Івченко І.Ю. Математичне програмування

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 

K  -! !  (-

, %++),  ’*    _-

`.

 :

1. K   , , , ’-

  ', .

2. K ,  ,   ’,

  , ,  

.

3. K ,     

; n-, (n     ).

 7

33 ? 4 

(339 

1. 8

(4?3823  (

! > :; ! ,  !  -

  ()  <

xcF

(7.1)

 

ijij

bxa

(i = 1, k) (7.2)

ijij

bxa

(i = k + 1, m) (7.3)

x (j = 1, l, l d n), (7.4)



jiij

cba ,,      k d m.

I (7.1) ! > !> ($ >

>, $ ! )  (7.1)—(7.4),   (7.2)—

(7.4) — $= <  ($ >  ).

"## ($ >) > :; ! -

,  !     <

(2.8)    (7.2)  (7.4),  k = m  l = n.

$%& ($ >) > :; ! 

 !     < (7.1)

   (7.3)  (7.4),  k = 0  l = n.

) ,

),...,,(

21 n

X # > 

(7.1)—(7.4), !  ’ ($ ,

$ >  :;).

>

mnmm

nmmm

aaa

iii

11211

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

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

  ! 

(7.2)—(7.3) > >  .

;  A

—   

.....

....



....

— - $=.

— j- - ,  (7.2)—(7.3) = 

 

BxAxAxA

d  ...

2211

 «=»   "

m ,  «d » —

 "

mm  .

C$ 

),...,(

1 n

>   (7.2)—

(7.4)   < . K

, # 

 x, >  x.

(-   !   <

 (7.1). O  $"  <  (# 

 max L),  «#» .

;

),,...,(



xxX  !  =

 <  L, !   -

 $ ’ .

O" , 



!  , 

),...,(



xxX —  ,



),...,(

1 n

X — $- "  < .

J#   min L,    $ ,

# ! 

)()( XFXF d



, # !  $  , ! >.

;    (7.2)—(7.4) =   -

 :

1. ? (7.2)—(7.4) , $  !  x

, x

......., x

>   (7.2)—(7.4),    -

! = .

2. ? (7.2)—(7.4) ,    -

  $=, $ >     -

  .

3. )  (7.2)—(7.4) ,    -

 $=.

2. 

:3  3BF ? 4  (  3C

C     "

   :;. %, " ,  -

 , = =    (,

)   2-

). ?    =    $

 , $    . O 

 max $ min < ,    maxmin $

minmax.

%   :; !   

’    :;. % !  $ -

    :;  "<  "<  -

= .

 "   (,   )

 :;    , # =    -

>   = $    -

"< . A !, # # ! $ = ’-

 !<   ,    = $ 

  $-<   .

Y$   !<    :;  "<,

$    , -",  -

 < <   <, -, -

  $=-  $=- , -

, -!, > , #  =, 

 .

J  ?

1. ?    !   <, =-

   =  <, =" 

(–1)  >:

 ...

2211

,  min F = –

max (–F).

2. $=- <  :;, # ! 

d », =   $=-  

 <  < < <,  $=-

-  «

t » —  $=- 

  <  < < <. % -

, $=-

ininii

bxaxaxa d ...

2211

  $= —   ...

2211

innin

bxxa 

1

n + 1

0), (7.5)

 $=-

ininii

bxaxaxa t ...

2211

 $=-

inninii

bxxaxaxa 

12211

... (x

n + 1 t

0). (7.6)

?  =  =   $=

ininii

bxaxaxa  ...

2211

=    :



d

d

ininii

bxaxaxa

...

,...

2211

(7.7)

K   , # , 

 $=-  $=- -

>!  > .

@ , # , >   -

 . %, #  $= <  :; -

$=!    $ ,  

 < <   , <   -

<,  $’! =  .

3. @   x

,    

>  $:

1-' . \<      u



" x

= u

– v

. (#   $"  ).

2-' . J# =     

 ,  =  $  ,

      "!.

;  >  .    -

  =! ,  ’ > ’!

  ",   =  !  > "-

 ,     $,   

.



(      :; (-

, , ).

1. P ,    = -

=    (,

)   2- (,

2. C   >  .

3. ?    =    $

 , $    .

4. O   max $ min < , 

  maxmin $ minmax.

=    >   =

$    "< .

 

1. 

   <  :;  : 

 <

5421

523



 



t

d

d

d

0,,,,

622

54321

541

5432

5431

xxxxx

xxx

xxxx

’:

?   !   <,  -

 $=   . =, #$  << 

 < , $   $= — -

  $= — .   , #

   $= ,  ,   -

 = $  

  -

 . ?  $= $> 

:











0,...,,

622

921

9541

85432

75431

65431

xxx

xxxx

xxxxx

=,   = $    < 

 :  >

5421

523



 :











0,...,,

622

921

9541

85432

75431

65431

xxx

xxxx

xxxxx

2. 

   <  :;  :

5421

55,235,75,6

 N max

 









0,,,,

632

14122

1243

54321

5421

5431

54321

xxxxx

xxxx

xxxxx

’.   > 

    <:   <

5421

55,235,75,6 xxxxF 

 







0,,,,

2011

26103

54321

431

543

432

xxxxx

xxx

      < , # -

!  =   <



N max

 



d

2011

26103

A     > -

 



<      $-

= .

  

1. [ ! ,    ?

K   ,  * & 

   ;

3. P , ' '!!   '?

 

  &  ()   (-



)    :

1. F = x

+ x

min

+ x

> = 8

+ 2x

> = 6

– x

< = 3

– 2x

= 6

, x

> = 0

. F = 5x

– 3x

min

+ 2x

> = 6

– 3x

– > = –6

– x

= 4

+ 7x

< = 8

> = 0

. F = 2x

+ x

max

– x

+ x

– x

+ x

, x

. F = 2x

– 4x

max

– 5x

+ 3x

+ 7x

, x

. F = 7x

– x

min

+ x

+ 5x

. F = x

+ x

min

+ x

–x

+ 2x

, x

. F = 2x

+ x

max

– x

+ x

– x

+ x

, x

. F = 2x

– 4x

max

– 5x

+ 3x

+ 7x

, x

. F = 7x

– x

min

+ x

+ 5x

, x

> = 0

. F = x

+ x

min

+ x

+ 2x

– x

, x

. F = 3x

+ 4x

+ x

+ 2x

4 –

min

+ 2 x

– 3x

+ x

– 5x

+ 3x

–5x

+ 2x

–7x

+ x

– 2x

+ 6x

+ 2x

, x

> = 0

. F = 10x

+ 2x

+ 5x

+ 3x

max

– 4x

– 5x

+ x

= 12

– 8x

+ 7x

+ x

+ 44x

– 2x

, x

, > = 0

. F = x

+ 2x

+ 8x

+ 3x

max

+ x

+ 5x

– 2x

+ 3x

, x

> = 0

+ x

+ 2x

+ 7x

= 3

. F = 3x

+ 2x

+ 8x

max

– 2x

+ 3x

– x