Теория оптимизации и численные методы

1 .1 : 1. 2. & . ., ."., . . . .' ( ..- ( .: .: ", 1998. ", 2000. * + 1 ' , I. * ( f (X) - X = ( x1, ...x n ) T - , , . !, " f (X) 1. f ( X ) = C const . % & ! . . – X. ! $ 2- ! ! , $ & & ! . f ( x1 , x 2 ) $ f ( X ) = C const . 2. ! ( x1 , x 2 & x2 f(x1,x2) & x2 x1 x1 & . . ., . 805 , 2005 ". 1 .2 (& f (X) ! & ! $ f ( X ) = C0 , ! & f (X) = C0 1 ) ) & & $ 1. % ! & ! & & . $! X 0 = (3,4) . ! : ! f ( X 0 ) = 32 + 4 2 = 25 . ! 2. , & : 5. 3. ) " – f ( X ) = x12 + x 22 , $ & X 0 = ( x10 , x 02 ) ! ! & x12 + x 22 = 25 - - (0,0) ! . x2 X0 1 x1 1 . f (X) 3. ! / , & : -- - f f f f (X) = , ,.., x1 x 2 xn ( n × 1) , n -! & &" T . : (1) (2) & & & & f (X) & " & " . f (X) (& & f (X 0 ) = ! f x1 ! ! , X = X0 f x2 X 0 = ( x 10 , x 02 ) f (X) = f (X) ! ! ! X0 & ! T X = X0 . . ., . 805 , 2005 ". " f f , x1 x 2 : T 1 .3 & ! $ & X . ! f ( X ) = x12 + 3x 22 2 ) ) & & 1. ) & $ X 0 = (0.5, 0.5) . X 0 = (0.5, 0.5) . ! 2. 2 ! , & : x12 + 3x 22 = 1 - - . f = 2 x1 ; x1 ) & ! 3. % ! ! : f ( X 0 ) = 0.52 + 3 0.52 = 1 , ! (0, 0) . ) & ! (0, 0) ! & ! f x1 X0 f x2 X0 2& f = 6x 2 , & x2 & f ( X ) = ( 2 x 1 , 6x 2 ) T . ! ! = 2 0.5 = 1 = 6 0.5 = 3 f ( X 0 ) = (1, 3) T . & 4. ) & ! $ X 0 = (0.5, 0.5) : ! & ! (0, 0) X0 . x2 f(X0) 1 X0 x1 . . ., . 805 , 2005 ". -&& 1 .4 . 4. , f (X) & ! (n × n) , , n - ! & : 2 2 f x 12 2 H(X) = . 5. . 6. 2 f x 1x 2 2 f x 2 x1 ..... 2 f x n x1 ..... f x 1x n 2 f ..... x 22 ..... 2 f xnx2 ..... ..... f x 1x n ..... 2 f x 2n A -det( A E ) = 0 ! : A>0 & A<0 A ! & 34 A= , ( ... a 1n a 21 a 22 ... ... a n1 a n 2 ... a 2 n ... ... ... a nn & ; & 1 = a 11 A & 3 a 12 & = >0 & j <0 & j & j () & & & & , & & & ! $ , , : 2 j A <> 0 " , a 11 A & " ! & & a11 a 12 a 21 a 22 ... . n = a 11 a12 ... a1n a 21 a 22 ... a 2 n ... ... a n1 a n2 . ... ... ... a nn ., . 805 , 2005 ". & ! & «-». 1 5 .5 ! 2- : f (X) = 6 & a 1x 12 + a 2 x1x 2 + a 3x 22 + a 4 x1 + a 5 x 2 + a 6 ! : a1x 12 + a 2 x 1x 2 + a 3 x 22 + a 4 x 1 + a 5 x 2 + a 6 = C 7 & & & ! : I = a1 + a 2 D = det a1 a2 2 a1 A = det a2 2 a4 2 a2 2 - & $ a3 a2 2 a4 2 a5 2 a3 a5 2 a6 C . . ., . 805 , 2005 ". 1 .6 7 8 9 5 ! D>0 & ( A <0 I I>0 (x 1 a) 2 A2 " + #) b) 2 (x 2 B2 =1 & 2 ( (x 1 a) 2 + & 2 b) 2 = R 2 (x 2 A=B & = - & , & ! D>0 -&& ) A <0 I I<0 % D<0 A 0 & (x 1 a) 2 ( A2 , + b) 2 (x 2 B2 & ) (x 1 a) 2 A2 (x 2 b) 2 B2 & D=0 A =1 x 2 = Ax 12 + Bx 1 + C , & x 1 = Ax 22 + Bx 2 + C . . ., . 805 , 2005 ". =1