generalized symmetric duality for multiobjective variational … · 2017-02-15 · results of [2]....

Journal of Mathematical Analysis and Applications 234, 147-164 (1999) Article ID jmaa.1999.6344, available online at http://www.ideaIibrary.com on IBEki@

Generalized Symmetric Duality for Multiobjective Variational Problems with Invexity*

Do Sang Kim and Won Jung Lee

Department of Applied Mathematics, Puhyong National UniversiQ, Pusan, 608-737, Republic of Korea

Submitted by Koichi Mizukami

Received March 25. 1998

We formulate a pair of multiobjective nonlinear generalized symmetric dual variational problems involving vector-valued functions which unify the Wolfe and Mond-Weir models. For a special case, our problems become the symmetric dual pair of D. S. Kim and W. J. Lee (1998, J . Math. Anal. Appl. 218, 34-48). Under invexity assumptions, we establish weak, strong, and converse duality theorems for our variational problems by using the concept of efficiency. Also, we give the static case of our multiobjective generalized symmetric duality results. Several known results are obtained as special cases. Q 1999 Academic Press

1. INTRODUCTION

Symmetric duality in nonlinear programming was introduced by Dorn [3] by defining a symmetric dual program for quadratic programs. Subse- quently, Dantzig, Eisenberg, and Cottle [ZI first formulated a pair of symmetric dual nonlinear programs in which the dual of the dual equals the prime, and established weak and strong duality for these problems concerning convex and concave functions. Mond and Weir [16] gave a different pair of symmetric dual nonlinear programming problems in which the pseudo-convexity and pseudo-concavity assumptions reduced to the convexity and concavity ones, and obtained weak and strong duality for these problems. Weir and Mond [Zll established two distinct pairs of multiple objective symmetric dual programs. Under additional assumptions the multiobjective programs are shown to be self-dual.

*Research supported by Korea Research Foundation, 1998, Project 1998-015-D00039.

147 0022-247)3/99 $30.00

Copyright 0 1999 by Academic Press All rights of reproduction in any form reserved.

148 KIM AND LEE

On the other hand, Mond and Hanson [141 extended the symmetric duality results to variational problems, giving continuous analogues of the results of [2]. Since the invexity conditions on functions were first defined by Hanson [5] as a generalization of convexity, many authors [l , 11, 13, 15, 18, 191 have extended the concepts of invexity and generalized invexity to those of the continuous versions of invexity and generalized invexity functions. Smart and Mond [191 extended the symmetric duality results to variational problems by using the continuous version of invexity.

Recently, Kim and Lee [6] presented a pair of symmetric dual variational problems in the spirit of Mond and Weir [16], different from the one formulated by Smart and Mond [19], using the continuous version of pseudo-invexity which is a generalization of the continuous version of invexity. Very recently, Kim, Lee, and Lee [8] extended Kim and Lee's symmetric dual results [61 to the multiobjective symmetric dual variational problems under pseudo-invexity assumptions.

Kim and Lee [9] formulated a pair of generalized symmetric dual variational problems. Weak, strong, and converse duality theorems are established under invexity assumptions for these problems. Also, we give the static case of our generalized symmetric duality results. Several known results [2, 12, 14, 191 are obtained as special cases.

In this paper, we formulate a pair of multiobjective generalized symmetric dual variational problems involving vector-valued functions which unify Wolfe and Mond-Weir models. For example, our model unifies [8] and [ 101. Under invexity assumptions, we establish weak, strong, and converse duality theorems for our variational problems by using the concept of efficiency. Also, we give the static case of our multiobjective generalized symmetric duality results. Several known results [2, 4, 7, 9, 10, 12, 14, 17, 191 are obtained as special cases.

2. NOTATIONS AND STATEMENT OF THE PROBLEMS

The following conventions for vectors in R" will be used:

x < y - x i < Y i , x sy - x i s y i , x < y e x i s y i , i = 1 , 2 ,..., n but x Z y ;

x 6 y is the negation of x I y .

i = 1 , 2 ,..., n; i = 1 , 2 ,..., n;

Let [ u , b] be a real interval and let f : [ u , b] X R" X R" X R"' X R"' + R P , N = { l , 2 ,..., n}, M = { l , 2 ,..., m}, A c N , I c M , N \ A = B , and M\I = J . Note that A , B, I , or J can be empty. We rearrange x, y as

GENERALIZED SYMMETRIC DUALITY 149

x = (x,, x,) and y = ( y I , y J ) , respectively. Consider the vector-valued function f ( t , x, x’, y , y ’ ) , where t E [a , b] , x and y are functions of t with x ( t ) E R” and y ( t ) E R”, and x’ and y’ denote the derivatives of x and y , respectively, with respect to t. Assume that f has continuous fourth-order partial derivatives with respect to x,, x,, x;, xh, y r , y J , y j , and y;. f , denotes the p x n matrix of first partial derivatives. If A E R P , then (ATf),, ( A T f ) x , , (ATf), , , and (ATf),b denote the gradient vectors of (AT f ) ( t , x, x’, y , y ’ ) at 6, x, x’, y , y ’ ) with respect to x,, x,, x;, and xb, respectively.

Similarly, ( ATf),, , ( ATf),,, ( ATf),;, and ( A‘f),; denote the gradient vectors of (AT f ) ( t , x, x’, y , y ’ ) at ( t , x, x’, y , y ’ ) with respect to y r , y J , y i , and y;, respectively.

We consider the problem of finding functions x: [a , bl + R” and y : [a , b] + R”, with ( x ’ ( t ) , y ’ ( t ) ) piecewise smooth on [a , b], to solve the following pair of our multiobjective generalized symmetric dual variational problems:

(GMSP) Minimize

d (A’ f ) , , ( t , x , x ’ , y , y ’ ) - - ( h T f ) , ; ( t , x , x ‘ , y , y ’ ) dt

subject to

(GMSD) Maximize

150 KIM AND LEE

subject to

.(a) =xo> u ( b ) =XI> .(a) = y o , u ( b ) = y 1 ,

( A T f ) , ( t 9 u 9 u’ I u 9 u ’) - - ( AT&( t , u , u’ , u , u ’) 2 0, d dt (3)

1 u; [ (A’f)&, u , u’ 9 u , u ’ ) - - ( A’f),h( t , u , u’, u , u ’ ) 5 0, (4) d dt

A > 0, A’e = 1,

where (11, (2), (3), and (4) may fail to hold at corners of ( x ’ ( t ) , y’(t>> and (u’(t>, u’(t)) , respectively, but must be satisfied for unique right-hand and left-hand limits, A E RP and e = (1,1,. . . , 1IT E RP.

When A = N and I = M , our pair of programs (GMSP) and (GMSD) is reduced to that of Kim and Lee [lo]: Minimize

d dt ( A T f ) , ( t , x , x ’ , y , y ‘ ) - - ( A ’ f ) , , ( t , x , x ’ , y , y ’ )

subject to

4.1 =TI> x(b) = X I , y ( a ) = y o , y ( b ) = y 1 ,

5 0 , d

( A T f ) , ( t J J ’ , Y J ’ ) - - ( A T f ) y ’ ( t , X , X ’ , y , y ’ ) dt

A > 0, ATe = I , Maximize

d - [ uT( ( A T f ) , ( t , u , u ’ , u , u ’ ) - -( dt A T f ) , , ( t , u , u ’ , u , u ’ )

subject to

.(a) =xo, u ( b ) =XI, u(.) = y o , u ( b ) = y , ,

( A T f ) , ( t , u , u ’ , u , u ’ ) - - ( A T f ) , , ( t , u , u ’ , u , u ’ ) 2 0 , d dt

A > 0, ATe = 1 .


When A = 4 and I = 4, our pair of programs (GMSP) and (GMSD) is reduced to that of Kim, Lee, and Lee [8]: Minimize

subject to

subject to

.(a) = x o , u ( b ) =x,, U(.) = y o , u ( b ) = y , , d

(A’f),(t, U , u’, U , u’ ) - -(A’f) , ,( t , u,u’, u , u ’ ) 2 0,

d dt

1 ~ ‘ [ ( A f ) ~ ( t , u , u ’ ,u , u’) - - ( A T f > , , ( t , dt u , u ’ , U , u ’ ) 5 0,

h > 0, ATe = 1 .

Remark. Observe that if p = 1 in (GMSP) and (GMSD), then (GMSP) and (GMSD) become (PI and (D) given by Kim and Lee [9], respectively.

3. GENERALIZED SYMMETRIC DUALITY

We consider the following multiobjective variational problem:

(MP) Minimize J b f ( t , x, x’) dt = ( f f ’ d t , J b f z d t , . . . , f f p d t ) a a

152 KIM AND LEE

subject to

x ( a ) = ff,

g ( t , x , x ' ) 5 0 ,

x(b) = p , t E [ u , b ] ,

where f : [a , bl X R" X R" +. RP, g: [a , b] x R" x R" +. R".

E [ a , bl} be the set of feasible solutions for (MP).

optimal) solution of (MP) if, for all x in K ,

Let K = {x E C([u, bl, R")lx(a) = a , x(b) = p, g( t , x ( t ) , x ' ( t ) ) 5 0, t

DEFINITION 1. A point x* in K is said to be an efficient (Pareto

/ d s ( t , x, x') dt g S b f ( t , x*, x*') dt a

(i.e., there exists no other x E K such that /,"f(t, x, x') dt I /abf(t, x*, x*') dt).

Now we define the invexity as follows.

DEFINITION 2. The functional /,"f is said to be invex in x and x' if, for each y : [a , b] + Rm, with y' piecewise smooth, there exists a function 7: [a , b] X R" X R" X R" X R" +. R" such that, V i = 1,. . . , p ,

for all x: [a , bl +. R", u: [a , bl +. R" with ( x ' ( t ) , u'(t)) piecewise smooth on [a , b].

DEFINITION 3. The functional -/,"f is said to be invex in y and y' if, for each x: [a , bl +. R", with x' piecewise smooth, there exists a function 6: [ u , bl X R" X R" X R" X R" + R" such that, V i = 1,. . . , p ,

for all u : [ a , b] +. R", y : [a , b ] + R"' with (u ' ( t ) , y ' ( t ) ) piecewise smooth on [a , bl.


In the following, we will write ~ ( x , u ) for 70, x, x’, u, u ’ ) and ( ( u , y ) for 5 0 , u, u ’ , y , y’) .

Remark. If f is independent of t , Definitions 2 and 3 reduce to the definition of invexity of the static case in [19].

Now we establish the generalized symmetric duality theorems for (GMSP) and (GMSD).

THEOREM 1 (Weak Duality). Let (x, y , A) be feasible for (GMSP), and (u , u , A) be feasible for (GMSD). Assume that /,“f is invex in x and x’, and - /,”f is invex in y and y’ , with ~ ( x , u ) + u ( t ) 2 0 and ((v, y ) + y ( t ) 2 0

for all t E [ a , bl (exceptperhaps at corners of (x ’ ( t ) , y’( t ) ) or (u’(t), u’(t)). Then

11 1 d - - ( ATf )y ; ( t , x, x’ , y , y ’) e dt dt

Prooj Assume to the contrary that

11 1 d dt - - ( ATf )y ; ( t , x, x’, y , y ’) e dt

d dt - - ( A T f ) & ( t t u , u’ , u , u ’ )

154 KIM AND LEE

Then, since h > 0,


so that

/ ’ { ( A T f ) ( t , x, x’, u , u ’ ) - (A’ f ) ( t , x, x’, y , y ’ ) } dt

Subtracting (7) from (6 ) and rearranging gives

156 KIM AND LEE

In order to prove the strong and converse duality theorems, we use the following multiobjective necessary optimality theorem which was proposed by Valentine [ZOI .

PROPOSITION 1. If (x* , y*, A*) is an eficient solution of (GMSP), then there exist a E RP, P : [ a , b] + R", y E R, and 6 E R P such that

satisfies

d d 2 dt dt2

d d 2 dt dt

H* - -HT + -H? = 0,

H,* - -H: + T H ~ , = 0,

throughout [ a , b] (except at corners of (x*'(t) , y*'(t)) where (8) and (9) hold for unique right-hand and left-hand limits). a , P(t), y , and 6 cannot be simultaneously zero at any t E [ a , b] , and P is continuous except perhaps at corners of (x*'(t), y*'(t)).

In the following theorems and proofs, h*'f* represents A*'f(t, x*, x* ' , y*, y*') and partial derivatives are similarly denoted.


THEOREM 2 (Strong Duality). Let (x*, y*, A*) be an ejjicient solution for (GMSP): j i x h = A* in (GMSD). Suppose that the system

only has the solution p ( t ) = 0 for all t E [a , b ] , and the set

is linearly independent. Then (x*, y*, A*) is feasible for (GMSD). I f , in addition, the inuexity conditions of Theorem 1 are satisfied, then

(x*, y*, A*) is an eficient solution for (GMSD), and the optimal values of (GMSP) and (GMSD) are equal.

Prooj By Proposition 1, if (x*, y*, A*) is an efficient solution of (GMSP), then there exist a E RP, p: [ a , b ] + R", y E R, and S E RP such that

satisfies

d d 2 dt dt2

d d 2 dt dt

H* - -HT + -H? = 0 ,

H,* - -H: + T H ~ , = 0 ,

d - vy:'( fy*, - - f ; ) dt - S = 0, (14)

158 KIM AND LEE

STA* = 0, (17)

( a , P , y J ) 20, ( 18) throughout [a , b ] (except at corners of (x*’(t) , y*’(t)) where (12) and (13) hold for unique right-hand and left-hand limits). a , P( t ) , y , and S cannot be simultaneously zero at any t E [ u , b] , and p is continuous except perhaps at corners of (x*’(t), y*’(t)).

Equation (1 2) now becomes

1

= 0. Equation (13) gives

d YYT - P J dt


d d dt

A*Tfy*Ixt - z A * T f $ x , - A *T fYiX * A*Tf:xt - -A*Tfy*,nxr - A*Tf$x

= 0. (20)

Multiplying (19) by

and then using (14), (15), (161, and (17) gives

Thus by the assumption (lo),

160 KIM AND LEE

From ( 191, we have

By the assumption (111,

a = yA*

If y = 0 in (221, then a = 0. By (21) and (141, p = 0 and 6 = 0. This contradicts (18). Hence, y > 0 and a > 0.

From (20, we have

y 2 0. (23)

By (20) and (22), we have

From (241, we have

Equation (15) with (21) gives

Hence, from (24) and (251, ( x * , y * , A*) is feasible for (GMSD).

(x*, y* , A*), namely, /:f(t, x*, x* ' , y*, y* ' ) dt. From (25) and (26), (GMSP) and (GMSD) have equal objective values at


Clearly, (x*, y*, A*) is an efficient solution for (GMSD); otherwise, there would exist a feasible solution (u*, u*, A*) such that

11 1 d - [ u : ~ ( A*TfxA(t ,u*,u*f ,u*,u*f) - -A*Tfxk(t,u*,u*f,u*,u*f) dt e dt

2 ~ ~ { f ( t , x * , x * f , y * , y * ’ )

11 1 d dt

--A*Tfxk(t,x*,x*f,y*,y*f) e dt.

- [ Y?T( A*Tfy,(t, x*, x*’, Y*, Y*’)

which also contradicts weak duality.

Remark. If f does not explicitly depend on y f , the system reduces to p(t)T(A*Tf,*,)p(t) = 0, which has only a zero solution iff A*Tf;y is negative definite for all t E [ a , bl.

A converse duality theorem may be stated; the proof would be analo- gous to that of Theorem 2.

I

162 KIM AND LEE

THEOREM 3 (Converse Duality). Let (x*, y*, A*) be an efSicient solution for (GMSD): Fix A = A* in (GMSP). Suppose that the system

T * T * ] d 2 + &T( - P W A fxY) P(t> = 0

only has the solution p ( t ) = 0 for all t E [ a , b] , and the set

d . f i* - - fiP i = 1 , 2 , . . . , p is linearly independent. x B dt

Then (x*, y*, A*) is feasible for (GMSP). I f , in addition, the invexity conditions of Theorem 1 are satisfied, then

(x*, y*, A*) is an efSicient solution for (GMSP), and the optimal values of (GMSP) and (GMSD) are equal.

4. THE STATIC CASE OF GENERALIZED SYMMETRIC DUALITY

If the time dependency of programs (GMSP) and (GMSD) is removed and f is considered to have domain R" X Rm, we obtain the symmetric dual pair given by [71:

(SP) Minimize

f ( x 9 Y 1 - [ YT ( hTf I,,< x 9 Y ,] e

( A T f ) y ( X * Y ) s 0,

Y T ( A T f ) , , ( X , Y > 2 09

subject to

A > 0, ATe = 1,

(SD) Maximize

f ( u , v > - [.:(ATf)x,<,,,,]e

( A T f ) x ( u , u > 2 0,

subject to

u i ( A T f ) , B ( u , u ) s 0,

A > 0, ATe = I .


The following duality theorems can be proved along the lines of Theo- rems 1, 2, and 3.

THEOREM 4 (Weak Duality). Let ( x , y , A) be feasible for (SP) and (u , u , A) be feasible for (SD). Assume that f is inuex in x , and -f is inuex in y , with ~ ( x , u ) + u 2 0 and ((v, y ) + y 2 0. Then

THEOREM 5 (Strong Duality). Let (x*, y*, A*) be an ejjicient solution for (SP): Fix h = A* in (SD). Suppose that h*'f;, is negatiue definite, and the set

is linearly independent. Then (x*, y*, A*) is feasible for (SD). I f , in addition, the inuexity conditions of Theorem 4 are satisfied, then

(x*, y*, A*) is an ejjicient solution for (SD), and the optimal values of (SP) and (SD) are equal.

Let (x*, y*, A*) be an efficient solution for (SD): jk h = A* in (SP). Suppose that h*Tf:x is positive definite, and the set

THEOREM 6 (Converse Duality).

is linearly independent. Then (x*, y*, A*) is feasible for (SP). I f , in addition, the inuexity conditions of Theorem 4 are satisfied, then

(x*, y*, A*) is an eficient solution for (SP), and the optimal values of (SP) and (SD) are equal.

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