Vol 6, No 12 (2015)

Available online throug

ISSN 2229 – 5046

SYMMETRIC DUALITY FOR MULTIOBJECTIVE OPTIMIZATION

UNDER CONVEXITY ASSUMPTION USING THE CONCEPT OF PROPER EFFICIENCY

ARUNA KUMAR TRIPATHY*

Department of Mathematics, Trident Academy of Technology,

F2/A, Chandaka Industrial Estate, Bhubaneswar, Orissa, India.

(Received On: 12-12-15; Revised & Accepted On: 30-12-15)

ABSTRACT

I

n this paper, a new class of symmetric duality for multiobjective optimization and their scalar parametric problems are formulated and the duality results are established under convexity assumption using the concept of proper efficiency. Also the duality relations with saddle point theory are presented.

Kew words: symmetric duality, proper efficiency, saddle point, convex function.

INTRODUCTION

The importance of convex function is well known in optimization theory. But for many mathematical model used in decision sciences, applied mathematics and engineering, the notion of convexity does no longer suffice. So it is possible to generalize the notion of convexity and to extend the validity of results to larger class of optimization problems. Consequently, various generalization of convex function has been introduced in the literature. The field of multiobjective programming, also known as vector programming has grown remarkable in different direction in the setting of optimality condition and duality theory since the 1980. It has been enriched by the application of various types of generalization of convex theory with and without differentiable assumption and in the frame work of continuous time programming, fractional programming, inverse vector optimization saddle point theory, symmetric duality, variational problem etc.

Symmetric duality in nonlinear programming problem was first introduced by Dorn [1] who defined a mathematical programming problem and it’s dual to be symmetric if the dual is the primal problems. Later Dantzing et al. [2] and Mond [3] formulated a pair of symmetric dual programs for scalar function f (x, y) that is convex in the first variable and that is concave in the second variable respectively.

Bazaars [1] and Dantzing et al. [2] studied on symmetric duality in nonlinear programming. Devi [4] studied symmetric duality for nonlinear programming problem involving 𝜂𝜂-bonvex functions. Dorn [5] formulated symmetric dual theorem for quadratic programs. Egudo established efficiency and generalized convex duality for multiobjective programs.

Geoffrion [6] gave the idea of proper efficiency and the theory of vector maximization. Gulati et al. [7] established second order symmetric duality with cone constraints where as Gulati and Geeta [8] established Mond-Weir type second order symmetric duality in multiobjective programming over cones. Gupta and Danger [9] duality for second order symmetric multiobjective programming with cone constraint. Kassem [10] established multiobjective nonlinear symmetric duality involving generalized pseudo convexity. Khurana [11] established symmetric duality in multiobjective programming involving generalized cone-invex functions. Kim et al. [12] established Multiobjective symmetric duality with cone constraint. Preda [13] studied on efficiency and duality for multiobjective programs. Suneja et al. [14] established multiobjective symmetric duality involving cones.

In this paper, we introduced a new class of symmetric duality for multiobjective optimization under convexity assumption using the concept of proper efficiency. Also the duality relations with saddle point theory are presented.

NOTATION AND DEFINITION

Let C1 and C2 denote closed convex cones with nonempty interiors in Rn and Rm respectively.

Let

C

_i∗(i=1,2,) be the polar of Ci ,i.e.

{

:

0

i

}

T

i

z

x

z

for

x

C

C

∗

=

≤

∀

∈

, where xT denote the transpose of x. Let f(x, y) be a vector valued function defined on an open set in R n+m.

Definition 1: A real-valued function

φ

is said to be convex if

)

(

)

(

)

(

)

(

x

φ

x

x

x

T

φ

x

φ

−

≥

−

∇

for all x,

x

∈

C

₁.

Definition 2: A real-valued function

f x y

( , )

=

(

f

₁

( , ),

x y

f

₂

( , ),. . . ,( , ))

x y

f

_k

x y

T is said to be convex in the first variable if each

f

_i

(.,

y

)

is convex for fixed

y

, i =1, 2, 3,……k.

Similarly

f

is said to be convex in second variable if

f

_i

(

x

,.)

is convex for fixed

x

,

i

=

1

,

2

,

3

,...

k

.

Primal (Pv):

Min

F x y

( , ) {

=

f

₁

( , )

x y

−

y

T

∇

_y

f

₁

( , ),. . . ,( , )

x y

f

_k

x y

−

y

T

∇

_{y k}

f

( , )}

x y

T

Subject to

(

x

,

y

)

∈

C

₁

×

C

₂

,

∇

_y

f

_i

(

x

,

y

)

∈

C

₂*

,

C

₁

⊂

R

m

,

C

₂

⊂

R

n

.

Dual (D):

Max

G x y

( , ) {

=

f

₁

( , )

x y

−

x

T

∇

_x

f

₁

( , ),. . . ,( , )

x y

f

_k

x y

−

x

T

∇

_{x k}

f

( , )}

x y

T

Subject to

(

,

)

,

(

,

)

,

1

,

2

.

* 1 2

1

n m

i

x

y

C

C

R

C

R

xf

C

C

y

x

∈

×

−∇

∈

⊂

⊂

Now we introduced the following scalar parametric problem:

Primal (

P

_λ): Min

λ

f

(

x

,

y

)

−

y

T

∇

_y

λ

f

(

x

,

y

)

,

Subject to

( , )

x y

∈

C

₁

×

C

₂

,

∇

_{y i}

f x y

( , )

∈

C

₂

*

,

C

₁

⊂

R

m

,

C

₂

⊂

R

n

,

λ

=

(

λ λ

₁

,

₂

,...,

λ

_k

)

T

∈

R

n

Dual

(

D

_λ

)

: Max

λ

G

(

x

,

y

)

Subject to

(

x

,

y

)

∈

C

₁

×

C

₂

,

−∇

_y

f

_i

(

x

,

y

)

∈

C

₁*

,

C

₁

⊂

R

m

,

C

₂

⊂

R

n

.

(

₁

,

₂

,...,

_k

)

T

R

n

λ

=

λ λ

λ

∈

We denote the set of feasible solution of P by S i.e

S

=

{(

x

,

y

)

∈

C

₁

×

C

₂

|

∇

_y

f

_i

(

x

,

y

)

∈

C

₂∗

Definition 3: A feasible point (

x

,

y

)

is said to be an efficient solution of (

P

_v) if

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

)

≥

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

)

)

,

(

)

,

(

)

,

(

)

,

(

x

y

y

f

x

y

f

x

y

y

f

x

y

f

_i

−

T

∇

_{y i}

=

_i

−

T

∇

_{y i}

⇒

for all i=1,2,3, ……k.

and

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

)

≤

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

)

)

,

(

)

,

(

)

,

(

)

,

(

x

y

y

f

x

y

f

x

y

y

f

x

y

f

_i

−

T

∇

_{y i}

=

_i

−

T

∇

_{y i}

Definition 4: A feasible point

(

x

,

y

)

is said to be properly efficient if it is efficient for

P

_v and if there exist scalars M1>0 and M2 >0, such that

1

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

(

)

,

(

)

,

(

(

M

y

x

f

y

y

x

f

y

x

f

y

y

x

f

y

x

f

y

y

x

f

y

x

f

y

y

x

f

j y T j j y T j i y T i y T i

_≤

∇

−

−

∇

−

∇

−

−

∇

−

for some j,

)

,

(

)

,

(

)

,

(

)

,

(

x

y

y

f

x

y

f

x

y

y

f

x

y

f

_j

−

T

∇

_{y j}

>

_j

−

T

∇

_{y j} and

)

,

(

)

,

(

)

,

(

)

,

(

x

y

y

f

x

y

f

x

y

y

f

x

y

f

_i

−

T

∇

_{y i}

<

_i

−

T

∇

_{y i} whenever

(

x

,

y

)

is a feasible for (

P

_v).

and ₂

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

)

,

(

(

)

,

(

)

,

(

(

M

y

x

f

y

y

x

f

y

x

f

y

y

x

f

y

x

f

y

y

x

f

y

x

f

y

y

x

f

j y T j j y T j i y T i y T i

_≤

∇

−

−

∇

−

∇

−

−

∇

−

for some j,

)

,

(

)

,

(

)

,

(

)

,

(

x

y

y

f

x

y

f

x

y

y

f

x

y

f

_j

−

T

∇

_{y j}

>

_j

−

T

∇

_{y j}

and

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

)

<

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

).

whenever

(

x

,

y

)

is feasible for

P

_v.

Lemma 3.1: If

( , )

x y

is a feasible solution of (

P

_λ) then

( , )

x y

is properly efficient solution of (

P

_v) where

k

i

=

1

,

2

,...

λ

Proof: Since

( , )

x y

is feasible solution of

P

_v then obviously it is efficient in

P

_v .Now we have to show that

( , )

x y

is properly efficient solution in

P

_v. If it is not properly efficient solution then

>

∇

−

−

∇

−

(

,

))

(

(

,

)

(

,

))

)

,

(

(

f

x

y

y

f

x

y

f

x

y

y

y

f

i

x

y

T i i y T i

[

( , )

T

( , )

( , )

T

( , )]

M

₁

f

_j

x y

−

y

∇

_{y j}

f

x y

>

f

_j

x y

−

y

∇

_{y j}

f

x y

Let us consider

(

) max(

/

)

,

1

1

M

k

_j

_i

i j

λ

λ

=

−

where

k

≥

2

for all j such that

)

,

(

)

,

(

(

)

,

(

)

,

(

x

y

y

f

x

y

f

x

y

y

f

x

y

f

y j

T j

j y T

j

−

∇

<

−

∇

.

Hence

(

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

))

−

(

f

_i

(

x

,

y

)

−

y

T

∇

_y

f

_i

(

x

,

y

))

>

(k-1) j

[(

_j

( , )

T _{y j}

( , ))

(

_j

( , )

T _{y j}

( , ))]

i

f

x y

y

f

x y

f

x y

y

f

x y

λ

λ

−

∇

−

−

∇

for all

j

≠

i

Multiplying both sides by

)

1

(

k

−

i

λ

then we get

[(

( ,

( , )

(

( , )

( , ))]

(

1

)

T T

i

i y i i y

f x y

y

f x y

f x y

y

f x y

k

λ

−

∇

−

−

∇

−

[(

f

( , )

x y

y

T

f

( , ))

x y

(

f

( , )

x y

y

T

f

( , ))]

x y

j

j

y j

j

y j

λ

>

−

∇

−

−

∇

for all

j

≠

1

.

By summing over

j

≠

i

,

_i

[(

f x y

_i

( , )

y

T

_{y i}

f x y

( , ))

(

f x y

_i

( , )

y

T

_{y i}

f x y

( , ))]

j i

λ

−

∇

−

−

∇

∑

≠

[(

( , )

T

( , ))

(

( , )

T

( , ))]

j

j

y j

j

y j

j i

f

x y

y

f

x y

f

x y

y

f

x y

λ

∑

≠

>

−

∇

−

−

∇

Lemma 3.2: Let

( , )

x y

is a is properly efficient solution of (

P

_v) where

λ

_i

=

1

,

2

,...

k

. If each

f

_iis convex in the

first variable and

−

f

_i is convex in second variable then

( , )

x y

is a feasible solution of

P

_λ.

Proof: Since

f

_iis convex in the first variable, from the properties of convexity, we obtain

(

( , )

( , ))

(

( , )

( , ))

₁

(

( , )

( , ))

i

f x y

y

f x y

f

x y

y

f

x y

M

f

x y

y

f

x y

i

T

i

T

T

i

y i

_j

j

y j

j

y j

j i

λ

−

∇

+

λ



_

−

∇

+

−

∇



_





∑

≠

( ( , )

f x y

y

f x y

( , ))

(

f

( , )

x y

y

f

( , ))

x y

(

f

( , )

x y

y

f

( , ))

x y

i

T

i

T

T

y i

j

y j

j

y j

j

j

j i

λ

λ





≥

−

∇

+

_

−

∇

−

−

∇

_





∑

≠

Since

λ

ij

≥

0

,

j

=

1

,

2

,

3

,...

k

and

1

1

=

∑

=

k

j i

j

λ

, from above inequation, we get

(f

_i

( , )x y y

T

_{y i}

f ( , ))x y M

₁

i

((f

_j

( , )x y y

T

_{y j}

f ( , ))x y

j

j i

λ

− ∇ +

_∑

− ∇

≠

( ( , )f x y y f ( , ))x y (f ( , )x y y f ( , )) (x y f ( , )x y y f ( , ))x y

i

T

i

T

T

y i

j

y j

j

y j

j

j

j i

λ λ  

≥ − ∇ + _ − ∇ − − ∇ _

 

∑

≠

summing over j yields after some rearrangement,

(1

₁

)(( ( , ) ( , )) (1

₁

)(( ( , ) ( , ))

1

1

M f x y y f x y M f x y y f x y

k

k

i

T

i

T

j

y j

j

y j

j

j

j

i

j

j

i

j

λ λ

+ − ∇ ≥ + − ∇

∑

∑

∑

∑

=

≠

=

=

).

,

(

)

,

(

))

,

(

)

,

(

(

))

,

(

)

,

(

(

y

x

F

y

x

F

y

x

f

y

y

x

f

y

x

f

y

y

x

f

y i

T i

i y T i

≥

⇒

∇

−

≥

∇

−

⇒

Similarly we can prove

F

(

x

,

y

)

≤

F

(

x

,

y

)

by taking

−

f

_i as convex in second variable.

Hence

(

x

,

y

)

is a saddle point of F.

Now

(

,

)

min

max

(

,

)

max

min

(

,

)

0 , 0 0

,

0

F

x

y

F

x

y

y

x

F

x y y

x≥ ≥

=

≥ ≥

=

This shows that the component x of a saddle point

(

x

,

y

)

solves the minimization problem

P

_λ.

Lemma 3.3: Let

(

x

,

y

)

be a saddle point of

λ

T

f

in

P

_λ , where λ >

_i

0,i=1 2 3, . .... ,k and 1 1 k

i j

λ = =

∑

.

If each

f

_i is convex in first variable and

−

f

_i is convex in second variable then it satisfies

( , ) ₁, ( , ) 0, 0

f x y C f x y x x

x i ∗ x i

∇ ∈ ∇ = ≥ and ∇_{y i}f x y( , )∈C₂∗,∇_{y i}f x y y( , ) =0,y≥0

Lemma 3.4: Suppose the feasible point

(

x

,

y

)

satisfies

( , ) ₁, ( , ) 0, 0

f x y C f x y x x

x i ∗ x i

∇ ∈ ∇ = ≥ and ∇_{y i}f x y( , )∈C₂∗,∇_{y i}f x y y( , ) =0,y≥0,i=1 2 3, , , ..., .k .

If

f

_i is convex in first variable and

−

f

_i is convex in second variable then it

(

x

,

y

)

is a saddle point of

λ

T

f

in

P

_λ

for all λ >

_i

0,i=1 2 3, . .... ,k and 1 1 k

i j

λ = =

Theorem 3.1 (Weak duality): Let

(

x

,

y

)

be feasible for

P

_v and

(

u

,

v

)

be feasible for (D). If each

f

_i is convex in

first variable and

−

f

_i is convex in second variable then

F x y

( , )

≥

G u v

( , ).

Proof: Since each

f

_i is convex in first variable and

−

f

_i is convex in second variable then

∑

∑

= =

∇

−

≥

−

k

i

k

i

i x i T i

i

i

f

x

v

f

u

v

x

u

f

u

v

1 1

)

,

(

)

(

))

,

(

)

,

(

(

λ

λ

(1)

and

∑

∑

= =

∇

−

≤

−

k

i

k

i

i y i T i

i

i

f

x

v

f

x

y

v

y

f

u

v

1 1

)

,

(

)

(

))

,

(

)

,

(

(

λ

λ

(2)

For all

1

0,

1, 2.3.... ,

1

k

i i

j

i

k and

λ

λ

=

>

=

∑

=

.

Multiplying-1 in inequality (2) and adding to inequality (1) we get

1 1 1 1

( ( , )

( , )

(

)

( , ) (

)

( , )

k k k k

T T

i i i x i i x i i y i

i i i i

f x y

f u v

x u

f u v

v

y

f x y

λ

λ

λ

λ

= = = =

−

∇

≥

−

∇

− −

∇

∑

∑

∑

∑

Since

(

x

−

u

)

∈

C

₁ and

(

v

−

y

)

∈

C

₂, therefore

F x y

( , )

≥

G u v

( , ).

Theorem 3.2 (Strong Duality): Let

(

x

,

y

)

be said to be properly efficient solution for

P

_v assume that each

f

_i is

convex in first variable and

−

f

_i is convex in second variable . Then

(

x

,

y

)

be said to be properly efficient solution for

(

D

)

and the objective values of

P

_v and (D) are equal.

Proof: Since

(

x

,

y

)

is properly efficient solution for

P

_v, Therefore from Lemma 3.2 and Lemma 2.3 it follows that

( , ) ₁ , ( , ) 0, 0

f x y C f x y x x

x i ∗ x i

∇ ∈ ∇ = ≥ and

( , ) ₂ , ( , ) 0, 0

f x y C f x y x y

y i ∗ y i

∇ ∈ ∇ = ≥ for i=1 2 3, , , ...,k

Thus ( , )x y ∈( )D

Now F x y( , )=λT( ( , )f x y − ∇_yf x y y( , ) =λT( ( , )f x y

and G x y( , )=λT( ( , )f x y − ∇_yf x y y( , ) =λT( ( , )f x y

Hence

F

(

x

,

y

)

=

G

(

x

,

y

)

.

CONCLUSION

In this paper, we present a new class of symmetric duality for multiobjective optimization and their scalar parametric problems and established the duality results under convexity assumption using the concept of proper efficiency. Also we discussed the duality relations with saddle point theory.

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Source of support: Nil, Conflict of interest: None Declared