Study on Fuzzy Quadratic Programming Problems

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2018 International Conference on Applied Mechanics, Mathematics, Modeling and Simulation (AMMMS 2018) ISBN: 978-1-60595-589-6

Study on Fuzzy Quadratic Programming Problems

Eer-dun BAI

1,*

and Yu-e BAO

2 1

College of Computer Science and Technology, Inner Mongolia University for Nationalities, China

2

College of Mathematics, Inner Mongolia University for Nationalities, China

*Corresponding author

Keywords: D-differentiability, Convex fuzzy programming, Fuzzy quadratic programming, Optimality conditions.

Abstract. Under the condition of D-differentiable, fuzzy quadratic programming, whose objective functions are fuzzy quadratic and constraint conditions are fuzzy linear, are studied. First, we prove that fuzzy quadratic programming are a special case of convex fuzzy programming, whose objective mappings and constraint conditions are both D-differentiable; Then by the KKT conditions of convex fuzzy programming, we propose the optimality KKT conditions of fuzzy quadratic programming.

Introduction

Fuzzy mathematics decision methods were originally put forward by Bellman and Zadeh[1] in 1970. Since then, with the efforts of many learned men, substantial achievements in the domain of fuzzy programming have been yielded[2,3,4,5,6] .In particular, fuzzy programming, whose objective functions are fuzzy quadratic and constraint conditions are fuzzy linear, are called fuzzy quadratic programming[6].

Being more practical in dealing vagueness and uncertainty, fuzzy quadratic programming have aroused great attention from scholars, and it has yielded a series of good results[7,8].It’s commonly known that Kuhn-Tucker conditions play a pivotal role in the classical solution of mathematical programming. Zhang et al. [8] introduced the concept of sub-differential and differential from the perspective of convex analysis in 2005.In Ref.[8], saddle-point and minimax theorem in vague sense were discussed and applied to the Lagrange duality theory of fuzzy programming, and Lagrange duality and KKT conditions of convex fuzzy programming were obtained; meanwhile, fuzzy quadratic programming were studied, besides, corresponding Lagrange duality and KKT conditions were proposed.

In Refs.[9,10], Bao et al. developed the D-differentiable of generalized fuzzy mapping and studied the convex fuzzy programming and its Lagrange duality problem, as well, a more reasonable and concise form of the KKT conditions of convex fuzzy programming had been derived. In this paper, based on D-differentiable, fuzzy quadratic programming will be further discussed; KKT conditions for general case will be established besides.

Preliminaries

Definition2.1[8,9]. LetR_{denote the set of all real numbers. A fuzzy nu}_mber[0,1]_{will be a}_{fuzzy set}

: [0,1]

u R _{with the following properties(i)-(iv):}

(i)u_{is normal, i.e., there exists an}x₀R, such thatu x

 

0 1;

(ii) uis convex, i.e.,







1



min



   

,



u x  y  u x u y wheneverx y, Rand 

 

0,1 ; (iii) u_{is upper semi-continuous;}

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The family of all fuzzy numbers will be denoted byF . For 

 

0,1 , the level sets of a fuzzy number uE is a closed interval

 

u  u

   

 ,u  _{. We call}u





u

   

 ,u  ,



0  1



_{is the} parametric expression of fuzzy numberuE.

Foru v, F and rR_{, the addition and scalar multiplication on}F_{can be represented as:}

       







, , 0 1 ,







   

, ,



0 1



u v u  v  u  v     ru ru  ru     ,

where



uv

      

 u  v  , uv

 

 u

     

 v  ,ru  ru

   

 ,ru  ru

 

 .

For uF and aR, we can easily obtain u a F ,

   

ua  u  a and ua

 

 u

 

 a for any 

 

0,1 .

Let F M: F , G Mi: F



i1, 2, ,n



_{are all convex fuzzy mappings, then we call}

(FCP)

 

min



 



. . _i 0, 1, 2, ,

F x

s t G x i n



 _ _

 , xM ,

is the convex fuzzy programming.

Definition2.2[8]. Let







ij

   

, ij ,



0 1





n n

A a  a   



  

be a fuzzy matrix. For any

 

0,1 ,

 



ij

 



_{n n}

A  a 





and A

 

 



aij

 





_{n n}_

are n n symmetric positive definite matrices, thenA_is

said to be symmetric positive definite.

Definition2.3[9,10]. Let F M: F be a fuzzy mapping,





0 0 0 0 1, 2, , n int

x  x x x  M

. If there exists



1, 2, , n



u u u u F n

such that

 





 









 

0

0 0 0

0

, , ,

lim 0

,

x x

D F x u x x F x u x x

d x x

 



   

 ,

then Fis said to be D-differential at x0, and

 





0

1, 2, , n

F x u u u

 

is referred to as the gradient of F at 0

x .

In [10], the following conclusions about convex fuzzy programming had been gotten.

Theorem2.1[10]. Le



  





0

0 1,2, , ,

j

X  x G x  j m x X

and F G, j



j1, 2, ,m



_is D-differential x0. If there exists





0 0 0 0 1, 2, ,

m

m R

      _

, such that

 

0 0 0 0 0 0

1 1

0 0 0 0

1 1

0

m m

j j j j

j j

m m

j j j j

j j

F x G x F x G x

G x G x

   

   

 



       

  

 _ _





(1)

Then





0 0

, m

x  MR_

is a saddle-point of L x

 

, , Conversely,





0 0

,

x 

is a saddle-point of

 

, L x 

, then x0is optional solution to problem (FCP) and





0 0

,

x 

satisfies the condition(1).

Theorem2.2[9]. Let F G, be fuzzy mappings(M F), and be D-differentiable at point x0intM , then FG is D-differentiable at point 0

x and 



FG



 

x0  F x

 

0  G x

 

0 .

Theorem2.3[8]. A point





0_, 0 m

x   M R_

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Main Results

In this section, a class of fuzzy quadratic programming problems will be discussed under the condition of D-differentiable. Next, let’s introduce the basic concepts about fuzzy quadratic programming problems[8].

Let n n fuzzy matrix Q is symmetric positive definite, bRm, b



b b1, 2, ,bm



, C and A_are

n-dimensional fuzzy vector and m n fuzzy matrix, respectively. Then fuzzy quadratic programming problems

(FQP)

1 min

2

. . ,

T T

n

C x x Qx

s t Ax b x R_

 _

 

 _ _



will be discussed in this section.

Let

 





1 2

T T n

F x C x x Qx xR_

, then by[8], we can easily obtain F x

 

is convex fuzzy mapping. Let G x

 

Axb, then G x

 

is convex and concave fuzzy mapping. So, (FQP) is a special case of convex fuzzy programming. Let

 



1

 

, ,

 





1, 2, ,



T T

m j j j jn

G x  G x G x ,A  x x x ,

then Gj

 

x A x bj  j



j1, 2, ,m



_{, (FQP) can be rewritten as follows}

(FQP)

  

min

 



. . _j 0 1, 2, ,

F x

s t G x j m



 _ _



and the corresponding Lagrangian of (FQP) is

 

1





,

2

T T T

L x  C x x Qx Ax b , where 



 1, 2, ,m





j 0, j1, 2, ,m



.

And we can easily obtain L x

 

, is a convex fuzzy mapping: RnRm F _.

The Lagrange duality problem of (FQP) denoted by (DFOP) max₀ d

 

  , where d

 

 minx0 L x



,



.

Proposition3.1.Let n n fuzzy matrix Qis symmetric positive definite, then the fuzzy mapping

 

T

h x x Qx

is D-differentiable at x0intRn_.

Proof. For any r

 

0,1 , we have

  

    

,

 

T T

h x r x Q r x h x r x Q r x

, and for i1, 2, ,n,

  

2



₁_i

 

₁ ₂_i

 

₂ _ni

 

_n



i

h x r a r x a r x a r x

x

 _ _ _{ }



  

2



₁_i

 

₁ ₂_i

 

₂ _ni

 

_n



i

h x r a r x a r x a r x

x

 _ _ _{ }



are continuous at





0 0 0 0 1, 2, , n

x  x x x

.Therefore real-valued functions h x r

  

and h x r

  

are differentiable at x0.Hence, by the concept of differentiability from real analysis, we can obtain

  

 

0

 

0

 



0





0



1

,

n

i i

h x r h x r h x r x x d x x

x 





   





(2)

  

 

0

 

0

 



0





0



1

,

n

i i

h x r h x r h x r x x d x x

x 





   





(4)

  

2



1i 10 2i 20 ni n0



 

,

  

2



1i 10 2i 20 ni n0



 

i i

h x r a x a x a x r h x r a x a x a x r

x x

 _ _  _ _

__    __ _    _

 

  ,

2



a x1i 10a x2i 02 a xni 0n



F



i1, 2, ,n



.

Take





0 0 0

1 1 2 2

2

i i i ni n

u  a x a x  a x

, then uiF



i1, 2, ,n



and for any r

 

0,1 , we get

_i

 

0

   

, _i

 

0

 

i i

u r h x r u r h x r

x x

 

 

 



i1, 2, ,n



.

Therefore, for u



u u1, 2, ,un



_{, by (2) and (3) we have}

 





 









 

 

  

 





 

  

 





 

0 0

0 0 0

0

0 0 0 0

1 1 0 0 0,1 , , , lim ,

= lim sup max 0.

, ,

x x

n n

i i i i i i

i i

x x r

D h x u x x h x u x x

d x x

h x r h x r u x x h x r h x r u x x

d x x d x x

      _                _{ }        



，

So by definition2.3, the fuzzy mapping h x

 

is D-differentiable atx0 and

 

0 0

2

h x Qx

 

.

Proposition3.2. Let A_bem n _{fuzzy matrix,}



1, 2, ,



n n

x x x x R_

, bjR j



1, 2, ,m



_{, then}

fuzzy mapping Gj

 

x A xj bj



i1, 2, ,m



_{is D-differentiable at}x0intR_n_.

Proof. For 

 

0,1 , we have Gj

  

x  Aj

 

 x b j , Gj

  

x  Aj

 

 x b j _{, and for}

1, 2, ,

j m_{. we can also obtain}

  

 

j ji

i

G x a

x  

 _

 , j

  

ji

 

i

G x a

x  

 _

 ,

are continuous at





0 0 0 0 1, 2, , n

x  x x x

.Hence real-valued functions Gj

  

x  and Gj

  

x  _are

differentiable at x0.

By the concept of differentiability from real analysis, we get

  

 

0

 



0





 

0



1

, ,

n

j j ji i i

i

G x  G x  a  x x  d x x



 



  (4)

  

 

0

 



0





 

0



1

, .

n

j j ji i i

i

G x  G x  a  x x  d x x



 



  (5)

Take uiaji_{, then}u_iF



i1, 2, ,n



_{and for}

 

0,1 _{we have}ui

 

 aji

   

 ,uj  aij

 

 _.

Hence, for u



u u1, 2, ,un



_{, by (4) and (5) we obtain}

 





 









 

0

0 0 0

0 , , , lim 0 , j j x x

D G x u x x G x u x x

d x x

 



   

 .

So, by definition2.3, it’s easy to know Gj

 

x _{is D-differentiable at} _x0 _and

 



1, 2, ,



j j j jn

G x a a a

 

.

Proposition3.3. Let Cis n-dimensional fuzzy vector, then

 

T

H x C x

, xRn_{is D-differentiable}

at





0 0 0 0 1, 2, , int

n n

x  x x x  R_ and H x

 

CT.

(5)

Theorem3.1. In fuzzy quadratic programming (FQP), Let





n

X  x R Ax_ b

, x0X . If there

exits





0 0 0 0 1, 2, ,

m

n R

      

, Such that

 







 



0 0

0 0 0 0

0

T T

C r Q r x A r

A r x b A r x b

 

   

 _ _ _

 

   

 _, ₍₆₎

then





0 0

, n m

x  R_R_

 

, ; conversely,





0 0

, n m

x  R_R_

 

, , then x0 is an optional solution of problem (FQP) and





0 0

,

x 

satisfies the conditions(6).

Proof. by proposition3.1 and proposition3.3, we derive

 

1 2

T T

F x C x x Qx

is D-differentiable at

0

x X_andF x

 

CT Qx0_{. And by proposition3.2,}Gj

 

x A xj bj_{is D-differentiable at}x0_and

 



1, 2, ,



j j j jn

G x a a a

 

. Besides, (FQP) is a special case of convex fuzzy programming. So, according to theorem2.1, this proof is completed.

Remark3.1. The conditions (5) in theorem3.1 are usually called KKT conditions for fuzzy quadratic programming (FQP).

Remark3.2. By theorem2.3, x0and0satisfying KKT conditions are the optional solutions of fuzzy quadratic programming(FQP) and dual fuzzy quadratic programming(DFQP). respectively, besides, they have equal optimal values.

Remark3.3. Under more general differentiable conditions, theorem3.1 is the generalization of the corresponding conclusions in[8].

Summary

Fuzzy quadratic programming (FQP) is a special kind of fuzzy programming problem. In this paper, based on D-differentiable, fuzzy quadratic programming is discussed. A more general KKT condition for fuzzy quadratic programming is given. For the further study of fuzzy quadratic programming problem has laid a good foundation.

Acknowledgement

This work is supported by the National Science Fund of China (11461052) and the Inner Mongolia Natural Science Foundation of China (2018MS01010).

References

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