Šepli folkman

7/24/2019 epli Folkman

1/33
http://goforward/http://find/http://goback/


2/33

C Rn x

, x

C

( )x + x C [ , ].
http://find/http://goback/


3/33

[

,

]

( )C+ C C.

Rn

B x

r :B(x

, r) =x

+rB {v

, v

, ..., vn} Rn R =x

+lin(v

, v

, ..., vn)

H =x

+lin(v

, ..., vn ) H(a, ) = {x Rn : a, x =}
http://find/


4/33

H+(a, ) = {x Rn : a, x }

H(a, ) = {x Rn : a, x } a, ( )x

+ x

( )a, x

a, x

. H = H+ H

K R

n x

K,

x

K

K K .

K

K + K K.

( )K + K K + K K na osnovu K K

K + K

K +

K = K.
http://find/


5/33

S Rn

C

S

S=

SC

C

S

T

C

S T S T, C = C, ( C) = C.

x

+ ... + kxk x ,..., xk S

,...,k R + ... + k= i k N

S k

kS= {k

i=

ixi :xi S,k

i=

i = , i }
http://find/


6/33

S.

S T kS kT

(

)

pS+

qS

p+qS

[ , ] C kC C

S R

n

S=

kN

kS

S Rn

coS=n+

k=

cokS
http://find/


7/33

C

C

Rn = x C

C

a, y a, x

C

H+(a, ),C H(a, ), H(a, ) C

C

x C

y C

a, y < < a, x

C Rn / C

C

{

}

C

C {}


8/33

C

C

R

n

[ yC

a, y, xC

a, x]

C

C

C

H(a, ) C

intH(a, ), iC intH+(a, ).


9/33

f

D(f) Rn

C D(f) f C x

, x

C [ , ]

f((

)x

+ x

)

(

)f(x

) + f(x

)

x

=x

f

f

a(x) = a, x + C = Rn

Cx, x + c, x C Rn

C

( )q(x

) + q(x

) q(( )x

+ x

) =( )C(x

x

), x

x

.


10/33

f C m N x

, ..., xm C, ,...,m + ... + m = f(

x

+ ... + mxm) f(x ) + ... + mf(xm)

f:D f Rn

f

epi (f) = {(x, ) D(f) R : f(x)}hypo(f) = epi(f)

lev (f, ) =

{x

D(f) :f(x)

}.

epi(f)

C

Rn

f

C

epi(f)


11/33

C Rn f:C R

intC.

C Rn. f:C R x C

f(x)

kf(xk)

{xk} C f C

f

C
http://find/


12/33

C Rn f:C R

f C

f

epi(f)

f

C
http://find/


13/33
http://find/


14/33

f I = [a, b]

x

I

f(x

) = xI

f(x)

a


15/33

f C

Rn

x

C f(x

) = xC

f(x)

x

x C

C

( )x + x=x + (x x ), [ , ] C

F() =f(x

+ (x x

)), [ , ] =

F( ) =

f(x

)(x

x

)

x

C.

x

x

C : f(x

)(x x

) x C.
http://find/


16/33

Rn

(Rn)

a : Rn R x a, x

C Rn F:C (Rn)

x C

F(x), y x y C.

C Rn

F:C (Rn) x

C:

F(x), y

x

y

R

C

C = R

f(x)(y x) y R f(x) =ex
http://find/


17/33

C

CR = C BR BR

R

R

n

F:C

(Rn)

xR CR F(xR), y xR y CR

CR=

C Rn F:C (Rn)

R>

xR CR |xR|


18/33

f C

(C), C Rn

F(x) = f(x) Rn (Rn)

x C

f(x) =

yC f(y).

x

x C : F(x), y x y C.

f

Rn+ = {x= (x , . . . , xn) Rn : xi )}

Rn

F: Rn+ Rn

x

Rn+ F(x ) Rn+ F(x ), x =


19/33

{ , } + { , } = { + , + , + , + } = { , , }

{Ci: i

I

} R

n

x

iI

Ci J(x) I n

x

iI\J(x)

Ci+

iJ(x)

co(Ci).
http://find/


20/33

x

x=iI

xi

xi co(Ci) i

x=

iI

xi xi n

Ci

R

:Ci = {( , ), ( , ), ( , ), ( , )} i= , ..., . co(C

+ ... + C

) = {x R : x

, x

}.

x co(C

+ ... + C

)

x= ( .

,

.

)

x

co(C

), ..., co(C

) x R

C

,...,

C

x = ( . , ) co(C ), x = ( , . ) co(C ), x = ( , ) C ,..., x = ( , ) C

, x

= ( ,

) C

, ..., x

C
http://find/


21/33

: RnI

R

n

((xi)iI) =

iI

xi

iI

Ci = (

iI

Ci).

( (

iI

Ci)) = ( (

iI

Ci)) = (

iI

(Ci))

iI C

i = iI

(Ci)

x iICi x

m

iICi m

m n+
http://find/


22/33

x

x=m

j=

jyj, yj

iI

Ci, j > m

j=

j = .

yj

yj =

iI

yij, yij Ci.

Fi {yij}

jm. yj

iI

Fi j

x

iI

Fi.

Ci Fi Ci

Fi Rn

iI

Fi RnI

H x P RnI :

P=H

iI

Fi = {(xi)iI: xi Fi

iI

xi =x}.
http://find/


23/33

x

iI

Fi P

iI

Fi H P (xi)iI

x=

iIxi xi Fi (xi)i I P n xi

Fi Fi Fi

(n+ ) (xi)iI Fi x , ..., xn+ xi,

i

n+ zi Rn i >

t [i, +i], xi+tzi Fi. =

in+i

n+

n

,...,n+

n+

i=

izi = .
http://find/


24/33

|i| , i n+ .

(xi)iI, (xi )iI R

nI

xi = xi+ izi,

i

n+ ,

xi = xi izi, i n+ ,

xi = xi = x

i .

xi xi Fi

iI

xi =

iI

xi+

n+

i=

izi =x

iI

xi =

iI

xi n+

i=

izi =x

(xi)iI (xi )iI P

(xi)iI =

(xi)iI+

(xi )iI,

(xi)iI P
http://find/


25/33

C rad(C)

xRn

yC|x y|

F

C Rn m>

rad(C) m C F x coCF C

yCF C |x y| m

n

F

rad(C) m

m

n
http://find/


26/33

x, x

, ..., xm Rn

x

{x

, ..., xm}

n

{x

, ..., xm}

Ci, (i= , ..., m) Rn C =

m

i=

Ci

x (C) x=m

i=

xi

xi Ci n

i
http://find/


27/33

x C x=mi=

yi yi Ci

i yi=

li

j=

ijyij ij >

li

j=

ij = yij Ci

Rn+m :

z = (x, , , ..., )

z j = (y j, , , ..., )

...............

zmj = (ymj, , , ..., ).
http://find/


28/33

z=

m

i=

li

j=

ijzij.

z z=m

i=

li

j=

ijzij ij

(n+m) ij>

x=

mi=

lij=

ijzij

lij=

ij = i.

xi =

li

j=

ijyij x=m

i=

xi xi Ci i

(n+m) ij ij> i

n

i ij> xi Ci

(m n) i


29/33
http://find/


30/33


31/33

x, y

Rn

x =max in|xi|, x =

n

i=

|xi|

u= ( , , ..., ) Rn; x y xi yi i= ,..., n; x y xi


32/33

: A

P R

n+

|A

|=m

M= {e(a ) + ... +e(an) : a , ..., an A} f C() p S

m

aA

|p(f(a) e(a))| Mm

maA |inf

{p(x

e(a)) : x

a f(a)

}| Mm

{x: px pe(a)
http://find/


33/33

Šepli folkman

Documents