Higher-Order Duality for Minimax Fractional Type Programming

Applied Mathematics, 2011, 2, 1387-1392 doi:10.4236/am.2011.211196 Published Online November 2011 (http://www.SciRP.org/journal/am)

Higher-Order Duality for Minimax Fractional Type Programming Involving Symmetric Matrices Caiyun Jin, Cao-Zong Cheng College of Applied Sciences, Beijing University of Technology, Beijing, China E-mail: {jincaiyun, czcheng}@bjut.edu.cn Received September 1, 2011; revised October 16, 2011; accepted October 23, 2011

Abstract Convexity and generalized convexity play important roles in optimization theory. With the development of programming problem, there has been a growing interest in the higher-order dual problem and a lot of related generalized convexities are given. In this paper, we give the convexity of ( F , ,  , d , b, )  vector-pseudo-

quasi-Type I and formulate a higher-order duality for minimax fractional type programming involving symmetric matrices, and give the weak, strong and strict converse duality theorems under the condition of higher-order ( F , ,  , d , b, )  vector-pseudoquasi-Type I. Keywords: Higher-Order ( F ,  ,  , d , b,  )  Vector-Pseudoquasi-Type I, Higher-Order Duality, Minimax

Fractional Type Programming, Positive Semidefinite Symmetric Matrix g ( x)  0.

1. Introduction

subject to

In this paper, we focus on the following nondifferentiable minimax fractional programming problem:

Under the optimality conditions of [1], Tanimoto [2] defined a first-order dual problem of ( P ) , which generalized the duality theorems for convex minimax programming problems considered by Weir [3] and relaxed the convexity assumptions in the sufficient optimality of [1]. Mishra and Rueda [4] introduced generalized second-order type I functions and considered the minimax programming problem ( P ) involving those functions and established second-order duality theorems for problem ( P ) . Husian, Anurag Jaysural and Ahmad [5] established two types of second-order dual models for problem ( P ) , which extends some previously known results on minimax programming. With the development of programming problem, there has been a growing interest in the higher-order dual problem. Mangasarian [6] first formulated a class of secondand higher-order dual problems for a nonlinear programming problem involving twice differentiable functions. In [7], Zhang considered the following nondifferentiable mathematical programming problem:

1

min sup xR n yY

f ( x, y )  ( xT Bx) 2 h( x, y )  ( x Cx) T

1 2

( P)

subject to g ( x)  0, x  R n , where Y is a compact subset of R l , f , h : R n  R l  R and g : R n  R m are continuously differentiable functions on R n  R l and R n , respectively, and 1

f ( x, y )  ( xT Bx) 2  0, 1

h( x, y )  ( xT Cx) 2 > 0, ( x, y )  R n  R l , B and C are two positive semidefinite n  n symmetric matrices. When B = C = 0 , (P) is a differentiable minimax fractional programming problem. The duality of programming problem involving symmetric matrix has been investigated widely. Schmitendorf [1] established necessary and sufficient optimality conditions for a particular case of the following problem ( P ) under convexity conditions. 1

min sup f ( x, y )  ( xT Bx) 2 yY

Copyright © 2011 SciRes.

( P )

1

Minimize f ( x)  ( xT Bx) 2 ( P ) subject to g ( x)  0, under higher-order invexity assumptions. Mishra and Rueda [8] generalized the results of Zhang [7] to higherAM

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1388

order type I functions. In [9], Ahmad, Husain and Sharma considered the nondifferentiable minimax programming problem ( P ) , and formulated a unified higher-order dual of (P ) , and established weak, strong and strict converse duality theorems under higher-order ( F ,  ,  , d ) -Type I assumptions. In [10], Jayswal and Stancu-Minasian formulated the weak, strong and strict converse duality of (P ) under generalized convexity of higher-order ( F ,  ,  , d ) Type I. For problem (P), H. C. Lai and K. Tanaka gave the necessary and sufficient conditions under the conditions of pseudo-convex, strictly pseudo-convex and quasi-convex [11]. In this paper, we will establish a higher-order dual of (P) and give the weak, strong and strict converse duality theorems under ( F ,  ,  , d , b,  )  vector-pseudoquasiType I assumptions. The convexity conditions in this paper generalized the convexity in [8], and hence, presents an answer of a question raised in [10].

2. Preliminaries

b( x, x )  f ( x, y )  xT   f ( x , y )  x T   w( x , y , p)  pT  p w( x , y , p)  < 0  F ( x, x , 1 ( x, x )( p w( x , y , p)   )) <  1d 2 ( x, x )

  e( x )  l ( x , p)  pT  p l ( x , p )   0  F ( x, x ,  2 ( x, x )( p l ( x , p)))    2 d 2 ( x, x ).

Definition 3: ( f , e) is said to be strictly higher-order ( F ,  ,  , d , b,  )  vector-pseudoquasi-Type I at x  X with respect to p  R n , if for all x  x  S and y  Y ( x) , b( x, x )  f ( x, y )  xT   f ( x , y )  x T   w( x , y , p)

Let R n be the n-dimensional Euclidean space, Rn be its nonegative orthant and X be an open subset of R n . Let S be the set of all feasible solutions of (P). Denote M = 1, 2, , m . For each ( x, y )  S  Y , we define

1  f ( x, z )  ( xT Bx) 2  = sup 1  zY T h( x, z )  ( x Cx) 2 



K ( x) = ( s, t , y )  N  Rs  R ls :1  s  n  1, t = (t1 , t2 , , ts )  Rs with s



i =1



ti = 1, y = ( y1 , y2 , , ys ) and yi  Y ( x), i = 1, 2, , s  Definition 1: A function F : X  X  R  R is said to be sublinear in its third argument, if x, x  X , 1) (subadditivity) a1 , a2  R n , n

F ( x, x ; a1  a2 )  F ( x, x ; a1 )  F ( x, x ; a2 );

<  1d 2 ( x, x )

Obviously, when  is subadditive function and satisfies a  0   (a)  0 , higher-order ( F ,  ,  , d , b,  ) Bu vector-pseudoquasi-Type I is the convexity condition of Theorem 3.1 in [10]. In the following section, we will use Lemma 1 and Lemma 2 which were given in [11]. Lemma 1: (Necessary Condition) If x is an optimal solution of problem (P) satisfying xT Bx > 0, xT Cx > 0 and g j ( x ), j  J ( x ) are linear independent, then there exist ( s , t  , y  )  K ( x ) , u  , v  R n and    Rm ,    R such that s

ti f ( x , yi )    (h( x , yi ))  Bu   Cv  i =1 m

(2.1)

   g j ( x ) = 0  j



j =1 1

1

f ( x , yi )  ( xT Bx ) 2    h( x , yi )    ( xT Cx ) 2  0,

2) (positive homogeneous)    R , a  R n ,

i = 1, 2, , s

F ( x, x ;  a) =  F ( x, x ; a ).

(2.2)

Let f : R  R  R ,  = ( 1 ,  2 )  R ,   R , 1 ,  2 : X  X  R \ {0} , b, d : X  X  R , Copyright © 2011 SciRes.

 F ( x, x , 1 ( x, x )( p w( x , y , p)   ))

 F ( x, x ,  2 ( x, x )( p l ( x , p)))    2 d 2 ( x, x ).

1  f ( x, y )  ( xT Bx) 2  Y ( x) =  y  Y : 1 T  2  h ( x , y ) ( x Cx ) 

l

 pT  p w( x , y , p)   0

 e( x )  l ( x , p )  pT  p l ( x , p )   0

J ( x) =  j  M : g j ( x) = 0

n

e : R n  R and  : R  R . Let w : R n  R l  R n  R , l : R n  R n  and k : R n  R n  R m be three differentiable functions. We assume that F is a sublinear functional throughout this paper. Definition 2: ( f , e) is said to be higher-order ( F ,  ,  , d , b,  )  vector-pseudoquasi-Type I at x  X with respect to p  R n , if for all x  S and y  Y ( x) ,

2

n

m

 j g j ( x ) = 0

(2.3)

i =1

AM

C. Y. JIN ET AL. s

ti = 1, ti  0, i = 1, 2, , s

(2.4)

j =1

u T Bu   1, vT Cv  1,  1  T  T  2 x Bu = ( x Bx ) ,   1  xT Cv = ( xT Bx ) 2 . 

1

Evidently, if (uT Bu )  1 , we have

 .

m      j g j ( z )   j k j ( z , p )  pT  p (  j k j ( z , p))   0,  j =1 





m   2 F  x, z;  p (  j k j ( z , p ))    d 2 ( x, z ).  2 ( x, z ) j =1  

s  0 = F  x, z; ti  p w( z , yi , p)  Bu i =1 

3. Duality Model

m    Cv   p (  j k j ( z , p ))  j =1 

We consider the following dual model (WD) .

,

sup

( s ,t , y )K ( z ) ( z ,u , v ,  ,  , p )H ( s ,t , y )

where H ( s, t , y ) denote the set of all ( z , u , v,  ,  , p )  R n  R n  R n  R  Rm  R n satisfying s

m

i =1

j =1

ti  p w( z, yi , p)  Bu  Cv   p ( j k j ( z, p))  0 (3.11) u Bu  1, v Cv  1, T

ti  f ( z, yi )   h( z, yi )  z

T

(3.12)

s  = F  x, z; ti  p w( z , yi , p)  Bu   Cv) i =1  m   F ( x, z;  p (  j k j ( z , p ))  j =1  s    F  x, z; ti  p w( z , yi , p )  Bu   Cv  i =1  



Bu   z Cv, T

i =1

(3.13)

 ti  w( z , yi , p)  pT  p w( z , yi , p )   0, i =1

  j g j ( z )   j k j ( z, p)  pT  p (  j k j ( z, p))  0. m

j =1

(3.14) If for a triplet ( s, t , y )  K ( z ) , the set H ( s, t , y ) =  , then we define the supremum over H ( s, t , y ) to be  . Next, we establish the duality of type (WD). Theorem 3.1 (Weak Duality) Let x and ( z , u , v,  ,  , s, t , y , p ) be feasible solutions of (P) and (WD), respectively. Assume that m  s  1)  ti  f (, yi )   h(, yi )  ,  j g j ()  is higher-order j =1  i =1  ( F ,  ,  , d , b,  ) Bu  Cv vector-pseudoquasi Type I at z , 2)  (a)  0  a  0, b( x, z ) > 0, Copyright © 2011 SciRes.

1 2

Since F is sublinear in its third argument, by (3.11) we can get

1

xT Bu  ( xT Bx) 2 .

s

h( x, y )  ( x Cx) T

then follows form 1) and  2 ( x, z ) > 0 , we have

1 2

s

yY

f ( x, y )  ( xT Bx) 2

Proof: From (3.14), we know that

The equality holds when Bx =  Bu for some   0 .

T

sup

(2.5)

xT Bu  ( xT Bx) 2  (u T Bu ) 2 .

max

1 2  0. 1 ( x, z )  2 ( x, z ) Then 1

Lemma 2: Let B be a positive semidefinite symmetric matrix of order n . Then for all x, u  R n , 1

1389

2 d 2 ( x, z ).  2 ( x, z )

Furthermore, by

1



2

1 ( x, z )  2 ( x, z ) 1 ( x, z ) > 0, we have

 0 and

  s  F  x, z; 1 ( x, z )  ti  p w( z , yi , p)  Bu   Cv    i =1   2   1d ( x, z ),

which implies that  s  b( x, z )   ti  f ( x, yi )   h( x, yi )  xT Bu   xT Cv     i =1  s    ti  f ( z , yi )   h( z , yi )  z T Bu   ( z T Cv)   i =1  s   ti  w( z , yi , p)  pT  p w( z , yi , p )   i =1   0. AM

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From 2) and (3.13), we can get

linearly independent, by Lemma 1, there exist ( s , t  , y  )  K ( x ), u  , v  R n and    Rm ,    R such that

s

ti  f ( x, yi )   h( x, yi )  xT Bu   xT Cv i =1 s

 ti  f ( z , yi )   h( z , yi )  z T Bu   ( z T Cv) i =1

s

ti f ( x , yi )     h( x , yi )   Bu   Cv  i =1

m

(2.1)

  g j ( x ) = 0

s

 ti  w( z , yi , p)  pT  p w( z , yi , p ) 

 j



j =1

i =1

 0.

1

Therefore, following from (3.12) and Lemma 2, 1 1    ti  f ( x, yi )  ( xT Bx) 2    h( x, yi )  ( xT Cx) 2   i =1      s

s

i = 1, 2, , s

(2.2) m

 j g j ( x ) = 0

 ti  f ( x, yi )   h( x, yi )  x Bu   x Cv T

1

f ( x , yi )  ( xT Bx ) 2    h( x , yi )    ( xT Cx ) 2 = 0,

T

(2.3)

i =1

i =1

s

 0.

ti = 1, ti  0, i = 1, 2, , s

Since t = (t1 , t2 , , ts )  0 , ti  0, i = 1, 2, , s , yi  Y 1

u T Bu   1, vT Cv  1,  1  T  T   x Bu = ( x Bx ) 2 ,  1  xT Cv = ( xT Bx ) 2 . 

and h( x, y )  ( xT Cx) 2 > 0,  ( x, y )  R n  R l , at least exists one q  {1, 2, , s} , such that f ( x, yq )  ( xT Bx) h( x, yq )  ( x Cx) T

1 2

1 2

 ,

which implies that 1

sup yY

f ( x, y )  ( xT Bx) 2 h( x, y )  ( x Cx) T

1 2

 .

Theorem 3.2 (Strong duality) Let x be an optimal solution of (P) satisfying xT Bx > 0, xT Cx > 0 and let g j ( x ), j  J ( x ) be linear independent. Assume that for any i = 1, 2, , s

1

sup







k j ( x , 0) = 0,





 p k j ( x , 0) = g j ( x ).

Then there exist ( s , t  , y  )  K ( x ) and ( x , u  , v ,   ,   , p )  H ( s , t  , y  ) such that ( x , u  , v ,   ,   , s , t  , y  , p = 0) is a feasible solution of (WD) and the two objectives have the same values. Furthermore, if the assumptions of weak duality hold for all feasible solutions of (P) and (WD), then ( x , u  , v ,   ,   , s , t  , y  , p = 0) is an optimal solutions of (WD). Proof: Since x is an optimal solution of (P) satisfying xT Bx > 0, xT Cx > 0 and g j ( x ), j  J ( x ) is Copyright © 2011 SciRes.

y Y

f ( x , y )  ( xT Bx ) 2 



1  2

T

= ;

h( x , y )  ( x Cx )

m  s  2)'  ti f (, yi )    h(, yi ) ,  j g j ()  is strictly j =1  i =1  higher-order ( F ,  ,  , d , b,  )    vector-pseudoquasiBu   Cv  Type I at z and; 3)'  (a ) > 0  a > 0, b( x , z  ) > 0,



and for any j  J ( x )

(2.5)

By (2.1) (2.2) (2.3) (2.5) and the conditions of theorem 3.2, we know that ( x , u  , v ,   ,   , p  =0)  H ( s , t  , y  ), that is ( x , u  , v ,   ,   , s  , t  , y  , p  = 0) is an feasible solutions of (WD). Furthermore, (3.2) implies that ( x , u  , v ,   ,   , s , t  , y  , p  = 0) is an optimal solutions of (WD). Theorem 3.3 (Strict Converse Duality) Let x be a feasible solution of (P) and ( z  , u  , v ,   ,   , p ) be a feasible solution of (WD). Suppose that 1)'

w( x , yi , 0) = 0,

 p w( x , yi , 0) = f ( x , yi )    h( x , yi ) ,

(2.4)

j =1



1



2

1 ( x , z )  2 ( x , z  ) 



0

Then z  = x ; that is z  is an optimal solution of (WD). Proof: Suppose that the contradiction is not true, that is z   x . Similar to the proof of Theorem 3.1 we

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obtain

1

  s  F  x , z  ; 1 ( x , z  ) ti p w( z  , yi , p )  Bu    Cv    i =1      2     1d ( x , z ), which implies that







  xT Bu     xT Cv  

which contradicts with 1)'. Remark: If we take place condition 2)' of this theorem by m  s  2)''  ti f (, yi )    h(, yi ) ,  j g j ()  is higherj =1  i =1  order ( F ,  ,  , d , b,  )    vector-pseudoquasi-Type

1





2

 0, the strict converse duality

holds too.

 t  w( z , y , p )  p  p w( z , y , p )    i =1  



1 ( x , z )  2 ( x , z  ) 

 z T Bu     ( z T Cv )    i

h( x , y )  ( xT Cx )

I at z , and take place condition 3)' by 3)''  (a )  0  a > 0, b( x , z  ) > 0,





> 

Bu   Cv



 i

1  2



 s   ti f ( z  , yi )    h( z  , yi )  i =1 

s

y Y



  s  b( x , z )   ti f ( x , yi )    h( x , yi )   i =1  

sup

f ( x , y )  ( xT Bx ) 2

T



 i

4. Acknowledgements



> 0.

From 3)' and (3.13), we can get

This work is supported by Youth Foundation of Beijing University of Technology (X1006011201002).

5. References

s

ti  f ( x , yi )    h( x , yi )  xT Bu    xT Cv

[1]

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[2]

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T. Weir, “Pseudoconvex Minimax Programming,” Utilitas Mathematica, Vol. 42, 1992, pp. 234-240.

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S. K. Mishra and N. G. Rueda, “Second-Order Duality for Nondifferentiable Minimax Programming Involving Generalized Type I Functions,” Journal of Optimization Theory and Applications, Vol. 130, No. 3, 2006, pp. 477-486. doi:10.1007/s10957-006-9113-9

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i =1 s





> ti f ( z  , yi )    h( z  , yi )  z T Bu  i =1 s

  ( z T Cv )  ti  w( z  , yi , p )  pT  p w( z  , yi , p )  i =1

 0.

Therefore, following from (3.12) and Lemma 2, s

1 1       T  2    T  2 t f ( x , y ) ( x Bx )  h ( x , y ) ( x Cx )     i  i i   i =1      s





 ti f ( x , yi )    h( x , yi )  xT Bu     xT Cv i =1

> 0. Since t  = (t1 , t2 , , ts )  0 , y  Y and

ti  0, i = 1, 2, , s ,

 i

1

h( x , y  )  ( xT Cx ) 2 > 0,  ( x , y  )  R n  R l , at least

exists one q  {1, 2, , s} , such that 1

f ( x , y  )  ( xT Bx ) 2 q

1  2

>  ,

h( x , y  )  ( xT Cx ) q

which implies that Copyright © 2011 SciRes.

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C. Y. JIN ET AL.

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Copyright © 2011 SciRes.

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