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Гомельский государственный университет им. Франциска Скорины

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лекции по спец. курсу(лубочкин)

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<<< < Предыдущая 1 2 3 45 / 65 6 > Следующая >>>

1) )3'*% $3 7,5%& ) ,&,8 x → x 6$5%&- ,%# 5%3$ &%')#+5, -&'"#+

) ,&, 6 .:8 β(x) ≤ β(x):

2))3'*% $3 7,5%& ')'3 8 J → J 6$5%&- ,%# 5%3$ &%')#+5, -&'"#+ ')'3 6 :8 β(J ) ≤ β(J ):

/$"#- {x, J } ')'3& + ) ,& /' " +#,%5

! &%"' (%4#'3 )'#%&*+, '(

u 6 .9:( (%4#'3 '*%&'4

= u 0 A − c 0

+ &%')'3& % 4'5)'&% "')3'(', , %"' )"%( ')

,&, 8

d j ,

)3+

j J +;

j J

−

dj , )3+

6 ):

)'"#3'+( )3% (,3+#% -&'

J + ( J −

6 ) : * +# (,!

5&', %"#(,

+& %4"'(

6 ) .:( 4 J

> 0( 4 J − +&/

'#&%"%5 ("% +& %4" j J ( #,4+%( #'

δj

%4" j J ( ! 4'#'3&

δj < 0( +& %4" j

J ( δj = 0 ( '#&%"%5

+#' 4

J + (

+#' 4 J −

+5 β(x, J ) 6 )-:

. &'(3%5%&&' ( +"

+ β(x, J ) ≤ ε( #' x ε/')#+5,

-&+ )

,& ( + &, #'5 )3'*%"" 3% %/

&+! )3%43,

,%#"!

,"&' 6 ) .: + &,+ %5 (% + +&$ µ

/$"#- β(x, J ) > ε /'"#3'+5 "'"

5,4"+5, -&'"'

( &' , 4'5)'&% (%4#'3, 7, "3,&+* )3!5 & '"3,&+/

%&+

+ 6 0:

( # %

max ρ

[

, d

]

= j J

+ µ = 0( # % ( )'

&! #"! $"

'(+! 6 )$:( #' ')#+5, -&+

) ,&

7, , + 6 .:2 6 0:( + (

#'5 " $ ,% )3'*%"" 3%

%&+! #,4,% )3%43, ,%#"!

/3+ 0 < µ ≤ µ0 ( " % µ0 ),3,5%#3 5%#' ,( )%3%+ %5 4 7,5%&% ')'3

6"5 ) ) ) ( ) 0: /3+ µ > µ0

)3+"#$),%5 4 7,5%&% )

,&, x

/'"#3'+5 &,)3,( %&+% l +75%&%&+! )

,&, x

l = − x.

6 :

'(+ )

,& x #$ %5 +"4,#- ( (+ %

x = x + θl,

θ ≥ 0,

" % +" ' θ ," ( ' - &,)3,(

%&+! l

0%4#'3 l 6 : &,)3,( %& +7 )

,&, x ( #' 4$ ( 4'#'3,! 8

1)!( !%#"! "')3'(', , +5 )"%( ') ,&'5

2)$ '( %#('3!%# $" '(+!5 ')#+5, -&'"#+ 6 1:

3)	'"#,( !%# 3% %&+% )3'+7(' &'+ 7, , % 6 )9:
4) '# , ,%# "('+"#('5 β( , J ) = 0
5) +5%%# 4'5)'&%$ ( "'(), , $ " (%4#'3'5 x		+ x ( &, 4'#'3'5
	'"#+",%# 5,4"+5$5, )3+3, %&+% % %('+ !$&4++ )3+ ( +" %&++

'*%&4+ "$#')#+5, -&'"#+	6 .:2 6 0: (,, & 5+ !( ! #"!
#' 4+ 73%&+! 5%#' , 3% %&+! 7, , +	6 .:2 6 0: (,, & 5+ !( ! #"!
% % (, "('+"#(, &,)3,( %&+! l 6 :8

1))3+ (+,%&++ ( ' - l )'"#'!&&' ( )' &!%#"! '"&'(&'% '"3,&+ %&+%

6 :( # 4 Ax = A(x+θl) = Ax+θAl = Ax+θA( −x) = (1−θ)Ax+θA = b

2)*% %(,! !$&4*+! )3+ (+,%&++ ( ' - l ('73,"#,%#8 df (x)/dθ =

= df (x+θl)/dθ = d(c 0 (x+θl))/dθ = d(c 0 x+θc 0l)/dθ = c 0 l = c 0

l +c 0

= {# 4 Al = 0}= −[c

−c

]l = −

l = −

( −x ) = {(2.48)} =

= β(x, J ) > ε ≥ 0

0 "+

$ )'"

% &%"' "('+"#(, ," θ ( '

- l "

% $%# ( #3,#- 5,4"+5, -&'

')$"#+5 5 (

# %

#,4+5

( #'#

&% &,3$ , +"- )3!5

% '"3,&+ %&+! 6 0:

/3+ #'5 θ < 1( # 4 )3+ θ = 1 )' $ ,%5 x = x + l = x +

− x = ( ,( )'

)3% )'

',%&+ 6µ > µ0 > 0:( "

$ ,+ d ≤ ≤ d &% 3%,

+7'(,

/3+ 0 ≤ θ < 1 &%')'3& % 4'5)'&% x

&% &,3$ , # )3!5 & '"3,/

&+ %&++ ( # 4

x = x

+ θl

= x + θ(

− x ) = (1 − θ)x + θ (

≤ x

≤ d ( d

≤

≤ d "

% $%#( #' d

≤ x

≤ d % '(,#%

-&' (

-4' )3!5 %

)3+ $(%

+ %&++ θ &, )'

$+%3(, % [0; 1) 5'"$# &,3$ +#-"! #'

')'3& % '"3,&+ %&+!

d ≤ x ≤ d .

6 9:

,+ %5 5,4"+5, -&'% +" ' θ0 ( )3+ 4'#'3'5 "'&3,&! #"! '"3,&+ %&+!

6 9: ! #'"' 7,)+ %5 6 9: )'4'5)'&%&' ( # %

d j ≤ xj + θlj ≤ dj , j J .

6 $:

.#'7&, +5 %3%7 θj 5,4"+5,

-&'% +" ' θ ( )3+ 4'#'3'5 ( )'

&!%#"! j/'%

'"3,&+ %&+% ( 6 $: ! 4,, '"' j ('75', & 0 " $ ,! 8 ,: lj > 0 #: lj

< 0

(: lj = 0

0 "

$ ,% ,: " ('73,"#,&+%5 θ 4'5)'&%, xj $(%

+ +(,%#"! + '"#+",%#

)3% %

-&' ')$"#+5'"' 7&, %&+! dj )3+ θ = θj

= (dj −xj )/lj &, '"+ &' ( (

$ ,% #: 4'5)'&%, xj $#(,%# + '"#+",%# )3% %

-&' ')$"#+5'"' 7&, %/

&+! d j

)3+ θ = θj = (d j −xj )/lj 0 " $ ,% (: 4'5)'&%, xj &% +75%&!%#"! 8

≡ xj ( # % 5,4"+5, -&' ')$"#+5+ ," θj 3,(%& #%"4'&% &'"#+

.# % +&!! #3+ " $ ,! ( )'

$ +5 !'35$ $

! ( +" %&+! θj

θj =		(dj − xj )/lj
θj =		(d j − xj )/lj



		∞

)3+

lj > 0;	6	-	:
lj = 0.
lj < 0;

,4"+5, -&' ')$"#+5+ ," θ0 )' "'('4$)&'"#+ 4'5)'&% xj ( j J (

3,(%& ( ' %(+ &' ( 5+&+5, -&'5$ +7 +"%	θj ( j J ( # %
θ0 = θj0 = min θj ,	j J .	6 ':

& %4" j0 J $4,7 (,%# 4'5)'&%$ xj0 ( 4'#'3,! )3+ $(% + %&++ θ )%3/

('+ '"#+",%# "3,&+* )3!5 & '"3,&+ %&++ "							+ {x, J } &%( 3', %&/
& + ')'3& + ) ,& ( #' θ0 > 0( # 4 ( #'5 "						$ ,% θj > 0 j J
'('5$ ) ,&$				0	l			6 91:
			x = x + θ		l			6 91:
)3+)+ %5 "#,3$ ')'3$ J + ( +" +5 '*%&4$ "$#')#+5, -&'"#+
β(x, J ) =	0	(x − ) =		0 (x + θ0l − ) =
=	0
=		(x + θ0( − x ) − ) =
	0
=	0	(1 − θ0)(x − ) = (1 − θ0)β(x, J ).						6 9.:

. %(+ &' ( β(x, J ) ≤ β(x, J )( + )' #'5$ 7,5%&, x → x $ '( %#('3!%#

)3+&!#'5$ ! +#%3,++ )3+&+)$ 8 '*%&4, "$		#')#+5, -&'"#+ &% ('73,"#,/
%#( , ( " $ ,% &%( 3',	%&&'"#+ "#3'"' $5%&-	,%#"!
" + '*%&4, 6 9.: $	'( %#('3!%# &%3,(%&"#($ β(x, J ) ≤ ε( #' )3'*%""

3% %&+! )3%43, ,%5 8 x ε/')#+5, -&+ ) ,& 0 )3'#+(&'5 " $ ,% )%3%/ &' +5 4' (#'3'+ ,"#+ +#%3,*++

2 2 4

.)'3& + ) ,& {x, J } &,7'(%5 ('+"#(%&&' &%( 3', %&& 5 ( %" +

	δj =	j 6= 0, j J .

0 "(!7+ " #+5 ')'3& + ) ,& {x, J }( '# , , ++ "('+"#('5 6 ..:( #$ %5
#%)%3- &,7	(,#- )3!5' &%( 3',	%&& 5 %( 3', %&& 5 , %% &,7 (,%#/
"! ')'3& + ) ,& ( 4'#'3 + + )3!5' ( + ('+"#(%&&' &%( 3', %&
6,4 #	' )'4,7,&' 3,&%%( " 4,, '+ ')'3'+ J "(!7,& "')3'(', , ++

%% ('+"#(%&& + ) ,& λ( " )'5' - 4'#'3'"' ( +" !%#"! 5%3, &%')#+/ 5, -&'"#+ ')'3 β(J ) 6 : ."&'( (,!"- &, #'+ "(!7+ ( 7,5%&$ ')'3 J → J '"$ %"#(+5 )' " % $ %+ "&%5%


J	J
l		l	,	6 9 :
λ →			,	6 9 :
λ →		λ

( 4'#'3'+ &'( + ('+"#(%&& + ) ,& λ )'"#3'%& #,4 ( #'# 8

.: ϕ(λ) ≤ ϕ(λ)

: λ "')3'(', , &%4'#'3$ ')'3$ J

/3+ #,4'

7,5%&% 5%3,0

&%')#+5,

-&'"#+0

')'3 $5%&- ,%#"! 8

β(J ) = ϕ(λ) −

ϕ(λ ) ≤

ϕ(λ) − ϕ(λ ) = β(J )

'" ,"'(,&& + ('+"#(%&& + ) ,&

(y, v, w)

6, "')3'(', , +%

('+"#(%&& % ) ,&

!( ! #"! "'

" ,"'(,&& 5+: )' &'"#-

')3% % !%#"!

)%3('+ 4'5)'&%'+ y 6( &, %5 "

$ ,% y = u: ,

%%( "''#(%#"#(+% 5%, $

4'5)'&%'+ y + 4')

,&'5 δ (7,+5&' ' &'7&, &' /' #'5$ 7, , +5 7,4'&

+75%&%&+! 4') ,&, ! %"' ')'3&'+ 4'5)'&% )'

', +5

δ = δ + σt ,

6 90:

" %

t = −ej0 sign lj0 ,

lj0 = j0 − xj0 ,

6 9):

+& %4" j

#'#( #'

&,+ %&

)3+

( +" %&++ ,", θ0

( ) ) . )3+

7,5%&% )

,&, x → x( +"

' σ ≥ 0 ,"

( '

- t

'" ,"&'

6 90:( 6 9):( "3% +

')'3& & 4'5)'&% 4')

,&, +75%&!%5

% +&"#(%&&$ δj0 (

"''#(%#"#($ $ 4'5)'&%% )

,&, xj0 ( )%3('+ '/

"#+" %+ "3,&+* )3!5 & '"3,&+ %&++ )3+ 7,5%&% )

,&, 6'5)'&%$ δj0

+75%&!%5 #,4 ( #'# %% &'('% 7&, %&+% δj0 )3+ σ ≥ 0

$ '( %#('3! ' $" '/

(+ ')#+5, -&'"#+ 6 1: ( "'('4$)&'"#+ " 4'5)'&%'+ )

,&, xj0 %+/

"#(+#% -&' (

≥ δj0

)3+

lj0

< 0

$ ,%#"!

δj0

= 0,

xj0

= d j0 ;

≤ δj0

= 0,

lj0

> 0

δj0

xj0

)3+

$ ,%#"!

= dj0 .

6 9 :

0 &,+ %5

$3,(&%&+! δ

= y

P − c

y 8

y 0

= (

+ c 0

)Q = (δ 0 +c 0

)Q + σt 0 Q = y 0 + σ y 0 ,

|{z}

" % y

= u

= c Q(

= t

Q = −ej0 Q sign lj0 = −Q(j0, I) sign lj0 .

%&,! y ( ( +" +5 &%')'3&$ 4'5)'&%$ "'" ,"'(,&&'"' 4')

6 99:

6 9$:

,&,

		0		0			0		0				0		0			0			0
				0			0		0				0		0			0			0
	δ	= y			A		− c	= y		A	− c			+ σ y		A	= δ		+ σt			,	6 9-:
" % %3%7 t '#'7&, %& (%4#'3
t 0 = y 0 A		= t	0			QA = −ej00 QA						sign lj0			= −Q(j0, I)A					sign lj0 .			6 9':

6'5)'&% v, w )3+ &%'#&' +5'"#+ 5', &' )'"#3'+#-( " % $! )3,(+ $

"'" ,"'(,&+! 6 )1: + 7,4'&$	6 90:( 6 9):( 6 9-:( 6 9': +75%&%&+! 4') ,&,
&, , , 3,""5'#3+5 " $ ,+ ( 4'" , {x, J } ('+"#(%&&' &%( 3', %&/
& + ')'3& + ) ,& 6'# ++ "	$ ,+ #$ %# 3,""5'#3%& ( " % $ %5 )$&4#%:

&%(

%&&'5 "

(

,"&' 7,4'&$

(

%#"! #,4'

, -/

& + $ ,"#'4 0 ≤ σ ≤

σ ( σ

> 0 ( &, 4'#'3'5 7&,4+ +"% δj

#$ $# "'(), ,#-

"' 7&,4,5+ +"%

δj

! ("%& j J 0 !"&+5 )'(% %&+% ('+"#(%&&'+ *%/

%('+ !$&4*++ &, #'5 $ ,"#4%8

ϕ(λ) =

b 0 y − d 0 v + d 0 w = {(2.40), (2.54), (2.64)} =

= b 0 y − 0

− d

0 + d w

0 v

0 =

{(2.66), (2.68), (2.65), (2.40), (2.63), (2.45)} =

b 0 y + σb 0

y − 0

− σ 0 t − σxj0 tj0

= {(2.40)} =

y − d v + d

w + σ(b

−

− xj0 tj0 ) = {(2.67), (2.69)} =

= ϕ(λ) + σ(t

− t

QA − xj0 tj0 ) =

ϕ(λ) + σ[t

Q(b − A

) − xj0 tj0 ] = {(2.41)} =

ϕ(λ) + σ(t

− xj0 tj0 ) = {(2.64)} =

ϕ(λ) + σ(tj0 j0 − xj0 tj0 ) = ϕ(λ) + σ( j0 − xj0 )tj0

= {(2.64)} =

ϕ(λ) − σ( j0 − xj0 ) sign ( j0 − xj0 )

ϕ(λ) − σ| j0 − xj0 |.

,4+5 '#

3,7'5

( &, &, , -&'5 $ ,"#4% ('+"#(%&&,! *% %(,! !$&4*+!

+&%+&, '#&'"+#%

-&' σ + $# (,%# " )'"#'!&&'+ "4'3'"#-

α0 = −| j0 − xj0 | < 0.

6 $1:

,"&' )3'(% %&& 5 ( +"

%&+!5 ( +&%+& +

7,4'& +75%&%&+! ϕ(λ)(

σ ≥ 0( "'&3,&!%#"! ' )'!(

%&+! )%3('"' &$

! σ1

$ &%')'3& & 4'5)'&%

( 4') ,&, %&, %&+% σ1

%"4' &,&' +#"! +7 6 9-:8

δj ( j J

σ1 = σj1	= min σj , j J ;

6 $.:

σj =	−δj /tj , %" + δj tj < 0;	6 $ :
	∞ ( '"#, -& & " $ ,!&.	6 $ :

						δj	j J
2
			δj					=
+0 δj = −0 d j dj				j

δj tj	< 0,	δj = +0, tj	< 0		δj = −0, tj > 0;
	> 0,	δj = +0, tj	> 0		δj = −0, tj < 0.

/' ', +5

= σ

= σj1

, J = (J \ j0)

j1.

/' )'"#3'%&+ ( +5%%5

ϕ(

) = ϕ(λ) + σ α0 < ϕ(λ).

6 $0:

/'4,,%5 ( #' J ')'3,( λ "')3'(', , ++ %% ('+"#(%&& + ) ,&

3% + +"% σj ( j J ( 6 $ : ("%" , &,+ $#"! 4'&% & %( # 4 ( )3'#+(&'5

"$ ,% ϕ(λ) → −∞ )3+ σ → ∞( #' &%('75', &' ( "+ $ "''#&' %&+! ('+/

"#(%&&'"#+ 6 09: + "'(5%"#&'"#+ '"3,&+ %&++ +"&' &'+ 6)3!5'+: 7, , + 6 .:2 6 0: 6'&% &'"#- +" , σj1 ( "'" ,"&' 6 $.:( 6 $ :( '7&, ,%#( #' tj1 6=

0 +" ' µ = |tj1 |( ( "+ $ 6 9':( +5%%# " % $ %% )3% "#,( %&+%

µ= |ej00 Qaj1 |,

#% µ 5' $ - 4' !! +*+%, )3+ aj0 ( 3,7 ', %&++ "#' #*, aj1 )' "#' #/

,5 aj ( j J ( ')'3&'+ 5,#3+ P = A(I, J ) 4,4 )' #,7+"					$ 0 +&%+/
&'+ , "%#3% )'4,7 (,%#"! ( #' )3+ µ 6= 0 + &%( 3', %&&'"#+					5,#3+* P
					P 7,5%&'+
&%( 3', %&, + 5,#3+*, P = A(I, J ) ( 4'#'3,! )' $ ,%#"! +7

"#' #*, aj0

&, aj1 /'4,,%5 #'

/3'(% %5 "&, ,

, )3% (,3+#% -& % 3,""$, %&+! /$"#- B &%'"'#,!

5,#3+*,( +7(%"#&, B−1 ( 5,#3+*,

A = B + uv 0 .

6 $):

'" ,( %" + +" ' µ = 1 + v 0 B−1u 6= 0( #' A &%'"'#,! 5,#3+*, +

A−1 = B−1 −

B−1uv 0 B−1.

%+"#(+#%

-&' (

)(B−1 −

B−1uv

B−1) =

(B + uv

= E + uv 0 B−1 −

uv 0 B−1 −

u v 0 B−1u v 0 B−1 =

µ−1

{z }

=E + uv 0 B−1 − µ1 uv 0 B−1 − µ −µ 1 uv 0B−1 =

=E + uv 0B−1 − µ1 uv 0 B−1 − uv 0 B−1 + µ1 uv 0B−1 = E.

75%&+5	#%)%3- ! '35$ $ 6 $): #,4 ( #'#	+75%&! "! #' -4' ' +& "#' /
#%* 5,#3+*	B 8

A = B + uej00 .

6 $ :

'" ,"&' 6 $ :( ( 5,#3+*% B +75%&!%#"! #' -4' j0 / + "#' #%* ( 4 %"' 4'5)'/ &%,5 '#,( ! #"! "''#(%#"#($ +% 4'5)'&% (%4#'3, u '" ,( %" + µ = 1 + ej00 B−1u 6= 0 ( #' A &%'"'#,! 5,#3+*, +

A−1 = B−1 − µ1 B−1uej00 B−1.

0%3&%5"! 4 )'"#,( %&&'+ 7, , % 5%%5

P = P + [A(I, j1) − A(I, j0)]ej00 = P + (aj1 − aj0 )ej00 ,

µ = 1 + ej00 Q(aj1 − aj0 ) = 1 + ej00 Qaj1 − ej00 Qaj0 = 1 + ej00 Qaj1 − ej00 ej0 =

= 1 + ej00 Qaj1 − 1 = ej00 Qaj1 6= 0 % '(,#%

-&' ( P &%'"'#,! 5,#3+*, +

= Q −

Q(aj1 − aj0 )e 0 Q.

,4+5 '#3,7'5 ( J ')'3, ."#,

'"- )'4,7,#-( #' ('+"#(%&&+ )

λ "')3'(', ,%# ')'3$ J

/' )'"#3'%&+ ( λ "'" ,"'(,&&+ ('+"#(%&& +

)

,& ''#(%#"#($/

++ %5$ 4')

'# , ,%# "('+"#('5

δ =

δ(J )

08 δj1 =

δ(

\ j1) = δ(J \ j0) = δ(J \ j0) = 0(

# %( "'"

,"&' 6 ) :( λ ('+/

"#(%&&+ )

,& ( "')3'(', , ++ ')'3$ J

,4+5 '

#3,7'5

( "&%5, 6 9 : )' &'"#- 3%, +7'(,&,

'5 4'3'#4'"' ,",

.)+",&&$

)3'*% $3$ 7,5%& ')'3 &,7'(%5 )3,(+

'" ,"&'

$0:

( )3+0

#'+ 7,5%&% 5%3, &%')#+5, -&'"#+ ')'3

$5%&- ,%#/

"! &, (%

+ +&$ |α |σ

( 4'#'3,! )' ', +#% -&,( %"

+ ')'3& + )

,& {x, J

}

('+"#(%&&' &%( 3'

#$ ,% (%

+ +&$( ( "+ $ 6 1:( $5%&- ,%#"! + '*%&4, "$#')#+5,

&'"#+

β(x,

) = β(x, J ) + α0σ < β(x, J ).

%,5%&'+ J →

7,4,& +(,%#"! (#'3,! ,"#- +#%3,*++ %#' 3% %/

&+! 7, , +

6 .:2 6 0:( ( 4'#'3'5 ')'3&+

) ,& 7,5%&!%#"! )' )3,(+

) ) ) .( ) ( &,7'(%5 , ,)#+(& 5 5%#' '5 " 4'3'#4+5

,"'5 , +#%3,/

*++ {x, J

} → {x, J } #'"' 5%#' , '*%&4, "$#')#+5,

-&'"#+ +75%&!%#"!

)' !'35$

β(x, J ) = (1 − θ

)β(x, J ) + α

.&, "#3'"' $5%&- ,%#"! ( %" + ')'3& + ) ,& {x, J } +

+ )3!5' ( + + ('+/

"#(%&&' &%( 3'

" + β(x, J )

≤ ε( #' )3'*%"" 3% %&+! 7, , + 6 .:2 6 0: )3%43, ,%#"!

&, ε/')#+5,

-&'5 )

,&% x 0 )3'#+(&'5 "

$ ,% )%3%&' +5 4 &'('+ +#%3,/

*++ " ')'3& 5 )

,&'5 {x,

}

)3%

%5 )$&4#% ')'3, 7,5%&!

,"- ( 5'5%

(

, )'!( ! "! )%3(

&$ - σ

$ &%')'3&&

4'5)'&% 4')

,&, δj ( j

#' "('+"#(' + , '

&,7(,&+% )3,(+

$ 4'3'#4'"' ,",

% %"- " *% -

)'( %&+! !! %4#+(&'"#+ 5%#' , 3,""5,#3+(,%#"! ('7/

5', &'"#- , -&%+ %"' (+,%&+! ( '

- t ' )'

&'+ 3%

,4",*++ ('+"#(%&/

&'+ *%

%('+ !$&4*++ ! #'"' )3% (,3+#% -&' ( !"&+5

&,3,4#%3 )'(%/

%&+! !$&4*++ ϕ(λ) ( σ ≥

0( )3+ ,

-&%+ %5 (+,%&++ ( '

- t

/3% )' ', +5

"&, , ,(

#' {x, J }

('+"#(%&&' &%( 3', %&& +

')'3& + ) ,& ( + ( #' 4% σ = σ1 (

'#3, ,%#"!

+ - ' &, 4'5)'/

&%, (%4#'3, δ

8 δj1 = 0 δj 6= 0( j J \ j1 '" , ( ('+"#(%&&'+ *% %('+

!$&4*++

ϕ(λ) = b

− d v + d

)3+ )3%&' % %3%7 #' 4$ σ = σ1 = σj1

&,3$ ,%#"!

+&%+& + 7,4'& )'(% %/

&+! #'

-4' $ ' &'"' ( 3,,%&+!

1 (σ) = −d

1 v 1

+ d w

1 .

6 $9:

% $%5

!$&4*+

* +# (,!

6 )1:( (+ +5

( #' 6 $

9: #' &%)3%/1 (

3 (&,! 4$"' &'/

+&%+&,! !$&4*+! ( %% "3,! +4 +5%%# +7

'5 ( #' 4% σ = σ

4'#'3'5$ "''#(%#"#($%# "4, '4 "4'3'"#+ +75%&%&+! ξj1 (σ)( 3,(& +

αj1 =	j1
ξ	(d	− d j1 )tj1	> 0
	(d j1 − dj1 )tj1		> 0

)3+

δj1 = δj1 =

< 0, tj1 > 0; > 0, tj1 < 0.

,4+5	'#3,7'5 ( )3+ )%3%&' % %3%7 σ = σ1 "4'3'"#- +75%&%&+! !$&4*++
6 $9: )'	$ ,%# )' ', +#% -&'+ )3+3, %&+%
	αξ = (d − d		1 )\|t	1 \|.
	j1	j1	j	j

,""5'#3+5 #%)%3- "+#$,*+ ( 4'" , &, ('+"#(%&&' &%( 3', %&&'5 ')'3&'5 ) ,&% {x, J } ( #' 4% σ = σ1 '#3, ,%#"! ( &$ - ' &'(3%5%&&'

&%"4' -4' 4'5)'&%# (%4#'3, δ ( σ ≥ 0 .#'7&, +5 %3%7

J1 = {j J : δj = 0 )3+ σ = σ1}.

%)%3- ( #' 4% σ = σ1 +75%&!%#"! 7,4'& )'(% %&+! $ ("%& !$&4*++

ξ (σ) = −d	j	v	j	+ d w	, j J	, σ ≥ 0.
j	j		j	j j	1

0 ('+"#(%&&$ *% %($ !$&4*+ $4,7,&& % !$&4*++ (&' !# , +#+(&' + ,&, '"+ &' !$&4*++ 6 $9: /' #'5$ "4, '4 α1 ( 4'#'3 + +") # (,%# (

#' 4% σ = σ1 "4'3'"#- +75%&%&+! !$&4*++ ϕ(λ) "4 , (,%#"! +7 "4, 4'( "4'3'"#%+ !$&4*++ ξj (σ)( j J1 ( # %

α1 =	αξ =	(d − d )\|t \|.
	X	X
	j	j j j
	j J1	j J1

('+"#(%&&,! *% %(,! !$&4*+! ϕ(λ) ( σ ≥ 0 ( )'" % #' 4+ σ = σ1 (% %# "%#! ( ' - t +&%+&' '#&'"+#% -&' σ + +75%&!%#"! "' "4'3'"#-

α1 = α0 +

α1,

" % α0 )' " +#,&&,! )' !'35$

% 6 $1: "4'3'"#- +75%&%&+! ('+"#(%&/

&'+ *% %('+ !$&4*++ &, $ ,"#4% 0 ≤ σ < σ1

#'# 7,4'& "'&3,&!%#"! '

)'!( %&+! " % $ %"' &$ ! σ2 $ !$&4*++

j ( j J

/$"#- σk−1 ( σk 0 < σk−1

< σk ( (,

)3'+7('

-& & "'"% &+& &$ !

j ( j J αk−1

"4'3'"#-

+75%&%&+!

('+"#(%&&'+ *%

%('+

!$&4*++

= {j J : δj = 0 )3+ σ = σk } 0

!$&4*++ &, $ ,"#4% σk−1 ≤ σ < σk ( Jk

#' 4% σ = σk "4'3'"#- +75%&%&+! ('+"#(%&&'+ *% %('+ !$&4*++ )'

$ ,%#

)' ', +#%

-&'% )3+3, %&+%

αk

− d

)|t |

j J

+ &, '#3%74% σk ≤ σ < σk+1 "#,&'(+#"! 3,(&'+

αk = αk−1 + αk .

%)%3- )3+"#$)+5 4 )'"#3'%&+

&'('

3%,

+7,*++ "&%5 6 9 : 6,4 )'/

4,7,&' ( ) 0( )3+ 3% ,4",*++ ('

+"#(%&&'+

%('

+ !$&4*++ $5%&-

,%#/

"! 5%3, &%')#+5, -&'"#+ "''#(%#"#($ %

')'3

/' #'5$ &,+ %5 #,4$

#' 4$( ( 4'#'3'+ !$&4*+! ϕ(λ) '"#+",%# 5+&+5$5, )3+ (+,%&++ ( '

t $&4*+! ϕ(

) '"#+",%# 5+&+5$5, ( #' 4% σ = σk0 ( " % k0 #,4'% +"

' (

−1

< 0, α

≥ 0

6 $$:

6 +" ' k0

("%" , "$ %"#($%#( # 4 inf ϕ(λ) > −∞:

/' )'"#3'%&+ (

k0−1

ϕ(λ) = ϕ(λ) +

αk (σk+1 − σk ), σ0 = 0.

6 $-:

k=0

."#, '"- )'"#3'+#- ')'3$ 8 J = (J \j0) j '"$# 3%, +7'(,#-"! (% ('75', &'"#+8 .: |Jk0 | = 1( : |Jk0 | > 1

0 )%3('5 " $ ,% 5&', %"#(' Jk0 "'"#'+# +7 ' &'"' %5%,( 4'#'3 + + ('7-5%5 ( 4, %"#(% j

6 0 " $ ,% : %5% 5&',%"#(, Jk0 $)'3! ' +5 )3'+7(' -& 5 '#3,7'5 &,)3+5%3 ( )' +& ('73,"#,&+ :8

	q
	[		< s2 < · · · < sq ,
s1, s2, . . . , sq ;	si = Jk0	; s1	< s2 < · · · < sq ,	6 $':
	i=1

# % " +#,%5 ( #' ( #' 4% σ = σk0

4'5)'&%, δs1

'"#+",%# &$ ! )%3('+ ( δs2

(#'3'+ + # /%3%#+3,! +& %4" +7 Jk0

( )'3! 4% 6 $':( &,+ %5 +& %4"

sp ( ! 4'#'3'"'

p−1

αs −1 = αk0−1 +

− d s )|ts | < 0;

i=1

αsp = αsp−1 + (dsp − d sp )|tsp | ≥ 0.

7 6 $$: "

% $%#( #' +& %4" sp

("%" , "$ %"#($%# /' ', +5 j = sp 6,4

+ ( ) ) )'4,7 (,%#"! ( #' J ')'3,( λ "')3'(', , ++

%% ('+/

"#(%&& + )

&%5,

6 9 : )'

&'"#-

3%,

+7'(,&,

.)+",&&$ ( ,&&'5 )$&4#% )3'*% $3

$ 7,5%&

')'3 &,7'(%5 )3,(+ '5

+&&'"' ,", .&, )3+ k0 = 1 )%3%&' +# ( )3,(+

' 4'3'#4'"' ,",

%,5%&'+

→

7,4,& +(,%#"!

(#'3,!

,"#-

+#%3,*++

{x, J } →

{x,

} , ,)#+(&'"' 5%#' , "

+&& 5 ,"'5

,"&' 6 $-:( '*%&4, "$#')#+5,

-&'"#+

−1

αk (σk+1 − σk ), σ0 = 0.

β(x, J ) = (1 − θ0)β(x, J ) +

k=0

+ β(x,

) ≤ ε(

#' )3'*%"" 3% %&+! 7,(%3 ,%#"! &, ε/')#+5, -&'5

)

,&% x /3+ β(x, J ) > ε )%3%&' +5 4 &'('+ +#%3,*++ " &'( 5 ')'3& 5

)

,&'5 {x,

}

/$"#- #%)%3- {x, J } ('+"#(%&&' ( 3', %&& + ')'3& + )

,& /3+

)'"#3'%&++ "')3'(', , %"' )"%( ') ,&, 5&',%"#(' {j J

: δj = 0}

3,7#+(, '"- &, (% ,"#+

6' &, ,"#- +& %4"'( (4

, ,"- ( J + ( 3

$",!

( J −: 6,, + )%3%&' %3%7 &$

- 8 δj

= 0 ( tj

< 0 ( j J + ( + + δj

= 0(

tj > 0( j J − ( ( 7 (,%# "4, '4 "4'3'"#+ +75%&%&+! ('+"#(%&&'+ *%

%('+

!$&4*++ $,% ( #' 4% σ = σ1 = σ0 = 0 8

α1 = α0 =

− d

)|t | ≥ 0,

j J1=J0

" % J0 = {j J +: δj = 0, tj < 0} {j J −: δj = 0, tj > 0}

/' #'5$

( "

$ ,% ('+"#(%&&'+ ( 3', %&&'"#+

&, , -&,!

"4'3'"#-

+75%&%&+!

('0

"#(%&&'1

+ !

"4'3'"#- &,

, -&'5

*% 2%(': 3,(&,$&4*++

# %

$ ,"#4% 0 = σ

= σ ≤

σ < σ

α1 = −| 0 − x 0 |

− d

)|t |.

j J0

" + J0 6= ( #' 5',%# '4,7,#-"! ( #' α1 ≥ 0 ( # % σ = σ1 = σ0 = 0

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