A New Generalized Derivative and Applications

2017 International Conference on Mathematics, Modelling and Simulation Technologies and Applications (MMSTA 2017) ISBN: 978-1-60595-530-8

 

A New Generalized Derivative and Applications

Guo-chen LIN

*

Department of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, China *Corresponding author

Keywords: Generalized derivative, Generalized subdifferential.

Abstract. We introduce a new generalized derivative approximated by subdifferentials of local convexifications of nonconvex functions, which overcomes "too large" Clarke subdifferential. At last we give its applications in optimization.

Definitions and Propositions

In 1970s Clarke made advance in differential of Lipschitzian functions[1-4]. People developed the nonsmooth analysis[5-6]. We introduce a new generalized derivative approximated by subdifferentials of local convexifications of nonconvex functions, which overcomes "too large" Clarke subdifferential.

In the paper g is a function on Banach spaceX x y, , X . B x r( , ) is a closed ball with center x

and radius r.

Definition 1.1: Generalized directional derivative of g at x is

0

0

0 ( , r)

( ) ( )

( ; y)

lim

sup

lim

rc rc

r

t z B x

z ty z x

t

g

g

g



 

  

where

rc

g

is a convexification of

( ) z B( , 2 )

( )

else.

r

g z x r

z

g

 _  

That is

 

 



1 _{1 1} 1 1



( ) inf n ( ) : , n , n , n 1, n

i i i i i

i i i

rc z r n i i i i X z

g









g x





_ 





x

_ 















x



. Obviously, when

z



B x

( , 2 )

r

,

 

1

 

1

1 1 1

( ) inf n ( ) : , n , n ( , 2 ), n 1, n

i i i i i

rc _i i i _{i i}

z n _{i i} B x r z

g

_



g x



 

x

 _



_



x

 

 _      _













.

Definition 1.2: Generalized subdifferntial of g at x is



* * 0 *



( ) : ( ; ) , , .

g x

x

X

g

x y

x

y y X

     

The following theorem shows



g x

( )

is approximanted by local convexifications.

Theorem 1.3: ( ) ( , )

o r rc

g x _{ }

g

B x r

 

 



, where

rc

g



is a classical subdifferential of rc

g

.

Proof: * ( , )

o r

g

rcB x r

x



 

 



, 0, 0 , ( , ), 0,

y X  r  z B x r t





          such that z ty B x r( , 2 ), we have

*

( ) ( )

, .

rc z ty rc z _y

t

g

g

x

0 *

, ( ; ) ,

y X

g

x y

x

y

    .

*

( )

g x

x

  .

Theorem1.4: Suppose g is locally Lipschitzian at

x

₀ with Lipschitzian constant L. Then

0

0

( ; )

g x

 is finite, positive homogeneous, subadditivity, Lipschitzian and |

g x

0( ₀; ) | 4 L || ||y  y .

Proof: It is obvious that it is positive homogeneous. Subadditivity is the result of convexity of .

rc

g

Since g is locally Lipschitzian at

0, 0, x y, B( , )0

x

   

x

 , we have

| g( ) g( ) |x  y L x|| y||. When 2r,

0

( , )|| ||[ ( _{|| ||} ) ( )] /

sup

z B r rc

r

y g z y z r

x y

g

  

 

 

0 1 0 1 1

( , ) 1 1

|| || sup [ ( ( )]: , , ( , 2 ), 1, /

|| ||

sup

_n n n _{n n}

i i i

i i i

z B r i i i i

r

y g z y g n _i B r z r

i

x



y

y



y

x





y



  

  

 

         















4rL y|| || /r



4 || ||L y

 .

Thus

0

0

0

0

0 ( , )

0 _{( , )}

( ; ) || || [ ( ) ( )] /

|| ||

|| || [ ( ) ( )] /

|| || 4 || || .

sup

lim

sup

lim

r z B r rc rc

r z B r rc

r

y y z y z r

x y

r

y g z y z r

x y

L y

g

x

g

g

g

 

 

  

  



According to subadditivity we have

0 0

0 0

( ; )y ( ; y) 4 ||L y|| 4 || || .L y

g

x



g

x

   



So

0 0

|

g x

( ; ) | 4 || || .y  L y Similarly we have

0 0

0 0

|

g

(

x

; )x 

g

(

x

; ) | 4 ||y  L xy||,x y, X.

Then 0 0 ( ; )

g x

 is Lipschitzian.

Theorem 1.5: Suppose g is locally Lipschitzian at x with Lipschitzian constant L. Then

a)



g x

( )

is nonempty, convex and *

w

compact. For all

x

*g x( ) we have ||

x

*|| 4 L,

b) 0



* *



, ( ; ) max , : (x)

y X

g

x y

x

y

x

g

    ,

c)



g

need not to be closed and upper semicontinous. Proof: a) Since 0

( ; )x

g

 is continuous and convex, and



g x

( )

is the classical subdifferenial of

0

( ; )x

g

 at 0. Then



g x

( )

is nonempty, convex and *

w

 compact. By theorem 1.4, For all

*

(x),

g

y

X

x

 

 

,we have

0 *

,y

g

( ; ) 4 || || .x y L y

x

 

Then *

||

x

|| 4 L.

b) If 0



_{* *}



, ( ; ) max , : (x)

y

g

x y

x

y

x

g

   , By the Theorem Hahn-Banach,

* * 0 *

, ( ; )x y ,y

y

x

g

y

   and

g

0( ; )x  

y

*, . Then

* (x)

g

0 * 0 ( ; )x y ,y ( ; ).x y

g



y



g

It is a contradiction. The proof is completed. c) Antiexamples: g :







is

0 ( )

0.

x x

g x

x x

 

 _{ }



Obviously,

 

 

 

1 0

( ) 0 0

1 0 .

x

g x x

x  



 _ 

  



Then



g

is neither closed nor upper semicontinous.

Theorem 1.6: Suppose g attains its local minimum at x, then 0g(x).

Proof: Since g attains its local minimum at x,

0 0

0 ( , ) ( , ) ( ).

rc

r r

B x y B x y g x

g

   



 









  .

The following propositions show the relationship between generalized subdifferential and Dini subdifferential when g attains its locally minimum at x.

Definition 1.7[5]: Dini directional derivative is

( ) ( )

( ; )

liminf

.

u y t o

g x tu g x g x y

t

d

  _{ }  

When g is locally Lipschitzian at x,

0

( ) ( )

( ; )

lim inf

.

t

g x ty g x g x y

t

d





  

Definition 1.8[5]: Dini subdifferential is



* * *



(x) : , ( ; ) .

g

x

X

x

y

d

g x y

 

  



Theorem 1.9: Suppose g is locally Lipschitzian and attains its local minimum at x, then

(x)

(x)

g

g



 



.

Proof: Since g attains its local minimum at x, _{( ) g( )}

rc x x

g

 for r small enough. Then

*

( ) 0,

o r

g

rc x r

x



 

 



     , such that

*

( )

rc x

g

x





*

2 ( ) ( )

, 0, , , ,

|| ||

r g x ty g x

y r t y

y t

x

   

        

* 0

( ) ( )

,

liminf

_t g x ty g x , .

y y

t

x



 

  

Since

0 r

g

rc( )x g x( ), g x( ) g x( )

 

 

 









 

 

.

The following theorem shows that the generalized derivative(subdifferential) we introduce is the generalization of classical derivative(subdifferential).

Theorem 1.10: Suppose g is continuous and convex, then



g x

( )

is the classical subdifferential

of g at x and 0 ( ; )x

g

 is the classical directional derivative

g

'( ; )x .

Proof: Since g is convex, |_{B( ,2 )x r} g |_{B( ,2 )x r}

rc

g

 . Obviously, 0 '

( ; )x ( ; )x

g

 

g

 . Now we

show

g

0( ; )x 

g

'( ; )x  . By the definition of 0

( ; ),x y 0, B x( , )

0 0

( ty) ( )

( ; ).

limlim

rc i rc i i t

x y t

g

z

g

z

g



  

Since *

i

z



is the classical subdifferential of g at

z

_isuch that

*

0 0

( ty) ( )

g( ty) ( )

,

lim

_{i i}

lim

_rc i _rc i .

i

t t

g y

t t

g

z

g

z

z

z

z

 _    _  

Then

0 *

, ( ; ).

lim

i i

y

g

x y

z



Since the classical subdifferential of g at x is locally bounded, then * *

x



X



,

x

* is the

*

w



cluster point of

 

*

i

z

. Then *

x is the classical subdifferential of g at x. Thus

0 * '

( ; )x y ,y ( ; ).x y

g



x



g

By Theorem 1.5 b), we have



g x

( )

is the classical subdifferential of g at x. The proof is

completed.

Now we naturally introduce the definition of strict differentiable and study its characters and propositions.

Definition 1.11: There exists a strict derivative of g at x, if

0 *

( ; ) ( ), , , ( ) .

s s

x y g x y y g x

g



D



D



X

Theorem 1.12: g is locally Lipschitzian at x, there exists strict derivative ( )

sg x

D

of g at x, then





( ) _s ( ) .

g x

D

g x

 

Conversely, if



g x

( )

is a single-point set

 

*

x

, then g is strict differentiable at x with strict

derivative *

( ) sg x

x

D

 .

Proof: Suppose that g is strict differentiable at x, then

0

( ; )x y _sg x( ),y ,for any y.

g



D

We now show

* *

( ), ( )

sg x g x

x



D

x

 _.

If not, *

, , _s ( ),

y

x

y

D

g x y

  . Then

0 *

, y _sg x( ), y

g

( ;x y).

x

 

D

  

It is a contradiction. The proof is completed.

Conversely,

 

y

X

, let *

 

* 0

:span y R, ,ky k ( ; )x y

y



y



g

. Now we show

 

* 0

, ( ; ) |x span y .

y  g 

If

k



0

, obviously * 0

,ky ( ;x ky)

y



g

. If k < 0; since

g

0( ; )x  is subadditivity, we have

0 0 0

( ; 0)x ( ;x ky) ( ;x ky).

g



g



g



0 0 * *

( ;x ky) ( ;x ky) , ky ,ky .

g

 

g

  

y

 

y

By the Theorem Hahn-Banach, *

y

can be extended to be a continuous linear functional on X,

without loss of generality, which is denoted by *

y

and

* 0

, ( ; ).x

y

 

g



Then

y

*



g x

( )

and

0 * *

( ; )x y ,y ,y .

g



y



x

If there exists y such that 0 * ( ; )x y ,y

g



x

, then

x

*

y

*. It is a contradiction with



g x

( )



x

*. Then

0 *

( ; )x y ,y .

g



x

Theorem 1.13: There exists the Frechet derivative *

x

of g at

x

0, then *

x

is the strict

derivative of g at

0

x

.

Proof: Since there exists Frechet derivative *

x

of g at

x

0,

 

y

X

,

 



0

,

 



0

,

2

r



 

,

 

x

B

(

x

₀

, 2 )

r

,

n





,

 

1

n

i

i

R







,

 

0 1 ( ,2 )

n i

B r i

x

 

x

,

1 1

n i i







such that

1

n i i i



x

x



, we have

*

0 0 0 0

2r ||

x x

_i || g( ) g(

x

_i

x

)

x x x

, _i ||

x x

_i || 2 .r

          

Then

*

0 0

1

2 n _i[g( ) g(_i ) , _i ] 2 .

i

r



x

x

x x x

r



 



   

Then

*

0 0

1

2 n _ig( ) g(_i ) , 2

i

r



x

x

x x x

r 

 



    .

Then

*

0 0

2 ( ) g( ) , 2

rc

r

g

x

x

x x x

r

      .

Then

*

0 0

( ) g( ) ,

2 || || 2 || ||

|| ||

rc x

y y

r y

g

x

x x x

    

   .

0 0 0 0 * 0 * 0 ₀ ( , ) * 0 ( , ) * *

0 _{0 0 0 0}

( , )

( ; ) ,

( ) ( )

lim lim ,

|| ||

lim [ ( ) ( )] ,

|| || || ||

lim [ ( ) g( ) , ] [ ( ) g( ) , ]

|| || || ||

sup

sup

sup

rc rc r _t

x B r

r

x B r rc rc

r

x B r rc rc

y y

x ty x y

x t

y r

x y x y

x r y

y r r

x y x y x x

x r y y

g x

x

g

g

x

g

g

x

g

x

x

x

g

x

x

x

 _                          0.      

Since 0 0 ( ; )

g x

 is subadditivity,

0 0 0 * *

0 0 0

0

g

(

x

; 0)

g

(

x

;y)

g

(

x

; )y 

x

,y 

x

,y 0.

Then

0 0 * *

0 0

( ; )y ( ; y) , y ,y .

g

x



g

x

 

x

  

x



Then 0 *

0

( ; )y ,y

g x



x

. Then 0 *

0

( ; )y ,y

g x



x

.The proof is completed.

Theorem 1.13 shows that the generalized subdifferential overcomes "too large" Clarke subdifferential[1-4]. We can obtain some propositions else.

Firstly, we have the following results: the generalized directional derivative



g x

( )

of locally Lipschitzian function g on n



is equal to

g

'( )x , a:e: Let [x; y] is the segment of n

 , '

' '

, ( )

x

y

y x

x

  

x

 . By Theorem Fubini, there exists

x

'x, g is Frechet differentiable

almost everywhere on ' ' ,

y

x

     . and

1 _' ' _'

0 g(

x

t y( x)),yx dtg( )

y

g( ).

x



Let

'

1 1 _'

0 ( ( )),

lim

0 ( ( )),

x

g x t y x y x dt g t y x y x dt

x

x

        





where

x

' is selected such that g is Frechet differentiable almost everywhere on ' ' ,

y

x

 

 

 . Then

'

1 1 ' '

0 ( ( )),

lim

0 ( ) ( ) ( ) ( ).

x

g x t y x y x dt g g g y g x

x

y

x

        _  _   





Corollary 1.14: Suppose g is locally Lipschitzian on an open subset D of a finite dimensional

space, for any x y, D, there exist some segments

0 x, 1

x

x

  

 ,

x x

1, 2 …

x

m1,

x

my which connect two points x and y, then g is uniquely determined by g excluding a constant.

Proof: Let x xD y, D,

0 x, 1

x

x

  

 ,  

x x

1, 2 …

x

m1,

x

my connect x and y. Then

1 1

1

1 1 1

1 0 ( ) ( ) ( ) ( ) ( ) ( ( )), m i i i m

i i i i i

i

f y f x f f

f x g t dt

x

x

x

x x

x x

          _  _      



 

for any f satisfying   f g. Then f is uniquely determined excluding a constant.

Corollary 1.15: Suppose g is locally Lipschitzian on a finite dimensional space, the generalized

Proof: Since g is locally Lipschitzian on a finite dimensional space, g is differentiable almost

everywhere and the Clarke subdifferential contains the Frechet derivative, by Theorem 1.13, the proof is completed.

The following is another characterization of generalized subdifferential. Definition 1.16:  DX x, D,

0 ( )

D

z D

z

else

 _ 





0



( ) : ( ; ) 0

D D

T x  yX



x y 



* * *



( ) : , 0, ( ) .

D D

N x 

x



X

x

y   y T x

Obviously,



0_D( ; ) 0x   or , and T_D( )x is a convex cone, N_D( )x is

w

* closed convex

cone.

Theorem 1.17: ND( )x  D( )x

Proof:  

x

* _D( )x ,  y T_D( )x ,we have *, 0( ; ) 0

D

y x y

x





 .Then

( ) ( ).

D D

N x   x

If there exists

x

*N_D( )x ,

x

*_D( )x ,then there exists yX, * 0

,y _D( ; )x y

x





.Then

0

( ; ) 0

D x y



 .Then y yT_D( )x .That is

x

*,y 0.It is a contradiction.

Theorem 1.18: Suppose g is a function on a finite dimensional space X and locally Lipschitzian

at

x

₀ with Lipschitzian constant L. Then

x

*₀g(

x

₀) if and only if

*

(g)

0 0 0

(

x

, 1) g_G (

x

, (g

x

)) where G g( ) is the graph of g.

Proof: Without loss of generality, the norm of X is || ( , ) || max || ||,| | /2x t 



x t L



.

Necessity: Since

x

*₀g(

x

₀),  s 0, 0   s, 2 r ,there exists y belong to B(

x

₀, )r ,

0

( , 2 )

x B

x

r

  ,we have

* 0

( ) ( ) , ,

rcx rc y x y

g



g



x



and

0 | ( ) ( ) | 2 .

rc y g rL

g



x



then ( , ( )) (( ₀, ( ₀)), )

rc

y

g

y B

x

g

x

r .

Note that B((

x

₀, (g

x

₀)), 2 )r G g( ) is compact, then co( ((B

x

₀, (g

x

₀)), 2 )r G g( )) is

compact. Then

0 0

( , ( )) co( (( , ( )), 2 ) ( )).

rc

y

g

y  B

x

g

x

r G g

Then

( )

( _{G g} ) ( ,_rc ( )) 0.

rc

y

g

y

 

( , )x t X

  ,without loss of generality, ( ₀, 2 ), t ( )

rc

*

( ) ( ) ( ) 0

( _{G g} ) ( , ) (_{rc G g} ) ( ,_rc ( )) ( _{G g}) ( , ) 0_rc ( , 1),( , ) ( , ( ))

rc rc

x t y

g

y x t

x

x t y

g

y

        

. Then

*

( )

0 0 0

(

x

, 1)  _{G g}(

x

, (g

x

)).

Sufficiency: Since *

( )

0 0 0

(

x

, 1)  G g (

x

, (g

x

)) , for any  ; there exist 2r and

0 0

( , )y t B((

x

, (g

x

)), )r ,  x B( , 2 )

x

₀ r ,we have

*

( ) ( ) 0

(G g ) ( ,rc x

g

_rc( )) (x  G g ) ( , )rc y t  (

x

, 1), ( , x

g

_rc( )) ( , ) .x  y t

Note that (_{G g( )}) ( , ) 0_rc y t  (if not, the left of inequality is ). Then ( )

rc

t

g

y . Then

* 0

( , 1),( , ( )) ( , ) 0.

rc

x

g

x y t

x

  

That is

* 0

( ) ( ) ( ) , .

rc x rc y rc x t x y

g



g



g

 

x



The proof is completed.

Applications

The section is about the applications of generalized derivative and subdifferential on optimization. P1:

 

0

m i

i

g



are locally Lipschitzian on n



, how to locally minimize

g

0 on



n: ( ) 0,1



i

D x



g

x   i m ?

Suppose z is the optimal solution of P1, without loss of generality,

0( ) 0z

g

 . If not,

0( ) 0( )z

g

 

g

is the substitute for

0( )

g

 .

P2: If g is a function on n



, how to locally minimize g on D of n



?

Theorem 2.1: If z is the optimal solution of P1 and

0( ) 0z

g

 , ( ) max



( ) : 0 i m



i

h 

g

   , then

there exists subset  with measure 0, such that









0 co ( ( _j) ) : 0 i m, ( _j), _j , _j , _{j h}

i

d

g z

i I

z z

z

z

z

         

where ( )_j



: ( )_j ( )_j



i

I

z

 i

g

z

h

z

, (



( ) )_j



i

d 

g z

is the set of the cluster points ( )_j i

g z

 , h is the set

of all points which are not Frechet differenntiable. Firstly, we need the following lemma:

Lemma 2.2 [5]: If z is the optimal solution of P1 and

0( ) 0z

g

 , then z is the locally minimum of h.

Lemma 2.3 [2]: If h is locally Lipschitzian at z on



n,  is a set with measure 0, then









( ) co ( ( ) ) : , , _h .

j j j j

ch z  d h

z

z

z

z

 

z

 



The proof of Theorem 2.1 is as followings: Proof: Since

i

g

is locally Lipschitzian, so as h, and (0 i m)

i

g

  is Frechet differentiable

excluding a set  with measure 0. By Lemma 2.2 and Lemma 2.3, z is a local minimum of h, then









Note that h,

i

g

is Frechet differentiable at zj and

( )_j ( ),_j ( )._j

i

h

z

g

z

i I

z

   

The proof is completed.

Competing interests statement: The author declare that the author has no competing financial interests.

Acknowledgement

The research was financially supported by the funding of young and middle-aged teachers' educational and scientific research project in the Fujian Education Department: JAT160371.

References

[1] F.H. Clarke, Generalized Gradients and Applications, Trans. Amer. Math. Soc, 205 (1975), 247-262.

[2] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience New York, 1983. [3] F.H. Clarke, Yu. S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Springer-Verlag New York, 1998.

[4] F.H. Clarke, R.J. Stern, P.R. Wolenski, Subgradient Criteria for Monotonicity, the Lipschitz Condition, and Convexity. Can. J. Math., 45 (1993), 1167–1183.

[5] G. Giorgi, S. Komlósi, Dini Derivatives in Optimization Part II, Decisions in Economics and Finance, 15(1992), no. 2, 3-24.