Weak Vector Saddle Point Theorem under Vector ρ,η – Convexity

Weak Vector Saddle Point Theorem under Vector , – Convexity

Weak Vector Saddle Point Theorem under Vector , – Convexity

Assistant Professor

DBS Engineering College, Kavali-524202, Andhra Pradesh

In this paper we derive sufficient optimality condition and weak vector saddle point theorem and also duality results for non smooth multiobjective fractional programming problem have been proved.

Key words:

weak vector saddle point, non smooth multiobjective fractional programming, vector – convexity-invexity for locally Lipschitz theorem.

Xu described saddle point optimality criteria and established duality theorems in terms of generalized Lagrangian functions. Jeya Kumar defined – invexity for non- smooth scalar-valued functions, studied duality theorem for non-smooth optimization problems and gave relationships between Saddle Points & optimality. But no serious attempt is made in utilizing the recent developed concept like Saddle Point Theorem under v- –convexity. Hence in this paper an attempt is made to fill the gap by developing vector valued functions under v- –convexity which is generalization of the concept of V convexity and (, ) convexity and establish sufficient optimality condition and weak vector saddle point theorems and also duality results for non- smooth multiobjective fractional programming problems are obtained.

The following are the definitions of Vector, v- –convexity -invexity for locally Lipschitz functions:

fi : Rn R gi

and hj

: Rn R be locally Lipschitz functions for i = 1, 2, p,

and j = 1, 2, .m, respectively

(i) fi

gi

, i 1, 2, …… p

is V- – -convex with respect to functions and

: Rn x Rn Rn if there exists i : Rn x Rn R+ \{ 0 } and R, i = 1, 2, p such that for

i

i

i

any x, u Rn and any

fi (u) ,

g

i

fi (x)

fi (u) 2

i (x, u) g

> (x, u)

(x) g (u)

(x, u) .

i i

i i

i i

(ii) hj, j = 1, 2,m is V- – -convex with respect to functions and

j

j

j

j

: Rn x Rn Rn if there exist : Rn x Rn R+ \ { 0 } and R , j = 1, 2, .m

Such that for any x, u Rn and any j dhj(u).

2

j (x, u) [hj(x) hj(u)] > j (x, u) + j (x, u)

Let u x is said to be a weak minimum of (FP) if there exists no xX such that

fi (x)

gi (x)

fi (u) , i = 1, 2, . P

gi (u)

Consider the following non-smooth multi objective fractional programming problems.

(FP) :

Min Max fi (x) ,

xX

1i p gi (x)

subject to hj(x) < 0, j = 1, 2, m,

where

fi : Rn R, i 1, 2, … p gi

and hj

: Rn R, i= 1, 2.p and hj

: Rn R, j = 1,

2, m are locally lipschitz function.

For the problem (FP), consider the dual problem (FD) :

g

g

i

i

(FD) max

fi (u)

(u)

p

p

fi (u) m

subject to O

g

g

i

i

i

i 1

(u)

j

j 1

hj (u)

where e = (1, 1, 1)t Rp

j hj (u) > 0, j = 1, 2, , m

i > 0, i = 1, 2, p

j > 0, j = 1, 2, m,

In this section we show that the generalized karush-kuhn-tucker conditions are sufficient for a weak minimum of (FP)

Theorem: – Let (u, , ) Rn x Rp x Rm satisfy the generalized karush-kuhn-Tucker conditions as follows.

p

p

fi (u) m

g

g

i

i

O i

i 1

(u)

j

j 1

hj (u)

If fi

gi

hj (u) < 0, j hj (u) = 0 , j = 1, 2, m,

i > 0, i = 1, 2, , p

t e > 0

j > 0 , j = 1, 2, .m

is V- – -convex and hj is v–convex with respect to the same functions

and and

p

p

i pi

i 1

m

m

• j j 0 , then u is weak minimum of (FP).

j 1

p

p

fi (u) m

g

g

i

i

i

i 1

(u)

j

j 1

hj (u) , there exist

fi (u) and j hj (u)

such that

g

g

i

i

i (u)

p m

ii j j 0

(4.1)

i 1 j 1

Suppose that u is not a weak minimum of (FP). Then there exists x X such that

fi (x)

gi (x)

fi (u) , i= 1, 2, .p,

gi (u)

since

i (x, u) > 0 we have

i

i

(x, u)

fi (x)

i

i

(x, u)

fi (u) , i 1, 2,….p

gi (x) gi (u)

by the V- – -convex of

fi , for all i ,

gi

(x, u) (x, u) 2 0 for each

fi (u)

i i i

(u)

g

g

i

i

Hence, we have

p p

(x, u)

(x, u)

2 0

i i

i 1

i i

i 1

p m

Since i i j j 0

it follows from (1)

i 1

j 1

m

m

j 1

j j

m

m

(x, u) j j

j 1

(x, u)

2 0

j

j

j

j

Then, by the v – – con vexity of hj, we have

m

m

j j 1

(x, u)

h j (x) j

h (u) 0

m

m

since j hj (u) = 0, j = 1, 2, , m, we have j (x, u) j

j 1

hj (x) 0

which

contradicts the conditions

j (x, u) > 0, j > 0 and hj (x) < 0.

Thus u is week minimum of (FP).

Hence the proof.

m

m

Let x be a feasible for (FP) and (u, , ) a feasible for (FD), assume that

p

p

i i i 1

j j j 1

0. If fi

gi

is V- –

-convex and hj is v –

-convex with

respect to same functions and , then

fi (x)

gi (x)

fi (u)

gi (u)

From feasibility conditions and

j (x, u) > 0, we have

j (x,u) jhj(x) < j (x, u) j hj (u). Then, by th v – – convexity of hj, we have

2

2

for each j hj (u).

j j (x, u) + j j

Hence we have

(x, u)2 < 0,

m

m

j j 1

m

m

j (x, u) j j

j 1

(x, u)

< for each

j hj

(u).

p

p

fi (u) m

Since O

g

g

i

i

j

j 1

(u)

j

j 1

hj (u),

There exists

fi (u)

and

h

(u)

such that

g

g

i

i

i (u) j j

p

p

i i j 1

m

m

j j (x, u) 0

j 1

p m

Hence, from the assumption i i j j 0

i 1 j 1

We have,

p

p

ii j 1

p

p

(x, u) i

i 1

i (x, u)

2 0

from the V- – -convex of

fi , we have

gi

p

p

fi (x)

fi (u)

i (x, u) i g (x)

i g (u) > 0

i 1 i i

Since i (x, u) > 0, i > 0, +e = 1 we have

fi (x)

gi (x)

fi (u)

gi (u)

Let x be a weak minimum of (FP) at which constraint qualification is satisfied

then there exists Rp and Rm (x, , )

is feasible for (FD).

If fi

gi

is V- – -convex and hj is v – – convex with respect to same function

and , then

(x, , )

is a weak maximum of (FD)

Since x is weak minimum of (FP) and a constraint qualification is satisfied x , from the generalized Karush-Kuhn-Tucker theorem there exist

i R and R such that

i R and R such that

p m

j

p f (x) m

O i

h (x)

i

i 1

gi

(x)

j j

j 1

j hj (x) 0 , j = 1, 2, . m

i > 0, i = 1, 2, p

+e > 0

j > 0, j = 1, 2, .m

Since i > 0 , i = 1, 2, .p and + e > 0,

we can consider that i and j as

i

i ,

p

p

i i 1

j

j

p

p

j i 1

Then

(x, , )

is feasible for (FD).

Since x is feasible for (FP), it follows from weak duality that

fi (x)

gi (x)

fi (u)

gi (u)

for any feasible u for (FD). Hence

(x, , )

is a weak maximum of (FD).

In this section, we prove Weak Vector Saddle Point theorem for the non smooth multiobjective fractional program (FP) in which functions are locally lipschitz. For the problem (FP), a point (x, , ) is said to be a critical point if, x is a feasible point for (FP), and

p f (x) m

O i h (x)

i

i 1

gi

(x)

j j

j 1

j hj (x) 0 , j > 0, j = 1, 2, . m

i > 0, i = 1, 2, p,

te = 1

Note, that

p f

(x) m

i

i

p f

(x) m

i

h (x) =

i

h (x)

j j

j j

i 1

gi (x)

j j

i

i

j 1

i 1

gi

(x)

j j

j 1

Let L (x, ) =

fi (x)

+ h (x) e,

gi (x)

Where x Rm and Rm +. Then, a point (x, ) Rn x Rm+ is said to be a weak vector Saddle Point if when ever we introduce L (x, , ) it means that L (x, , ) has p

j

j

fi (x) t

components like

g

g

i (x)

hj (x) hj (x)e,

i = 1, 2, p, j = 1, 2,m

L (x, )

> L(x, )

> L(x, )

for all x Rn and Rm+

(x, , )

be a critical point of (FP) assume that

fi (x)

g (x)

• h

  j (x) e is V- – -convex with respect to function and and

  p

  p

  i

  i 0 .

  i

  Then

  i 1

  (x, ) is a weak vector Saddle Pont of (FP).

  (x, , )

  is a critical point for (FP), there exists

  p

  fi (x) m

  i g (x)

  j g j (x)

  i 1 i j 1

  such that

  p

  i 0

  p

  since

  i 0

  i1 i i1 i

  p p

  i (x, x) i (x, x)

  2 0

  j

  j

  i 1 i i 1 i

  Then, by the V- – invexity of

  fi (x) th gi (x)

  (x)e,

  p

  we have

  i (x, x)

  fi (x)

  g (x)

  fi (x)

  g (x)

  j hj (x)

  j hj (x)

  > 0 for any x

  Rn. Since i

  i 1

  i i i

  (x, x ) > 0, i 0 and t e 1

  j

  j

  fi (x) gi (x)

  h j (x) e

  fi (x)

  gi (x)

• j

  h j (x) e

  (2)

  for any x Rn ,that is L (x, ) L(x, ) , for any x Rn.

  Now, since j

  hj (x)

  < 0 for any Rm+ .

  j h j

  (x) j h j

  (x) 0, forany Rm

  fi (x)

  fi (x) p

  Thus,

  g (x)

• j h j (x) e g (x)

• j h j (x) eR

i i

and hence, L (x, ) L(x, ), for any Rm+.

Therefore,

(x, ) is a weak vector Saddle Point of (FP).

Theorem :- If there exists Rm such that (x, ) is a weak Vector Saddle Point, then

x is a weak minimum of (FP).

(x, ) is a weak Vector Saddle Point from left of 2nd Equation.

fi (x)

j j

j j

h (x) e

> fi (x) h

(x)e , for any Rm .

j j

j j

gi (x) gi (x)

+

+

Thus j h j (x) e

> jh j (x)e

for any Rm+ ,and hence we have

j h j (x) j h j (x)

, for any Rm+ (3)

Since j can be taken arbitrary large, hj(x)

< 0. Hence j

hj (x) 0 .

Let j = 0 in (3),

j hj (x)

> 0. Therefore, j

hj (x) 0

. Now, from the right

inequality of (2) equation and

j hj (x)

= 0, we have for any feasible x for (FP),

fi (x) >

gi (x)

fi (x)

gi (x)

Hence x is a weak minimum for (FP).

Hence the proof.

REFERENCES:

1. Xu, Z.K. (1988). Saddle point type optimality criteria for generalized fractional programming. J. Opt. Theory and Appl. 57, 189 196.

2. Jeya kumar, V. (1998). Equivalence saddle points and optima and duality for a class of non-convex problem. Journal of mathematical analysis and Application, 130: 334-343.

3. Varalakshmi, G and Reddy, P.R.S (2007). Multi-objective fractional minimax problem involving locally lipschitz functions vinvexty. International conference on statistical science, OR & IT, Tirupati; OR: 47.