(ii) A function φ is said to be pseudo-invex with respect to a vector function
η = η ( x,u ) , if
η T ( x,u ) ∇ φ ( u ) ≥ 0 ⇒ φ ( x ) ≥ φ ( u ) .
(iii) φ is said to be the strictly pseudoinvex with respect to η if x ≠ u,
η T ( x,u ) ∇ φ ( u ) ≥ 0 ⇒ φ ( x ) > φ ( u )
Equivalently, if
φ ( x ) ≤ φ ( u ) ⇒ η T ( x,u ) ∇ φ ( u ) < 0.
We shall make use of the generalized Schwartz inequality [3]
(x
T
B ω ) ≤ ( xT B x ) 1 / 2 ( ω T B ω ) 1 / 2
The equality holds if, for λ ≥ 0 , B x = λ B ω . The function φ ( x ) = ( xT B x ) ,
1/2
48
On Nondifferentiable Nonlinear Programming
being convex and everywhere finite, has a subdifferential in the sense of convex
analysis.
The subdifferential of
∂ ( xT B x )
1/2
=
(x
T
B x)
1/2
is given by
{Bw x Bw = ( x B x )
T
T
1/2
}
, where w ∈ R n , and wT B w ≤ 1 .
We also require the Mangasarian-Fromovitz constraint qualification which is
described as the following:
Let x ∈ Ω be the set of feasible solution of the problem (EP), that is,
Ω=
{x ∈ R
n
}
g ( x ) ≤ 0, h ( x )= 0 , and by A ( x ) , the set of inequality active
=
A( x )
constraint indices, that is,
j , g j ( x ) 0} ,
{=
where x ∈ Ω .
We say the Mangasarian-Fromovitz constraint qualification holds at x ∈ Ω when
the
equality
constraint
gradients ∇h1 ( x ) , ∇h2 ( x ) ,..., ∇hp ( x ) are
linearly
0 and
independent and there exist a vector d ∈ R n such that ∇h ( x ) d =
∇g j ( x ) d < 0, for all j ∈ A ( x ) .
3 Optimality Conditions
In this section, optimality conditions for the problem (EP) are obtained.
Theorem 3.1 (Fritz John Necessary Optimality Condition) If x is an optimal
solution of (EP), then there exist r ∈ R, y ∈ R m , z ∈ R p and w ∈ R n such that
I. Husain and Santosh K. Srivastava
49
r∇f ( x ) + B ω + ∇yT g ( x ) + ∇z T h ( x ) = 0,
yT g ( x ) = 0,
xT B w=( xT B x ) ,
1/2
ω T B ω ≤ 1,
( r, y ) ≥ 0
( r , y, z ) ≠ 0.
Proof: The problem (EP) may be written as
Minimize φ=
( x ) f ( x ) +ψ ( x ) ,
Subject to
− g ( x ) ∈ R+m ,
h ( x ) = 0.
Where R+m is the non-negative orthant of R m and the nondifferentiable convex
function ψ = ( xT B x ) .
1/2
A Fritz John theorem [4] shown that the necessary condition for a minimum of
(EP) at x are the existence of Lagrange multipliers r ∈ R, y ∈ R m , z ∈ R p and
ω∈ R n such that
0 ∈ r ∂ φ ( x ) + r ∂ψ ( x ) + ∂ yT g ( x ) + ∂z T h ( x )
This implies
0 ∈ r {∇f ( x )} + r{B=
ω ω ∈ Rn , x T B w
(x
T
B x ) , ω T B ω ≤ 1} + {∇yT g ( x ) + ∇z T h ( x )}
1/2
which implies
r ( ∇f ( x ) + B ω ) + ∇ yT g ( x ) + ∇ z T h ( x ) = 0,
yT g ( x ) = 0,
xT B w=( xT B x ) ,
1/2
ω T B ω ≤ 1,
( r , y ) ≥ 0,
( r , y, z ) ≠ 0.
50
On Nondifferentiable Nonlinear Programming
Thus the theorem follows.
Karush-Kuhn-Tucker type optimality conditions can be deduced from the above
Fritz John optimality condition if Mangasarian-Fromovitz Constraint Qualification
holds at x ∈ Ω.
The following theorem gives the Karush-Kuhn-Tucker types optimality
conditions.
Theorem 3.2 (K-K-T optimality condition) If x is an optimal solution of (EP)
and satisfies Mangasarian-Fromovitz constraint qualification, then there exist
r ∈ R, y ∈ R m , z ∈ R p and ω ∈ R n such that
∇f ( x ) + B ω + ∇yT g ( x ) + ∇z T h ( x ) = 0,
yT g ( x ) = 0,
xT B w=( xT B x ) ,
1/2
ω T B ω ≤ 1,
y ≥ 0.
4 Mond-Weir type Duality
In this section, we present the Mond-Weir type dual to (EP). Using
Karush-Kuhn-Tucker necessary optimality conditions, Wolfe type dual to the
following problem was formulated in [5].
Problem: (P)
Minimize
f ( x ) + ( xT B x )
1/ 2
Subject to
g ( x) ≤ 0 .
Dual (WD):
Maximize f ( u ) + yT g ( u ) − u T ( ∇yT g ( u ) + ∇f ( u ) )
I. Husain and Santosh K. Srivastava
51
subject to
∇f ( u ) + B ω + ∇y T g ( u ) = 0 ,
ω T B ω ≤ 1,
y ≥ 0.
The problem (WD) is a dual to (P) assuming that f and g are convex:
Further, Chandra et. al. [1] in order to weaken the convexity requirements in [5]
for duality to hold, formulated the following Mond-Weir type dual to the problem
(P).
(M-WD): Maximize f ( u ) + u T B ω
subject to
∇f ( u ) + B ω + ∇y T g ( u ) = 0 ,
yT g ( u ) ≥ 0,
ω T B ω ≤ 1,
y ≥ 0.
and established duality results assuming that f (.) + (.) B ω is pseudoconvex for
T
all ω ∈ R n and yT g (.) is quasiconvex. Later Mond and Smart [6] generalized
the results by Zhang and Mond [7] and Chandra et. al. [1] to invexity conditions.
Here, we propose the following Mond-Weir type dual to the problem (EP) to
study duality:
(M-WED): Maximize f ( u ) + u T B ω
Subject to
∇f ( u ) + B ω + ∇yT g ( u ) + ∇z T h ( u ) = 0,
(3)
ω T B ω ≤ 1,
(4)
yT g ( u ) ≥ 0,
(5)
z T h ( u ) ≥ 0,
(6)
y ≥ 0.
(7)
52
On Nondifferentiable Nonlinear Programming
( u, y, ω )
Theorem 4.1 (Weak Duality) Let x be feasible for (EP) and
feasible for (M-WED). If for all feasible ( x, u , y, ω ) , f (.) + (.) B ω
T
is
pseudoinvex, yT g ( .) and zT h ( .) are quasi-invex with respect to the same η ,
then
infimum (EP) ≥ supremum (M-WED)
Proof: Since x is feasible for (EP) and ( u , y, ω ) is feasible for (M-WED), we
have
yT g ( x ) ≤ yT g ( u )
(8)
zT h ( x ) ≤ zT h ( u )
(9)
By quasi-invexity of yT g (.) and zT h ( .) with respect to the same η , (8) and (9)
respectively yield
η T ( x , u ) ∇y T g ( u ) ≤ 0
(10)
η T ( x , u ) ∇z T h ( u ) ≤ 0
(11)
Combining (10) and (11), we have
η T ( x,u ) ( ∇yT g ( u ) + ∇z T h ( u ) ) ≤ 0
which because of (3) gives
η T ( x , u ) ( ∇f ( x ) + B ω ) ≥ 0
By pseudoinvexity of f ( .) + ( .) B ω , this implies
T
f ( x ) + xT B ω ≥ f ( u ) + u T B ω
Since ω T B ω ≤ 1, by the generalized Schwartz inequality, it follows that
f ( x ) + ( xT B x )
1/2
≥ f ( u ) + uT B ω
giving
infimum (EP) ≥ supremum (M-WED).
I. Husain and Santosh K. Srivastava
53
Theorem 4.2 (Strong Duality) If x is an optimal solution of (EP) and MFCQ
holds at x , then there exist y ∈ R m , z ∈ R p and ω ∈ R n such that ( x , y, ω , z ) is
feasible for (M-WED) and the corresponding values of (EP) and (M-WED) are
equal. If, also f (.) + (.) B ω is pseudoinvex for all ω ∈ R n , yT g ( .) and zT h ( .)
T
are quasi-invex with respect to the same η . Then
( x , y, ω , z )
is an optimal
solution of (M-WED).
Proof: Since x is an optimal solution of (EP) and MFCQ holds at x , then from
Theorem 3.2, then there exist y ∈ R m , z ∈ R p and ω ∈ R n such that
∇f ( x ) + B ω + ∇yT g ( x ) + ∇z T h ( x ) = 0,
yT g ( x ) = 0,
xT Bω =( xT B x ) ,
1/2
ω T B ω ≤ 1,
y ≥ 0.
Since x is feasible for (EP) and (M-WED), h ( x ) = 0, which implies
z T h ( x ) = 0,
∇f ( x ) + B ω + ∇yT g ( x ) + ∇z T h ( x ) = 0,
yT g ( x ) = 0,
ω T B ω ≤ 1,
y ≥ 0.
and
If
f ( .) + ( .) B ω
T
is pseudoinvex for all ω ∈ R n , yT g (.) and
z T h (.) are
quasi-invex with respect to the same η . Then from Theorem 4.1, ( x , y, ω , z ) is an
optimal solution for (M-WED).
The following is another dual to the following (EP):
(M-WED1): Maximize
subject to
f ( u ) + uT B ω
54
On Nondifferentiable Nonlinear Programming
∇f ( u ) + B ω + ∇yT g ( u ) + ∇z T h ( u ) = 0,
yi gi ( u ) ≥ 0, i =
1, 2,..., m
z T h ( u ) ≥ 0,
ω T B ω ≤ 1,
y ≥ 0.
This is a dual of (EP) if
f ( .) + ( .) B ω is pseudo-invex and each yi gi ,
T
i = 1, 2,..., m is quasi-invex and zT h ( u ) is quasi-invex with respect to the same η .
It is remarked here that if gi ( .) is quasi-invex with respect to same η , yi ≥ 0,
then yi gi is quasi-invex with respect to η . The problem (M-WED1) is a dual to
(EP) if f (.) + (.) B ω is pseudoinvex and each gi ,i = 1, 2 ,..m is quasi-invex and
T
zT h ( .) is quasi-invex with respect to the same η .
Theorem 4.3 (Strict Converse duality) Assume that f (.) + (.) B ω for all
T
ω ∈ R n is strictly pseudoinvex, yT g (.) is quasi-invex and z T h (.) is quasi-invex
with respect to the same η . Assume also that (EP) has an optimal solution x
which satisfies Mangasarian-Fromovitz constraint qualification. If ( u , y, z , ω ) is
an optimal solution. Then u is an optimal solution of (EP) with u = x .
Proof: We assume that u ≠ x and exhibit a contradiction. Since x is an optimal
solution of (EP), it implies from Theorem 4.2 that there exists y ∈ R m , z ∈ R p and
ω ∈ R n such that
( u , y, z , ω )
( x , y, z , ω )
is an optimal solution of (M-WED1). Since
is also an optimal solution of (M-WED1), it follows that
f ( x ) + xT Bω =
f (u ) + u B ω
This, by strict pseudoinvexity of f (.) + (.) B ω yields
T
I. Husain and Santosh K. Srivastava
55
η T ( x , u ) ( ∇f ( u ) + B ω ) < 0
(12)
From the constraints of (EP) and (M-WED), we have
yT g ( x ) ≤ yT g ( u ) ,
zT h ( x ) ≤ zT h ( u ) ,
which by quasi-invexity of yT g (.) and z T h (.) with respect to the same η , yield
η T ( x , u ) ∇y T g ( u ) ≤ 0
(13)
η T ( x , u ) ∇z T h ( u ) ≤ 0
(14)
From (12), (13) and (14), we have
η T ( ∇f ( x ) + B ω ) + ∇ yT g ( u ) + ∇ z T h ( u )  < 0,
contradicting the feasibility of ( u , y, z , ω ) for (M-WED).
Hence
u = x.
Theorem 4.4 (Converse duality) Let
( x , y , z , ω ) be
optimal to (M-WED) at
which
( A1 ) : the matrix
∇ 2 ( f ( x ) + y T g ( x ) + z T h ( x ) ) , is positive or negative definite
and
( A 2 ) : the vectors
∇ y T g ( x ) and ∇ z T h ( x ) are linearly independent.
If, for all feasible
( x, u , y , z , ω ) ,
f (.) + (.) B ω is pseudoinvex, yT g ( .) and
T
z T h (.) are quasi-invex with respect to the same η , then x is an optimal solution
of (EP).
Proof: By Theorem 3.1, there exist τ ∈ R, θ ∈ R n , α ∈ R, β ∈ R, η ∈ R m and
γ ∈ R such that
56
On Nondifferentiable Nonlinear Programming
τ ( ∇f + B ω ) + θ T ( ∇ 2 f ( u ) + ∇ 2 y T g ( u ) + ∇ 2 z T h ( u ) )
+ α ∇y T g + β ∇z T h ( u ) =
0,
θ ∇g ( u ) + α g + η =
0,
(15)
(16)
Implies
θ T ∇y T g ( u ) =
0,
(17)
θ T ∇z T h ( u ) =
0,
(18)
0,
τ ( xT B ) + θ B − 2 γ B ω =
(19)
α yT g = 0,
(20)
β zT g = 0,
(21)
γ (1 − ω B ω ) =
0,
T
(22)
η T y = 0,
(23)
(τ , α , β , γ ,η ) ≥ 0,
(24)
(τ , α , β , r ,η ,θ ) ≠ 0.
(25)
Using (3) in (15), we have
τ ( ∇y T g ( u ) + ∇z T h ( u ) ) + θ ∇ 2 ( f ( u ) + y T g ( u ) + z T h ( u ) )
0,
+ α ∇y T g ( u ) + β ∇ 2 ( z T h ( u ) ) =
0,
( α − τ ) ∇y T g ( u ) + ( β − τ ) ∇ 2 z h ( u ) + θ T ∇ 2 ( f ( u ) + y T g ( u ) + z T h ( u ) ) =
⇒ (α − τ ) θ T ∇y T g ( u ) + ( β − τ ) θ T ∇ 2 z T h ( u ) + θ T ∇ 2 ( f ( u ) + y T g ( u ) + z T h ( u ) ) =
0,
Which because of (17) and (18), yields
θ T ∇ 2 ( f ( u ) + yT g ( u ) + zT h ( u ) ) =
0,
By ( A1 ) , it follows that θ = 0.
Then (15) implies
0.
(α − τ ) ∇ y T g ( u ) + ( β − τ ) ∇ z T h ( u ) =
I. Husain and Santosh K. Srivastava
57
0,
By the linear independent of ∇ y T g ( x ) and ∇ z T h ( x ) , this gives (α − τ ) =
0.
and ( β − τ ) =
Let=
θ 0, τ > 0 . Then =
α 0,=
β 0. Consequently (16) implies η = 0.
The relation (19) and (22) together imply that γ = 0. This leading to a
contradiction to (25). Hence
τ > 0, α > 0 and β > 0. Also=
θ 0, τ > 0 and (19) give
xT B =
2γ
τ
B ω. Hence x T B ω = ( x T B x )
1/2
(ω
T
Bω) .
1/2
If γ > 0, then (22) give ω T B ω = 1 and so x T B ω = ( x T B x ) .
1/2
If γ = 0, (19) gives B x = 0. so we still get
(x
T
Bω) =( xT B x ) .
1/2
(26)
Thus, in earlier case, we obtain
(x
T
Bω) =( xT B x ) .
1/2
Therefore, from (26), we have
f ( x ) + ( xT B x )
1/2
=
f ( x ) + ( x T B ω ).
The equality of objective values follows.
If, for all feasible
( x , u, y, z , ω ) ,
f (.) + (.) B ω is pseudoinvex, yT g (.) and
T
z T h (.) is quasi-invex with respect to the same η , then by Theorem 4.1 , it implies
x is an optimal solution of (EP).
5 Mixed type duality
=
Let M
2,..., m} , L {1, 2,...l} , Iα
{1,=
⊆ M, α =
0,1, 2,..., r
58
On Nondifferentiable Nonlinear Programming
r
with Iα =
Iβ φ, α ≠ β
Jα ∩ J β= φ , (α ≠ β ) and
and
 Iα = M
and Jα ⊆ L, α =
0,1, 2,3..., r with
α =0
r
 Jα = L .
α
=0
In relation to (EP), consider the problem.
Mix (ED): Maximize f ( u ) + u T B ω +
∑ y g (u ) + ∑ z h (u )
i
i∈I 0
i
j
j∈J 0
j
subject to
∇f ( u ) + B ω + ∇ y T g ( u ) + ∇z T h ( u ) = 0
(27)
∑ y g ( u ) ≥ 0,
α=
1, 2,..., r.
(28)
∑ z h ( u ) ≥ 0,
α=
1, 2,..., r.
(29)
i∈Iα
j∈Jα
i
i
j
j
ωT B ω ≤ 1
(30)
(31)
y≥ 0
Theorem 5.1 (Weak Duality) Let x be feasible and
(ED). If for all feasible
( u, y, z, ω ) feasible
( x, u, y, z, ω ) , f (.) + (.)T B ω + ∑ yi gi (.) + ∑ z j h j (.)
i∈I 0
pseudoinvex,
∑ y g (.), α = 1, 2,..., r
i∈Iα
i
i
is
quasi-invex
for
is
j∈J 0
∑ z h (.),
and
j∈Jα
j
j
α = 1, 2,..., r is quasi-invex with respect to the same η , then
f ( x ) + ( xT B x )
1/2
≥ f (u ) + u T B ω +
∑ y g (u ) + ∑ z h (u )
i∈I 0
i
i
j∈J 0
j
j
That is,
infimum (EP) ≥ supremum Mix (ED).
Proof: Since x is a feasible for (ED) and
we have
( u, y, z , ω )
is feasible for Mix (ED),
I. Husain and Santosh K. Srivastava
∑ y g ( x ) ≤ ∑ y g (u )
i
i∈Iα
i
i∈Iα
By quasi-invexity of
i
59
, α=
1, 2...r
i
∑ y g (.) , α = 1, 2,..., r ,
i∈Iα
i
i
this implies


η T ( x, u ) ∇  ∑ yi gi ( u )  ≤ 0, α =
1, 2,..., r
∈
i
I
 α



η T ( x, u ) ∇  ∑ yi gi ( u )  ≤ 0, α =
1, 2,..., r
∈
−
i
M
I


T
f (.) + (.) B ω + ∑ yi (.) gi (.) + ∑ z j (.) h j (.) is pseudo-invex.
(32)
0
i∈I 0
Also
j∈J 0
1, 2,...r .
∑ z h ( x ) ≤ ∑ z h (u ) , α =
j
j∈Jα
j
j∈Jα
j
j
By quasi-invexity of ∑ z j h j (.) , α = 1, 2,...r
j∈Jα

This gives η T ( x, u ) ∇ 

η T ( x, u ) ∇ (
∑
j∈L − J 0
∑ z h (u )
j∈Iα
j
j

1, 2,..., r
 ≤ 0, α =

1, 2,..., r
z j h j ( u )) ≤ 0, α =
(33)
From the (27), it follows that
η T ( x, u ) [∇ f ( u ) + B ω + ∇∑ yi gi ( u ) + ∇ ∑ z j h j ( u )] ≥ 0
i∈I 0
The pseudoinvexity of f (.) + (.) B ω +
j∈J 0
∑ y g (u ) + ∑ z h (u ) ,
i∈I 0
i
i
f ( x ) + ( xT B ω ) ≥ f ( u ) + u T B ω +
j∈J 0
j
this yield
j
∑ y g (u ) + ∑ z h (u )
i∈I 0
i
i
j∈J 0
j
j
Since ω T B ω ≤ 1, by the generalized Schwartz inequality, this implies
f ( x ) + ( xT B x ) 1/2 ≥ f ( u ) + u T B ω +
∑ y g (u ) + ∑ z h (u )
i∈I 0
i
i
implying,
infimum (EP) ≥ supremum Mix (ED).
j∈J 0
j
j
60
On Nondifferentiable Nonlinear Programming
Theorem 5.2 (Strong Duality) If x is an optimal solution of (EP) and
Mangasarian-Fromovitz Constraint Qualification is satisfied at x , then there exist
y ∈ R m , z ∈ R l and ω ∈ R n such that ( x, u , z , ω ) is feasible for Mix (ED) and the
corresponding
value
of
(EP)
and
Mix(ED)
are
equal.
f (.) + (.) B ω + ∑ yi gi (.) + ∑ z j h j (.) is pseudoinvex for all ω ∈ R n ,
i∈I 0
and
j∈J 0
If,
also,
∑ y g ( .)
i∈Iα
i
i
∑ z h (.) are quasi-invex with respect to the same η , then ( x, y, z, ω )
j∈Jα
j
j
is
optimal for Mix (ED).
Proof: Since x is optimal solution of (EP) and MFCQ is satisfied at x , then
from Theorem 3.2, there exist y ∈ R m , z ∈ R l and ω ∈ R n such that
f ( x ) + B ω + ∇ yT g ( x ) + ∇z T h ( x ) = 0,
yT g ( x ) ≥ 0,
ω T B ω ≤ 1,
x Bω = ( x T B x ) ,
1/2
y ≥ 0.
The relation yT g ( x ) ≥ 0, and z T h ( x ) ≥ 0, are obvious. From the above it implies
that
( x , y, z , ω )
is feasible and the corresponding value of (EP) and (Mix ED)
are equal. If f ( .) + ( .) Bω + ∑ yi gi ( .) +
T
i∈I 0
∑ z h ( .)
j∈J 0
j
j
is quasi-invex, yT g (.) and
z T h (.) are quasi-invex with respect to the same η , then from Theorem 5.1,
( x, y , z , ω )
must be an optimal solution of (Mix ED).
We now consider some special cases of (Mix ED).
If=
I 0 φ=
, Iα M
for some α ∈{1, 2,..., r} and =
Jα φ=
, α 0,1, 2,..., r. Then
(Mix ED) be (WD) and (WD) is dual to (EP) if f (.) + (.) B ω + yT g (.) is
T
pseudoinvex with respect to the same η .
I. Husain and Santosh K. Srivastava
61
If =
Jα φ=
, α 0,1, 2,..., r. Then (Mix ED) become the Mixed dual to the problem
(P) considered by Zhang and Mond [7] and generalized dual to (P) if
f ( .) + ( .) B ω + y T g ( .)
T
is
pseudoinvex
∑ y g (.) , α = 1, 2,..., r
and
i∈Iα
i
is
i
quasi-invex with respect to the same η .
Theorem5.3 (Converse duality) Let ( x , y,z,ω ) be an optimal solution of Mix
(ED) at which
( A1 ) : the
matrix ∇ 2 ( f ( x ) + yT g ( x ) + z T h ( x ) ) is positive or negative definite
and
( A2 ):

the vectors  ∑ ∇yi gi ( x ) ,
i∈Iα


1, 2,..., r  are
∑ ∇ z h ( x ), α =

j∈Jα
j
j
independent. If f (.) + (.) B ω is pseudoinvex,
T
linearly

∑ y g ( .) and ∑ z h (.)
i∈Iα
i
i
j∈Jα
j
j
are
quasi-invex with respect to the same η . Then x is an optimal solution of (EP).
Proof:
(
τ (∇f ( x ) + B ω + ∑ ∇ yiT gi ( x ) + ∑ ∇z j h j ( x )) + θ T ∇ 2 ( f ( x ) + yT g ( x ) + z T h ( x ) )
i∈I 0
j∈J 0
r
r




+ ∑ pα  ∑ ∇yi gi ( x )  + ∑ qα  ∑ ∇z j h j ( x )  =
0
∈
=
∈
α=
α
i
I
j
J
1
1
 α

 α

T
τ B x +θ B − 2γ B ω =
0
)
(34)
(35)
τ gi ( x ) + θ ∇g i ( x ) +η =0, i ∈ Iα , α =
1, 2,..., r
(36)
τ h j ( x ) + θ T ∇h j ( x )= 0, j ∈ Jα , α= 1, 2,..., r
(37)
pα ( ∑ yi gi (=
x )) 0,=
α 1, 2,..., r
(38)
α 1, 2,..., r
qα ( ∑ z j h j (=
x )) 0,=
(39)
η
T y= 0
(40)
γ (1 − ω T B ω ) =
0
(41)
T
i∈Iα
j∈Jα
62
On Nondifferentiable Nonlinear Programming
(τ , p1 , p2 ,..., pr , γ ,η , q1 , q2 ,..., qr ) ≥ 0
(τ , p1 , p2 ,..., pr , γ ,θ ,η , q1 , q2 ,..., qr ) ≠ 0
(42)
(43)
From (36), it follows that
θT
∑ ∇y g ( x ) =0,
i
i∈Iα
α =1, 2,..., r
i
(44)
From (37), it follows that
θT
0, α =
1, 2,..., r
∑ ∇Z h ( x ) =
J
j∈ jα
(45)
j
Using equality constraint of the problem Mix(ED) in (34), we have
r
r
( pα − τ ) ( ∑ ∇y g ( x )) + ∑ ( qα − τ )( ∑ ∇z h ( x ))
∑
α
α
=
1
i
i∈Iα
+ θ T ∇2
i
=
1
j
j∈Jα
j
0
( f ( x ) + y g ( x ) + z h ( x )) =
T
T
(46)
Multiplying (46) by θ , and then using (44) and (45), we have
θ T ∇ 2 ( f ( x ) + yT g ( x ) + z T h ( x ) )θ =
0.
By ( A1 ) , from this we obtain
θ = 0.
(47)
Then (46) implies
r
∑ ( pα − τ ) ( ∑ ∇yi gi ( x )) +
1
α=
Since
the
i∈Iα
vector
( qα − τ ) ( ∑ ∇z h ( x ))
∑
α
{∑ ∇yi gi ( x ),
i∈Iα
r
=
1
j∈Jα
j
1, 2,..., r} are
∑ ∇z h ( x ) , α =
j∈Jα
j
j
(48)
j
linearly
independent, (48) yields
pα − τ= 0, qα − τ= 0, α= 1, 2,..., r
If τ = 0, (49) implies pα= 0= qα , α= 1, 2,..., r .
(49)
From (36) implies η = 0 and from (35) together with (41), we have γ = 0.
Hence (τ , θ , p1 , p2 ,..., pr , γ ,η , q1 , q2 ,..., qr ) = 0 contradicting to (43), hence τ > 0.
Consequently
pα > 0 and qα > 0, α =
1, 2,..., r.
Multiplying (36) by yi , i ∈ I 0 and using (40), we have
I. Husain and Santosh K. Srivastava
63
τ yi gi (=
x ) 0, i ∈ I 0
(50)
Multiplying (37) by z j , j ∈ I 0 , we have
τ z j h j (=
x ) 0, j ∈ J 0
Then from τ > 0, (47) and (48) implies that
(51)
yi gi ( x=
) 0, i ∈ I 0
(52)
z j h j (=
x ) 0, j ∈ J 0
(53)
Also =
θ 0,τ > 0 and (35) implies
Bx=
2r
τ
Bω
(54)
Hence
xT Bω = ( xT B x)
1/2
(ω
T
Bω)
1/2
(55)
If γ > 0, then (41) implies ω T B ω = 1.
xT Bω =( xT B x )
1/2
consequently, (55) yields
If γ = 0, then (35), yields B x = 0. So we obtain
(x
T
B x ) 1/2 = x T B ω .
Thus in either case, we obtain x T B ω = ( x T B x ) .
1/2
Therefore from (49) and (50), with x T B ω = ( x T B x ) , we have
1/2
f ( x ) + ( xT B x )
1/2
=f ( x ) + x T B ω +
∑ y g (x) + ∑ z h (x) .
i∈ I 0
i
i
j∈ J 0
j
j
If the hypotheses of Theorem 5.1 hold then x is an optimal solution of (EP).
References
[1] S. Chandra, B.D. Craven and B. Mond, Generalized concavity and duality
with a square root term, Optimization, 16, (1985), 654-662.
[2] S. Chandra and I. Husain, Symmetric dual nondifferentiable programs,
64
On Nondifferentiable Nonlinear Programming
Bulletin of the Australian Mathematical Society, 24, (1981), 295-307.
[3] F. Riesz and B. Sz.-Nagy, Functional analysis, (translated from the 2nd
French edition by Leo F. Boron), Fredrick Ungar Publishing Co., New York,
1955.
[4] B.D. Craven and B. Mond, Lagrangean conditions for quasidifferentiable
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Research Report No. 7, (1976), Survey of Mathematical Programming,
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[5] B. Mond, A class of nondifferentiable mathematical programming problems,
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