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7/24/2019 epli Folkman
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C Rn x
, x
C
( )x + x C [ , ].
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7/24/2019 epli Folkman
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[
,
]
( )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 =}
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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.
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7/24/2019 epli Folkman
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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 }
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S.
S T kS kT
(
)
pS+
qS
p+qS
[ , ] C kC C
S R
n
S=
kN
kS
S Rn
coS=n+
k=
cokS
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7/24/2019 epli Folkman
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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 {}
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C
C
R
n
[ yC
a, y, xC
a, x]
C
C
C
H(a, ) C
intH(a, ), iC intH+(a, ).
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7/24/2019 epli Folkman
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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
.
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7/24/2019 epli Folkman
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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)
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7/24/2019 epli Folkman
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C Rn f:C R
intC.
C Rn. f:C R x C
f(x)
kf(xk)
{xk} C f C
f
C
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7/24/2019 epli Folkman
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C Rn f:C R
f C
f
epi(f)
f
C
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f I = [a, b]
x
I
f(x
) = xI
f(x)
a
-
7/24/2019 epli Folkman
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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.
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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
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7/24/2019 epli Folkman
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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|
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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 =
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{ , } + { , } = { + , + , + , + } = { , , }
{Ci: i
I
} R
n
x
iI
Ci J(x) I n
x
iI\J(x)
Ci+
iJ(x)
co(Ci).
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7/24/2019 epli Folkman
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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
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: 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+
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7/24/2019 epli Folkman
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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}.
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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 = .
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7/24/2019 epli Folkman
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|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
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7/24/2019 epli Folkman
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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
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7/24/2019 epli Folkman
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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
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7/24/2019 epli Folkman
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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, , , ..., ).
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7/24/2019 epli Folkman
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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
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x, y
Rn
x =max in|xi|, x =
n
i=
|xi|
u= ( , , ..., ) Rn; x y xi yi i= ,..., n; x y xi
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7/24/2019 epli Folkman
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: 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)
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http://find/http://goback/