Введение в специальность / Vv_v_sp_11

3

4

4

3

"

=)#-=/%'/) *0

+&*+,) m ,/#+, =)&(=&+, Ri i =

1, m

:&#=1),#) :03

')=/0$#

&/'/;)&9/

- (-=0,$)*;)&9/- '=(# ,+':+.

*+&'/

** + %$)9'=+3

%*)=0

/% / '!# !; 9+'+=#/ /:))'&% , *0$/;// bi i = 1, m )#/*/. &++',)'3

&',)**+ #+$. *+ -=+/',+#/'1 -=+#(9./ n '/-+, Pj j =

1, n

! &',)&'*0

*+=:0 =0&/+#0 aij i =

1, m

j =

1, n

i30+ =)&(=&0 *0

)#/*/.( j3+- -=+3

#(9.// ! 8&&)9'/,*+&'1 ,#-(&90 )#/*/.# j3+- -=+#(9

.// /0=09')=/'()'3

j =

1, n

-

9+$/;)&',+ -=+#(9.// j30+ ,/#0 -$0*/=():+) 9 -=+/',+#&',(!

=+#(9./% j30+ ,/#0 :+.)' -=+/',+#/'1&% ! xj > 0 /$/ *) -=+/',+#/'1&% !

xj = 0 '!)!

/:)): +0=0*/;)*/%

n

· · ·+ ainxn = Pj=1 aij xj )#/*/. ! 8'+' =0&/+# *) :+.)' -=),#&/'1 bi )#/*/.

'!)! -+$(;0): +0

=0*/;)*/%

n

X aij xj ≤ bi, i =

1, m.

:)!);

n

-=/%'/% c1x1 + c2x2 + · · ·+ cnxn = Pj=1 cj xj ! ,) '=)<()'&% :09&/:/'/=+,0'1

n

X cj xj → max.

:)!4;

j=1

2,)#): ,)9'+=# x = (xj , j =

1, n

) =

(x1, x2, . . . , xn) c = (&j

*+0#0 '0#0;0 :)!2; :)!4; -=/:)' ,/#

c 0x → max, Ax ≤ b, x ≥ 0.

:)!(;

A

60#0;0 :)!(; :/$/ ;'+ '+ .) &0:+) '0#0;0 :)!2; :)!4;; *0'#,0)'&%

%

,&$/ , -+&'0*+,9) %9+*+:/;)&9+- '0#0;/ -+'=)<+,0'1 -+$*+0

+ /'=0&3

/+#+,0*/% /:) //

&% =)&(=&+, :/'3'0 *),+':+. *+&'/ // 7/=0*)*/% >; '+

,:)&'+ +

0=0*/;)*/-

:)!);

-+$(;/: +0=0*/;)*/%

n

X aij xj

= bi, i =

1, m

:)!+;

j=1

/ ,:)&'+ '0#0;/ :)!(; '0#0;(

c 0x → max,

Ax = b,

x ≥ 0,

:)!6;

9+'+=0% *0'#,0)'&%

%

,&$/ #+-+$*/')$1*+ /',)&')* :/*/:0

$1*

#- -$0* -=+/',+#&',0 -=+3

#(9.// -+ ,/#0: d j j = 1, n :9+'+=#- :+.)' +'*0;0'1 +<%'0')$1*+) ,#3

-+$*)*/) -+$(;)**#/ '090'+,

:/*/:0$1*+) 9+$/;)&',+ -=+#(9.// 90.#+3

0+ ,/#0

*)+</+#/:+) #$%

&

(*9.

/+*/=+,0*/% -=)#-=/%'/% / '!# !;

0 '09.)

:09&/:0

$1*#- -$0* dj j = 1, n :&-=+& *0 -=+#(9./ :+ *+&'1 -=)#-=/3

%'/% -+ -=+/',+#&',( 90.#+0+ ,/#0 -=+#(9.// / '!# !; '+ ,:)&'+ +0=0*/3

;)*/- :)!2; -+$(;/: +0=0*/;)*/%

d j ≤ xj ≤ dj , j =

1, n

/ '0#0;(

c

0

x → max,

Ax = b,

,

:)!<;

d ≤ x ≤ d

0#) d = (d 1, d 2, . . . , d n) d

= (d1, d2, . . . , dn)! 60#0;0 :)!<; *0'#,0)'&%

"

(&'1 #0$)) /',)&'*+

'09.)

:/*/:0$1*+) 9+$/;)&',+ =)&(=&+, b =

&/:0$1*+) #+&'(-*+) )0+ 9+$/;)&',+ b = (b1, b2, . . . , bm) ! *+0#0 ,:)&'+ :)!);

/$/ :)!+; -+$(;/: +0=0*/;)*/%

n

b i ≤ X aij xj ≤ bi , i =

1, m

/ '0#0;(

j=1

0

x → max,

,

,

c

b ≤ Ax ≤ b

d ≤ x ≤ d

9+'+=0% *0

'#,0)'&%

0&'+ 0&&+='/:)*' -=+#(9.// #+$.)*

(#+,$)',+=%'1 (&$+,/ 9+:3

-$)9'*+&'/ #$% &<+=9/ ! x1

: x2 : · · · : xn =

l1 : l2 : · · · : ln ! 8'/ (&$+,/%

:+. *+ '0-/&0'1 '09

:) '+0+ /'

*+=: '0'=0' +<+=(#+,0*/% &$)#( ' +0=0*/;)*/% ,/#0

:)!);!

n

X aij xj ≤ bi, i =

1, m

j=1

,/#0 '=)<():+) *0 /'0+'+,$)*/) xj

)#/*/. /'#)$/- j

30+ '/-0

n

Pj=1 aij xj

i =

1, m

-

9+$/;)&',+ &'0*9+3;0&+, +<+=(#+,0*/% i30+ ,/#0 '=)<():+) *0

/'0

+'+,$)*/) ,&)/ '/-+, /'

#)$/- / %'0 ,)$/;/*0 *) :+.)' -=),#&/'1

&+*3

#0 +<+=(#+,0*/% i30+ ,/#0;! 2#$) <#$0 ,#;/&$)*0 '00=('90 +<+=(#+,0*/%

i

n

30+ ,/#0!

m

n

Pi=1 Pj=1 aij xj ! *=)<+,0*/) :09&/:0

$1*+- '00=('9/ +<+=(#+,0*/% &+&'+/'

+;),#*+ , :09&/:/

0.//

'+

-

,)$/;/*

#

m n

'n

m %

n

X X aij xj

= X X aij

xj → max /$/

X cj xj → max,

:)!1;

i=1 j=1

j=1 i=1

j=1

0

#) 9+

%&&

/./)*'

#

m

cj

= X aij , j =

1, n

:)!=;

'/-0

!

,&$/ =0&&:0'=/,0'1 '0#0;( :/*/:/'0.// -=+&'+), +<+=(#+,0*/% '+

-+$(;/:

m

m n

n

X bi − X X aij xj → min /$/

α − X cj xj → min,

i=1

i=1 j=1

j=1

0

#)

m

α = X bi - +< /- &+*# ,&)/ ,/#+, +<+=(#+,0*/%

i=1

cj ,#=0.0 '&% &+=:($+- :)!=;! +&9+$19( α - -+&'+%**0% ,)$/;/*0 '+

-+&$)#*%% '0#0;0 /:))' '+' .) &:#&$ ;'+

/ ,#$) - :09&/:/'0./% /&3

-+$1

'+,0*/% +<+=(#+,0*/% !

*09/: +<=0'+: :0'):0'/;)&90%

:+#)$1

'0#0;/ + =0./+*0$1*+: /&3

-+$1

'+,0*// +<+=(#+,0*/% /:))' ,/#

:)!2; :)!);

:)!1;!

n

n

X cj xj → max,

X aij xj ≤ bi,

i =

1, m

;

xj ≥ 0, j =

1, n

j=1

j=1

3

7

4

2

+'#)$) ')/*/;)&9+0+ 9+*'=+$% :**+ ;

*)9+'+=+- &/=:# =0<+'0 '

9+*'=+$)=# n =0'=%#+,! *< 0% *+=:0 ,#=0<+'9/

**+ '0 =0<+;( &:)3

*( &+&'0,$%)' *) :)*)) b /'#)$/- ! ++*'=+$)= j30+

=0'=%#0 -=+,)=%)' aj

/'#)$/-

'0 =0<+;( &:)*( -=/;): '+;*+&'1

)0+ =0<+'# &+&'0,$%)' βj %

:*) +$/<0)'&% , βj % &$(;0),;! 6

0=0<+'*0% -$0'0 9+*'=+$)=0 j30+ =0'=%#0

=0,*0 wj '0 =0<+;( &:)*(! =/

90.#+- +$/<9) 9+*'=+$)=0 &/=:0 *)&)'

(<#'9/ , =0':)=) γ ! '/=:0 /:))' ,+':+. *+&'1 /&-+$1'+,0'1 #+ dj 9+*'=+3

$)=+, j30+ =0'=%#0! 60#0;0 &/=:# - *0-'/ +-'/:0$1*#- &+&'0, **+ -=/

**+ ! *;),/#*+

0 ≤ xj ≤ dj , j = 1, n

n

9+*'=+$1 <(#(' =0,*#

Pj=1 cj xj / // *)+</+#/:+ :/*/:/'/=+,0'1! *<$3

n

n

X cj xj → min,

X aj xj ≥ b, 0 ≤ xj ≤ dj , j =

1, n

j=1

j=1

7

4

8 8

7

4

&,%

'0*0 & -=+<$):+- %9+*+:*+0+ -/'0*/% $ #)- :/$/ .

/,+'*#/; *+

-=/ (&$+,// -+'=)<$)*/% -/'0')$1*#/ ,) )&',

:,/'0:/*+,

90$+=/- :/3

*)=0$1*#/

,) )&', <)$9+,

/

#= !; , 9+$/;)&',0

/

*) :)*))

')/ 9+'+=#)

*)+</+#/:#

#$% *+=:0$1*+0+

. /'*)+<)&-);)*/% !

(&'1 , ');)*/) '0#

0**+0+ ,=):)**+0+ -)=/+#0 -/ 0 -=/0

+'0,$/,0)'&%

j =

1, n

9+$/;)&',+ )#/*/. j30+

-=+#(9'0

'09(-0):+0+ *0 =0&&:0'=/,03

):#- -)=/+# ! *;),/#*+ xj ≥ 0

j =

1, n

:j3- -=+#(9' :+. *+ -+9(-0'1

/$/ *) -+9(-0'1;! 2 xj )#/*/.0/ -=+#(9'0 Pj &+#)=. /'&% aij xj -/'0')$13

*+0+ ,) )&',0 Vi ! *+0#0 ,+ ,&)/ -=+#(9'0/

&+#)=.0*/) ,) )&',0 Vi =0,*+

n

Pj=1 aij xj ! 8'+ &+#)=.0*/) *) :+.)' <#'1 :)*1$) bi '!)! -+$(;0): +0=0*/3

;)*/%

n

X aij xj ≥ bi, i =

1, m.

*0'#,0 ' " '+/:+&'1 -=+#(9'0 Pj =0,*0 cj xj

0 ,&)/ -=+#(9'+, -

n

Pj=1 cj xj )) *)+</+#/:+ :/*/:/'/=+,0'1 ! *09/: +<=03

'+: :0'):0'/;)&90% :+#)$1 '0#0;/ + #/)') /:))' ,/#

n

n

X cj xj → min,

X aij xj ≥ bi, i =

1, m

; xj ≥ 0,

j =

1, n

j=1

j=1

/$/ , ,)9'+=*+3:0'=/;*+- &+=:)

c 0x → min,

Ax ≥ b, x ≥ 0.

+$(;)**( '0#0;( *0

'#,0 '

"

7

3

4

2

:0&')=&9( -+ /

'0

+'+,$)*/ *)9+'+=

/ #)$/

-+&'(-0)' :0')=/3

0

2

#/:0')=/0'

$0-! "$% /'0+'+,$)*/%

$ ,

=($+*0/ , 90.#+: /' 9+'+=#/ α

90.#+0

+ /'#)$/% '=)<( '&% '00+'+,9/ m ,/#+, -+ γi

2 90.#0% i =

!

1, m

-=/ j3+: =0&9=+) :j =

1, n

;! )+</+#/:+ +-=)#)$/'1

909+) 9+$/;)&',+ =(3

$+*+, :0')=/0$0 &$)#()' =0&9=+/'1 90.##: /' (90

'0**#/ &-+&+<+, #$%

/'0+'+,$)*/% N /'#)$/- ;'+<# +'/+## <#$/ :/*/:0$1*#:/ !

+&'=+/: :0'):0'/;)&9( :+#)$1 '0#0;/ ! *<+'*0;/: ;)=)' xj 9+$/3

;)&',+ =($+*+,

:0')=/0$0 9+'+=+) *)+</+#/:+ =0&9=+/'1 j3: &-+&+<+:

:j =

1, n

;! &*+

;'+ xj ≥ 0 j =

1, n

:=($+*# j3: &-+&+<+: :+. *+ 9=+/'1

/$/ *) 9=+/'1;!

++$/;)&',+ '00+'+,+9 i30+ ,/#0 -+$(;)**+) , =)'($1'0')

=0&9=+% xj =($+*+, -+ j3:( &-+&+<( =0,*+ aij xj 0 9+$/;)&',+ '00+'+,+9

i30+ ,/#0 -+$(;)**+) , =)'($1'0') =0&9=+% ,&)/ =($+*+, :,&):/ &-+&+<03

n

:/; -

:i = 1, m;! +&9+$19( *0 /'0+'+,$)*/) +#*+0+ /'#)$/%

Pj=1 aij xj

/#)' βi '00+'+,+9 i30+ ,/#0 '+ #$% N /'#)$/- *)+</+#/:+ bi = N βi '00+'+3

:)*1$) %'+- ,)$/;/*# ! n

X aij xj

≥ bi, i =

1, m.

j=1

0-#): ,)$/;/*( +'/+#+,! *0;0$0 +'/+#0:/ <(#): &;/'0'1 9(&9/ :03

')=/0$0 9+'+=#) *) %,$% '&% '0

0+'+,90:/ ! $+ 0#1 '00+'+,+9 i30+ ,/#0

-=/ j3+: =0&9=+) +#*+0+ =($+*0 =0,*0 aij γi 0 -$+ 0#1 '00+'+,+9 ,&)/ ,/#+,

-

m

Pi=1 aij γi -$+ 0#1 =($+*0 =0,*0 α! $)#+,0')$1*+ +'/+## -=/ =0&9=+)

+#*+0+ =($+*0 -+ j3:( &-+&+<( =0,*#

m

cj = α − X aij γi, j =

1, n,

i=1

+'/+## -=/ =0&9=+) xj =($+*+, -+ j3:( &-+&+<( - cj xj &(::0=*#) +'/+##

n

+' =0&9=+% ,&)/ =($+*+, :,&):/ &-+&+<0:/; -

Pj=1 cj xj ! 8'( &(::( *)+<3

/+#/:+ :/*/:/'/=+,0'1

! *09/: +<=0'+: :0'):0'/;)&90% :+#)$1

'0#0;/ +

=0&9=+) , =0&&:0'=/,0):+: &$(;0) /:))' ,/# !

n

n

X cj xj → min,

X aij xj ≥ bi, i =

1, m

; xj ≥ 0, j =

1, n

j=1

j=1

n

,&)/ $//'+,#/ :0')=/0$0/ -

Pj=1 aij xj ! 8'+ 9+$/;)&',+ #+$. *+ <#'1 =0,3

n

,) *)+</+#/:+ :/*/:/'/=+,0'1!

*09/: +<=0'+:

:0')=/0$+, -

Pj=1 cj xj !

:0'):0'/;)&90% :+#)$1

'

0#0;/ + &-$0,0/ /:))' ,/# !

n

n

X cj xj → min,

X aij xj = bi, i = 1, m;

xj ≥ 0,

j =

1, n

j=1

j=1

/$/ , ,)9'+=*+3:0'=/;*+- &+=:)

c 0x → min,

Ax = b, x ≥ 0.

50&&:+'=/: *)9+'+=#) /':)*)*/% , -+&'0*+,9) '0#0;/

! 2

+3-)=,#/ <(3

#): &;/'0'1 ;'+ 9+$/;)&',+ &-$0,0

9+'+=+) *)+<

/

+#/:+ -=+/

'

,)&'/ #+$

.

3

*+ <#'1 *) :)*1$) +<$):0 α / *) <+$1$) α

! 2)$/;/*0 α

:+.)' /0=093

=():#- &<#' -=+/',+#/:+0+ &-$0,0 :α ≥ α

;! *+0#0 -+$(;/: &$)#( /)

+0=0*/;)*/%

n

b i ≤ X aij xj ≤ bi , i =

1, m,

j=1

0#) b

= α β b = α β i =

!

i

1, m

i

i

i

2

+3,'+=#/ -=)#-+$+. /: ;'+ $//'+,#- :0')=/0$ Aj :+. *+ '09(-0'1

, 9+$/;)&',) *) -=),#$0 ): ,)$/;/*( dj

j =

1, n

! & 9=+:) '+0+ (

&0:+0

+ -

=)#-=/%'/%

/:))'&% 90.##- $//'+,#- :0')=/0$ ,

9+$/;)&',)

d j j

= 1, n 9+'+=+) *)+</+#/:+ -+$*+&'1

/'=0&/+#+,0'1!

*+0#0 -+$(3

;/: +0=0*/;)*/%

d j ≤ xj ≤ dj , j = 1, n.

2

3'=)'1// -(&'1 ,:)&'+ &'+/:+&'/ $//'+,#/ :0')=/0$+, '0#0*0 -=/3