Прямые методы поиска безусловного экстремума ФМП

3 .11 3 f (X) n , n * H(X ) . ! $ , , ! " , ! $ ! $ , . . & ! ! # f (X) $ # " $ # k {X } ! {X k } 1. % ! ! ! " # f (X) , $ f (X) . $ k ! . lim f ( X k ) = f ( X * ) , , k f (X) . $ ! " {X k } 2. % k # ! $ . . ' $ ! . % & # . ! ! ! , * lim X = X . k & ! : X " Xk - ! ; ! ! k +1 k ) < f (X ) , ( .% , $ k X k +1 Xk ! , $ ! ! ! . ! ! & ! dk ! ! ! tk . # dk , f (X) , $ ! ! $ : 3. 1 ! k >0). !– 1. 1 2. 1 # $ ! ! ( f ( X ), d ) < 0 ; ! , # " – $ & # ; # % = X k + t kd k , X0 – , X k +1 dk f (X tk - k +1 ! ! , , # ! $ 1$ # # $ f (X) . $ 1- 2- f (X) . ! , # . $ . # ., . 805 !, 2005 #. $ f (X) . ! 3 .12 (1) 4 : X k +1 = X k t k f (X k ) (1.1) : dk = tk f (X k +1 5 f (X k ) - # ! ! !# $ ; $ : k ) < f (X ) . (1.2) ! $ ! : x2 X0 X1 1 X* x1 f(X0) 6 % ! : ! k f (X ) < , , - 7 : $ X k < ~, : " ! X k +1 ! : . (1.3) k=M f (X k +1 ) f (X k ) < ~ , (1.4) : ~- " . (1.5) (2) : X 4 dk = tk - k +1 = Xk t k f (X k ) f (X k ) ! (2.1) # ! $ ! !# (2.1): t k = arg min[f ( X k +1 )] . . ., $ , (2.2) . 805 !, 2005 #. 3 .13 5 ! $ ! x2 X0 1 * X X1 x1 f(X0) " d0 = X1 " , , ! ! , , , ! f (X ) f (X k +1 : ) ( f ( X ), f ( X ! " , : k +1 ! : )) = 0 (2.3) . ! B ! ! t k , . . f (X : k +1 " A X1 ! , k tk & ! , ! k !, f (X 0 ) : f ( X k +1 ) , # $ # # d (t k ) $ : = 0; dt k $ ! ! # $ ) = (t k ) , ! , # $ ! f (X) ! ! , # $ ! f (X) d 2 (t k ) > 0. < d2tk . ! (2.3) ! ! .: " " . f ( X k +1 ) = ( t k ) C [a , b ] II, §5, . 5.1.4.) " min ( . [2], D. (3) 4 : : dk = X k +1 = X k [ f ( X )] k "# ! x i - tk f (X tk ! k +1 [ ] f ( X k ) "# $ ! ! (3.1) ! xi xi !# # $ $ : k ) < f (X ) (3.2) . . ., . 805 !, 2005 #. 3 5 .14 ! $ ! x2 X0 $ ! - f(X0) $ X1 $ ! - f(X0) 1 X* f(X0) ! " , (4) 4 " : : dk = tk [ f ( X )] k "# ! x i - ! ( [ ] f ( X k ) "# $ ! ! ) (4.1) ! xi xi # ! (4.1): t k ! . - X k +1 = X k tk - 5 x1 !# $ $ , = arg min[f ( X k +1 )] (4.2) $ ! x2 X0 $ ! - f(X0) $ 1 f(X0) ! & " , tk $ ! - f(X0) X X* 1 x1 . " A C, . . . ., . 805 !, 2005 #. 3 .15 5 $ -B ( ! !) , . . C # $ ! f (X) , f ( X) ! f ( Y) LX % k . X* = lim X k ! k , $ , # $ " " , , ! : ! (5) 4 : $ : X, Y R , L > 0 , Y, f (X k ) D n & ! '( # ) (* $ . -+ ) ) X k +1 = X k + t k d k . (5.1) : d0 = f (X 0 ) dk = f (X k ) + k 1 (5.2) f (X k ) = f (X k 1 ) tk - k 1d k 1 (5.3) 2 (5.4) 2 ! ! ! (5.1): t k E 5 # (5.2) $ (5.5) !# $ , = arg min[f ( X k +1 )] , ! (5.5) $ ! !" . ! $ ! x2 X0 f(X1) 1 d0 d1 X* X1 d0 = ! & f(X0) " , tk x1 . " A C, . . . ., . 805 !, 2005 #. 3 .16 7 ! # 7 , $ !# !" $ ! , , F # , 7 ! # ! % $ , ( - . .[2], ! # .II, §6. .6.5). . ., F . ! $ f (X) . $ ! . 805 - !" !, 2005 #. ,