processding59.pdf - page 75

1275
6WXG\LQJ DQG LQYHVWLJDWLQJ RQ WKH YDULDWLRQ LQHTXDOLW\ SUREOHPV
6WXG\LQJ DQG LQYHVWLJDWLQJ RQ WKH UHODWLYHO\ QRQH[SDQVLYH PXOWL YDOXHG PDSSLQJ
(VWDEOLVKLQJ WKH QHZ WKHRUHP IRU WKH YDULDWLRQ LQHTXDOLW\ SUREOHPV DQG IL[HG SRLQW SUREOHPV LQ %DQDFK
VSDFHV
,Q WKLV VHFWLRQ ZH LQWURGXFH DQ LWHUDWLYH VFKHPH ZKLFK FRQYHUJHV VWURQJO\ WR D FRPPRQ VROXWLRQ RI
WKH YDULDWLRQDO LQHTXDOLW\ SUREOHPV DQG D IL[HG SRLQW RI D UHODWLYHO\ QRQH[SDQVLYH PXOWL YDOXHG PDSSLQJ LQ D
UHDO XQLIRUPO\ VPRRWK DQG XQLIRUPO\ FRQYH[ %DQDFK VSDFH
7KHRUHP
/HW
C
EH D QRQHPSW\ FORVHG FRQYH[ VXEVHW RI D XQLIRUPO\ FRQYH[ DQG XQLIRUPO\ VPRRWK %DQDFK
VSDFH
E
/HW
T
:
C
N
(
C
)
EH D UHODWLYHO\ QRQH[SDQVLYH PXOWL YDOXHG PDSSLQJ DQG OHW
A
EH D FRQWLQXRXV
PRQRWRQH PDSSLQJ RI
C
LQWR
E
'HILQH D PDSSLQJ
F
r
n
:
E
C
E\
F
r
n
u
=
{
x
C
:
y
x, Ax
+
1
r
n
y
x, Jx
Ju
0
,
y
C
}
.
$VVXPH WKDW
Θ :=
F
(
T
)
V I
(
A, C
) =
.
)RU DQ LQLWLDO SRLQW
x
1
C
OHW
{
x
n
}
EH WKH VHTXHQFH JHQHUDWHG
E\
u
n
=
F
r
n
x
n
,
x
n
+1
= Π
C
J
1
(
α
n
Ju
n
+ (1
α
n
)
Jz
n
)
,
ZKHUH
z
n
T
(
x
n
)
IRU DOO
n
1
$VVXPH WKDW
{
α
n
}
LV D VHTXHQFH LQ
[0
,
1]
VXFK WKDW
OLP LQI
n
→∞
α
n
(1
α
n
)
>
0
DQG
{
r
n
} ⊂
[
d,
)
IRU VRPH
d >
0
.
7KHQ
{
x
n
}
FRQYHUJHV VWURQJO\ WR VRPH SRLQW RI
Θ
0HWKRGRORJ\
0DLQ UHVXOW
1...,65,66,67,68,69,70,71,72,73,74 76,77,78,79,80,81,82,83,84,85,...300
Powered by FlippingBook