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!! (3:3):513.4 &$$8 " +Q -!) + ! *) , *! x0 | !)( / !! (3:3) * ;i i = 0 1 : : : m + 1 .D*! G"*-8! :8 !8 ci 2 ;i i = 0 1 : : : m +1 !, c0 + : : : + cm+1 = 0: H ! : /! !! *! ! !!*!,.0! 8:8 !!:min f (x)(3.4)'i(x) 0 i = 1 : : : s(3.5)'i(x) = 0 i = s + 1 : : : k(3.6)nx2GR :(3.7)C!, ," f 'i { ,,",G { , ! -, intG 6= : ! !, - G 8 . !, G = fx j xi 0 i = 1 : : : ng:!-, /+ -+ - ) !) + /! 3.6 (G"*-8!. G /* ! !! (3.4)-(3.7) !! (3:3) 8: -! G0 = fx j f (x) <f (x0)g, Gi = fx j 'i (x) 0g i = 1 : : : s Gm = G Gm+1 = fx j'i (x) = 0 i = s +1 : : : kg, * m = s +1 x0 | . !! (3.4)-(3.7).
G! !) ;i i = 0 : : : m + 1 ! ! 8:+ -+. 3.7 . f (x) | - - f 0(x0) 6= 0: 5 ;0 ;0 = fe 2 Rn j hf 0(x0 ) ei < 0g:52 3.8 . 'i(x) | - , 'i(x0) < 0 '0i(x0) 6= 0:5 ;i (n'i(x0) < 0;i = fe 2 Rn j h'0 (x ) ei <R0g 'i(x0) = 0:i 0 3.9 . G | Rn , intG 6= x0 2 G: 5 ;m - ;m = fe 2 Rn j e = (x ; x0 ) > 0 x 2 intGg:L!, x0 2 intG ;m = Rn : 3.10 . 'i(x) i = s + 1 : : : k | - . 5 ;m+1 ;m+1 = fe 2 Rn j h'0i (x0) ei = 0 i = s + 1 : : : kg:G! !) 3.7-3.10 - !( ?1]. 3.7 (I: ! -( 1!*!-!) . x0 | % (3.4)-(3.7).
5 ' 0i i = 0 : : : k , , 0i 0 i = 0 : : : s0i 'i(x0) = 0 i = 1 : : : sh f 0(x ) + X i '0i(x ) x ; x i 000k00i=100 x G: (+ 0i -.)53. # !!!, *! :- - !*!) 8: :1) f 0 (x0) 6= 0,2) 8* i = 1 : : : s 'i (x0) < 0 '0i (x0) 6= 0:G(), ) f 0(x0 ) = 0: D*! ! ! 00 = 1 0i = 0 i 1: - ), 'i (x0) = 0 '0i (x0) = 0 i = 1 : : : s: D*! ! ! 0i = 1 0j = 0 j 6= i: 3.7-3.10 , + )+ !!( x0 * 8: ! :;0 = fe 2 Rn j hf 0(x0 ) ei < 0g 8* i=1,. .
. ,s;i =(Rn 'i (x0) < 0i < 0g 'i(x0) = 0fe 2 j h i;m = fe 2 Rn j e = (x ; x ) > 0 x 2 intGg;m = fe 2 Rn j h'0i (x ) ei = 0 i = s + 1 : : : k g:Rn'0 (x0) e0+10 / . D! ! x0 | !) . !! (3.4)-(3.7), \mi=0+1 ;i = : D*! G"*-8! , :8 ! 8 ci 2 ;i i = 0 : : : m + 1 !, c0 + c1 + ::: + cm+1 = 0:, !8 ( !!! ! - . - * !-, ;0 = fc0 j c0 = ;0 f 0 (x0) 0 0g: -* ! , c0 !- ;0 *! ) *!, *! 8* e 2 ;0 hc0 ei 0. I8! !, ;0 - - ! ;0f 0 (x0) * 0 0:54!-, ;0 fc0 j c0 = ;0 f 0 (x0) 0 0g:) c !- ! - fc0 j c0 = ;0 f 0 (x0) 0 0g: D*! ( : ( e !(, h;0f 0 (x0) ei >hc ei 8 0 0: !*! 0 = 0 hc ei < 0.Q ! 0 > 0 0 ! 1 ! !hf 0(x0) ei 0: L!, c !- ;0:D ! -;i = fci j ci = ;i '0i (x0) i 0g + i s, + 'i(x0 ) = 0.
P - 'i(x0 ) < 0, ;i = f0g. C!), + i s ;i = fci j ci = ;i '0i (x0) i 0 i 'i (x0) = 0g:1* ), ;m+1 = fcm+1 j cm+1 = ;kXi=s+1i'0i (x0)g:D! ! , ! ! -+ ! c0 + c1 + ::: + cm+1 = 0 !:cm = 00f 0(x0) +kXi=10i '0i (x0) 2 ;m 0i 0 i = 0 : : : s0i 'i(x0) = 0 i = 1 : : : s* 0i i = 0 : : : k { -, -) ! 8 ( !) !! ci 0 i m + 1).55 8 ;m c, 8:+ ! hc (x ; x0 )i 0 + > 0 8+x 2 intG.
D! ! 8! ! * -! G )( ( -! intG c 2 ;m + x 2 G hc x ; x0 i 0 , !),h00f 0(x0)+kXi=10i '0i (x0) x ; x0i 0 8* x G: D! ! !!.+ !) 3.7 ! !8: ! -( 1!*!-!, ! 0i i =0 : : : k | - 1!*!-!. 3.7 ) 8) !(, *! -) 00 *! "( ," f 0 (x0) ! 8, *! + /! ! ( ,". D! !! ! - x0: I , 8:+ ) ! -+ !!, ! ! G"*8!. P 00 = 0, c0 = 0 c1 + ::: + cm+1 = 0. 3.7-3.10 , ;1 ;2 : : : ;m | -!.
C!), G"*-8! ;1 \ ;2 \ ::: \ ;m+1 = :L!, !* -! , ! 00 = 0 8!. ! , *!!8: 00 : ! -( 1!*!-!, !8 + !! (3.4)-(3.7), +G = Rn . 3.8. ( + !-D!!) . x0{ % (3.4)-(3.7), G = Rn f'0i(x0) j 'i(x00) = 0g { . 5 ' i i = 1 : : : k , :0i 0 i = 1 : : : s56(3.8)0i 'i (x0) = 0 i = 1 : : : sf 0(x0) +kXi=10i '0i (x0) = 0:(3.9)(3.10).
G = Rn , !hc x ; x i 0 - ) + x 2 G ) 0c = 0. C!), 3.7 00f 0 (x0) +PkXi=10i '0i(x0 ) = 0:(3.11)PP 00 = 0, ki=1 0i '0i (x0 ) = f0i '0i(x0 ) j 'i (x0 ) = 0g = 0 8 (( ! -! f'0i(x0) j 'i (x0) = 0g. / ! !! (3.11)- ! ) ! 00 > 0: , - 0i =00 i =1 : : : s 8 + (3.8)-(3.10). D! ! !!.0! !! (3.9), (3.10) *! 'i(x0 ) =0 i = s +1 : : : k: I ! 8 k + n !( k + n 0i i = 1 : : : kB xj j = 1 : : : n: ) !() ( . + , - !( .
(0 x0): I!, *, (3.8)-(3.10) 8 ! !), ! x0- !- ) !) / !! (3.4)-(3.7).+ ! !) ! ) + ! (+ !!. !, !! * *!!, !!!8 -.3.5 5$6 4 740! !! * *!! !min f (x)(3.12)57'i(x) 0 i = 1 ::: mx 2 G Rn (3.13)(3.14)* G { ! -, 8: , f 'i { ,". -, x0 2 G !-( ,"( f (x) ; f (x0) 'i (x) i = 1 ::: m :8 -! !! Rn , ! + !8 "!) !. + !!( x0 - G0 = fx 2 Rn j f (x) < f (x0)gGi = fx 2 Rn j 'i (x) 0g i = 1 : : : m G ! !* 8: ! :;0 = fe j e = (x ; x0 ) > 0 f (x) < f (x0 )g;i =(Rn 'i (x0) < 0fe j e = (x ; x0) > 0 'i(x) < 0g 'i(x0) = 0; = fe j e = (x ; x0 ) > 0 x 2 intGg: )+ !!( *!( {! ) ! Rn .&" 3.5 + h 2 Rn f x0, x 2 Rn f (x) ; f (x0) hh x ; x0i:- + *! ," f x0 !!) @f (x0) ! !) ,,"! f x0 . 3.11 # ;0 ;i i = 1 : : : m -;0 = fc0 j c0 = ;0h h 2 @f (x0) 0 0g;i = fci j ci = ;i h h 2 @'i(x0) i 0 i'i (x0) = 0g i = 1 : : : m:.
8 " " * -* ! *!!. !- !588. ) c0 2 ;0 : D*! 8 -*! + x, 8:+ ! f (x) < f (x0 ), + > 0 ! hc0 x ; x0 i 0:2 ; hc x ; x i 0:C!), 8* c0f (x0) 00!! f (x) <00! ! R2 -Y = f(y 1 y2) j 9x 2 Rn : y1 = hc0 x ; x0i y2 f (x) ; f (x0)g: ," f ) -! Y: D!! c0 2 ;0 - Y ! "!)!R2; = f 2 R2 j 1 < 0 2 < 0g: 3.2 : ( 2 R2 !(, + 2 R2; + y 2 Y (. 7) !1 1 + 22 1 y1 + 2 y 2:0. 7. -! Y , R2; 8:! *)59- y 1 = hc0 x ; x0 i y 2 = f (x) ; f (x0): D*! 8+x 2 Rn 1 < 0 2 < 0 1 1 + 2 2 1 hc0 x ; x0i + 2 (f (x) ; f (x0)):H - .) !, *! 1 0 2 0 +x 2 Rn1 hc0 x ; x0 i + 2(f (x) ; f (x0 )) 0:P c0 6= 0 / ! - ) +x 2 Rn ) 2 > 0: C!),f (x) ; f (x0) h;1 =2 c0 x ; x0 i + x 2 Rn : H !!, h = ;1 =2 c0 !- @f (x0): D! ! :8 x, + f (x) <f (x0), 1 > 0: /c0 = ;0h* 0 = 2 =1 > 0:!* !!!, ;i = fci j ci = ;i h h 2 @'i(x0 ) i 0g !, 'i (x0) = 0 i m.
#!, 'i (x0 ) < 0 ! ! ;i = f0g !;i = fci j ci = ;i h h 2 @'i (x0) i 0 i'i(x0 ) = 0g 8* i = 1 ::: m: 1! ! !!.&" 3.6 :L(x ) = f (x) + h '(x)i = f (x) +60mXi=1i'i (x)(3.15)L(x 0 ) = 0f (x) + h '(x)i = 0f (x) +mXi=1i 'i(x)(3.16) - ' - (3.12)-(3.14).&" 3.7 . (x0 0) x0 2 G 0 = (01 : : : 0m) 0 -, (x 2 G 0 L(x 0) L(x0 0) L(x0 ):! 3.7, ! ! (x0 0) ! ," 1!*!-! x ,! ! = 0 ! ,! ! x = x0 (.
8).0. 8. C! ! ," 1!*!-! . ; , (3.12)-(3.14) 1 , ' x0 2 G , 'i (x0) < 0 ( i = 1 : : : m:61P C(! , - +.( - 88 . I!, : *, (. 9). 3.9 (!-D!!) . 1- x0 2 G. 5 x0 | % (3.12)-(3.14) , ' 0i 0 i = 1 ::: m , (x0 0) - L.0. 9. . +). ) x0 | !) -. !! (3.12)-(3.14).
D*! \mi=0 ;i \ ; = G"*-8! :8 ! 8 ci 2 ;i i = 0 : : : mPc 2 ; !, c0 + c1 + ::: + cm + c = 0.C!), c = ; mi=0 ci . D), ! ;0 ;i i = 1 : : : m, !(* 3.11, , :8 ! 8 - 0i 0 i = 0 : : : m ! 8 hi i = 0 : : : m !, 0i 0 i = 0 : : : m0i 'i(x0) = 0 i = 1 ::: mh0 2 @f (x0) hi 2 @'i(x0 ) i = 1 ::: m62h h00 0+mXi=10i hi x ; x0i 0P 8* x G: 00h0 + mi=1 0i hi *! ," L(x 0 0) x0 :+! :L(x 00 0) ; L(x0 00 0) h00h0 +mXi=10i hi x ; x0 i 0 + x 2 G: * !! , +x 2 G !L(x 00 0) L(x0 00 0):D! ! x0 | !) . !! (3.12)-(3.14), 0 (x ) = 0 i = 1 : : : m 0f (x ) 0f (x ) +!00000Pm ' (xi ')i8*0.C!),i=1 i i 0L(x 00 0) L(x0 00 0) L(x0 00 ) 8+ x 2 G 0: P 00 6= 0 , / !!! 00, ! .
G!-, 00 6= 0. C(! !( x0 2 G !, 'i (x0) < 0 i = 1 ::: m:!! x = x0 ! L(x 00 0) L(x0 00 0) ! f (x0 ) 0000Xf (x0) + 0 ' (x0)mi=1i i! 00 = 0 ! 0i 0 'i(x0) < 0 i =1 : : : m ! , - 0i i = 1 : : : m - !8. H -, ! ! 0i i = 0 : : : m- ) -).G!). !-, :! ( (x0 0) ," 1!*!-! L(x ) ! !)x0 !! (3.12)-(3.14). 8 ( L(x0 ) = f (x0 ) +63mXi=1i'i(x0) f (x ) + X i 'i(x ) = L(x )m0 + 0: D*!mXi=100i=1i 'i(x0) mXi=1000i 'i(x0):(3.17)L!,P 'i(x0) 0 + i = 1 : : : m: ! mi=1 i 'i (x0) *!! + ! - "!)+ , ! (3.17). C!),mXi=10i 'i(x0) 0(3.18) x0 | P. !!. - !(3.17) = 0: D*! mi=1 0i 'i (x0) 0: #! (3.18), !mXi=10i 'i(x0) = 0:(3.19) !! L(x 0) L(x0 0) (3.19) , L(x0 0) = f (x0) f (x) +mXi=10i 'i (x) + x 2 G: P ! x .!! (3.12)-(3.14), f (x0 ) f (x) +mXi=1D! ! !!.640i 'i (x) f (x): 4.
$%& '& "0! . !! ! * ! ," f (x), !!( ! ! Rn : 0!!" ! !8 , + !)) x0 x1 : : : xk : : : 8:! 8f (x0) f (x1) ::: f (xk ) : : :: /+ !+ xk 8 ,xk+1 = xk + k pk * pk | !! !, k | ! .!*! ) /* !-!.!-(.( +!!( + +) +. " !! ! * (( +, k = 0 1 : : :kxk ; xk q kxk ; xk+1 + )8 *( *kxk ; xk qk kx ; xk0* x | ," f (x), ! q | ! !!,0 < q < 1.