WOLFE TYPE HIGHER ORDER SYMMETRIC DUALITY UNDER INVEXITY†

Journal of Applied Mathematics & Informatics.
2014.
Jan,
32(1_2):
153-159

- Received : March 06, 2013
- Accepted : July 15, 2013
- Published : January 28, 2014

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In this paper, we introduce a pair of higher-order symmetric dual models/problems. Weak, strong and converse duality theorems for this pair are established under the assumption of higher-order invexity. Moreover, self duality theorem is also discussed.
AMS Mathematics Subject Classification : 90C46, 49N15, 90C30.
F
-convexity. Yang et al.
[19]
obtained the duality results for multiobjective higher-order symmetric duality under invexity assumptions.
Recently, Ahmad
[1]
discussed higher-order duality in nondifferentiable Multiobjective Programming. In this paper, we introduce a pair of higher-order symmetric dual models/problems. Weak, strong and converse duality theorems for this pair are established under the assumption of higher-order generalized invexity. Moreover, self duality theorem is also discussed.
Definition 2.1.
A function
ϕ
:
R^{n}
↦
R
is said to be higher-order invexity at
u
∈
R^{n}
with respect to
η
:
R^{n}
×
R^{n}
↦
R^{n}
and
h
:
R^{n}
×
R^{n}
↦
R
, if for all (
x, p
) ∈
R^{n}
×
R^{n}
,
Let ∇
_{xy}f
denote the
n
×
m
matrix and ∇
_{xy}f
denote the
m
×
n
matrix of second order derivative. Also ∇
_{xx}f
and ∇
_{yy}f
denote the
n
×
n
and
m
×
m
symmetric Hessian matrices with respect to
x
and
y
, respectively.
Primal Problem (WHP):
Minimize
L
(
x, y, p
) =
f
(
x, y
) +
h
(
x, y, p
) −
p^{T}
∇
_{p}h
(
x, y, p
) −
y^{T}
[∇
_{y}f
(
x, y
) + ∇
_{p}h
(
x, y, p
)] subject to
Dual Problem (WHD):
Maximize
M
(
u, v, r
) =
f
(
u, v
) +
g
(
u, v, r
) −
r^{T}
∇
_{r}g
(
u, v, r
) −
u^{T}
[∇
_{u}f
(
u, v
) + ∇
_{r}g
(
u, v, r
)] subject to
where
f
:
R^{n}
×
R^{m}
↦
R, g
:
R^{n}
×
R^{m}
×
R^{n}
↦
R
and
h
:
R^{n}
×
R^{m}
×
R^{m}
↦
R
are twice differentiable functions.
Theorem 3.1
(Weak Duality).
Let
(
x, y, p
)
and
(
u, v, r
)
be feasible solutions for primal and dual problem, respectively. Suppose that
(i) f(., v) is higher-order invexity at u with respect to η
_{1}
and g
(
u, v, r
),
(ii)
−[
f
(
x, .
)]
is higher-order invexity at y with respect to η
_{2}
and
−
h
(
x, y, p
),
(iii) η
_{1}
(
x, u
) +
u
+
r
⪴ 0,
(iv) η
_{2}
(
v, y
) +
y
+
p
⪴ 0.
Then
Proof
. It is given (
x, y, p
) is feasible for (WHP) and (
u, v, r
) is feasible for (WHD), therefore by the hypothesis (iii) and the dual constraints (4), we get
or
which on using the dual constraint (5) implies that
Now by the higher order invexity of
f
(
., v
) at
v
with respect to
η
_{1}
and
g
(
u, v, r
), we get
Similarly, hypothesis (iv) along with primal constraints (1) and (3) yields
Therefore, by higher-order invexity of −[
f
(
x, .
)] at y with respect to
η
_{2}
and −
h
(
x, y, p
), we obtain
Adding inequalities (9) and (11), we get
or
Thus the result holds.
Theorem 3.2
(Strong Duality).
Let
be a local optimal solution of (WHP). Assume that
(i)
is negative definite,
(ii)
(iii)
(iv)
Then (I)
is feasible for (WHD) and
(II)
Also, if the hypotheses of Theorem
(3.1)
hold for all feasible solutions of (WHP) and (WHD), then
are global optimal solutions of (WHP) and (WHD), respectively
.
Proof
. Since
is a local optimal solution of (WHP), there exist
α, δ
∈
R, β, ξ
∈
R^{m}
and
μ
,∈
R^{n}
such that the following Fritz-John conditions
[13
,
17]
are satisfied at
Premultiplying equation (14) by
which along with equations (15), (16) yields
Now from equations (1), (19) and (20), we obtain
Using hypothesis (i) in inequality (21), we have
This, together with hypothesis (ii) and equation (14), yields
Now, we claim that
α
≠ 0. Indeed if
α
= 0, then equations (22) and (23) give
Therefore equations (12), (13) and (23), imply
μ
= 0 and
ξ
= 0. Hence (
α, β, δ, μ, ξ
), a contradiction to (19). Thus
Using equations (13), (22) and (24), we have
Therefore hypothesis (iii) implies
Moreover, equation (12) along with (22), (26) and hypothesis (iv) yields
or
Also using equation (17)
Thus
satisfies the constraints (4)-(6), that is, it is a feasible solution for the dual problem (WHD). Now using equations (19), (26), (27) and hypothesis(iv), we get
i.e
Finally, by Theorem (3.1),
are global optimal solutions of the respective problems.
Theorem 3.3
(Strong Duality).
Let
be a local optimal solution of (WHP). Assume that
(i)
is positive definite,
(ii)
(iii)
(iv)
Then (I)
is feasible for (WHP) and
(II)
Also, if the hypotheses of Theorem
(3.1)
are satisfied for all feasible solutions of (WHP) and (WHD), then
are global optimal solutions of (WHD) and (WHP), respectively
.
Proof
. Follows on the line of Theorem (3.2).
f, g
and
h
. If
f
:
R^{n}
×
R^{m}
→
R
and
g
:
R^{n}
×
R^{m}
×
R^{n}
→
R
are skew symmetric, i.e
as shown below. By recasting the dual problem (WHD) as a minimization problem, we have Minimize
M
(
u, v, r
) = −{
f
(
u, v
)+
g
(
u, v, r
)−
r^{T}
∇
_{r}g
(
u, v, r
)−
u^{T}
[∇
_{u}f
(
u, v
) + ∇
_{r}g
(
u, v, r
)]} subject to
Now as
f
and
g
are skew symmetric, i.e
then the above problem rewritten as :
Minimize
M
(
u, v, r
) =
f
(
v, u
) +
g
(
v, u, r
) −
r^{T}
∇
_{r}g
(
v, u, r
) −
u^{T}
[∇
_{u}f
(
v, u
) + ∇
_{r}g
(
v, u, r
)] subject to
Which is identical to primal problem, i.e., the objective and the constraint functions are identical. Thus, the problem (WHP) is self-dual.
It is obvious that (
x, y, p
) is feasible for (WHP), then (
y, x, p
) is feasible for (WHD) and vice versa.
Khushboo Verma is currently pursuing her Ph.D. degree in Mathematics at Indian Institute of Technology Roorkee, India. She received her Master of Science in Mathematics from Banaras Hindu University, Varanasi, India.
Department of Mathematics, Indian Institute of Technology, Roorkee-247 667, India.
e-mail: 1986khushi@gmail.com
T.R. Gulati is a Professor in Department of Mathematics, Indian Institute of Technology Roorkee, India. His research interests are in the areas of single and multiobjective mathe-matical programming and generalized convexity. He has published more than 75 research papers in journal of intermational repute.
Department of Mathematics, Indian Institute of Technology, Roorkee-247 667, India.
e-mail: trgmaiitr@rediffmail.com

1. Introduction

The concept of symmetric duality was first introduced by Dorn
[8]
for quadratic programming. Later, in non linear programming this concept was significantly developed by Dantzig et al.
[7]
, Mond
[15]
and Bazarra and Goode
[5]
. Mangasarian
[14]
introduced the concept of second and higher-order duality for nonlinear programming problems, which motivated several authors to work on second order dualty
[3
,
4
,
9
,
10
,
11]
. Subsequently, higher-order symmetric duality for nonlinear problems has been studied in
[1
,
2
,
12
,
18]
. The study of second and higher-order duality is significant due to the computational advantage over the first-order duality as it provides tighter bounds for the value of the objective function when approximations are used. Mond and Zhang
[16]
discussed the duality results for various higher-order dual problems under invexity assumptions. For a pair of nondifferentiable programs, Chen
[6]
also discussed the duality theorems under higher-order generalized
2. Preliminaries

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3. Higher order symmetric duality

We consider the following pair of higher order symmetric duals and establish weak, strong and converse duality theorems.
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4. Self Duality

A mathematical problem is said to be self dual if it formally identical with its dual, that is, the dual can be rewritten in the form of the primal. In general, (WHP) is not self-dual without some added restrictions on
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BIO

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Citing 'WOLFE TYPE HIGHER ORDER SYMMETRIC DUALITY UNDER INVEXITY†
'

@article{ E1MCA9_2014_v32n1_2_153}
,title={WOLFE TYPE HIGHER ORDER SYMMETRIC DUALITY UNDER INVEXITY†}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2014.153}, DOI={10.14317/jami.2014.153}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={VERMA, KHUSHBOO
and
GULATI, T. R.}
, year={2014}
, month={Jan}