Unidimensional Search Methods
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1 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Uniiensionl Serch Methos Dr. José Ernesto Rys Sánchez 1 Outline Uniiensionl optiiztion proles Well-ehve n ly-ehve functions Multiol n uniol functions Methos for optiizing uniol functions Golen Section etho Fioncci etho Qurtic interpoltion etho Aville con in Mtl 2 1
2 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Uniiensionl Optiiztion Proles Mny ultiiensionl optiiztion strtegies require one-iensionl techniques to serch long soe fesile irection t ech itertion Given u: n n x n, when solving * x rg in ( x) x u we cn select t the i-th iterte x i serch irection i, n the next iterte x i+1 cn e foun y solving then x * rg inu( x x * i1 i i i ) rg inu( ) The ove prole is clle exct line serch i 3 Well-Behve n Bly-Behve Functions Well-ehve functions: continuous with continuous erivtives u( ) Bly-ehve functions: iscontinuous with iscontinuous erivtives u( ) 4 2
3 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Uniol n Multiol Functions Multiol functions: severl ini t the selecte intervl Uniol functions: only one iniu t the selecte intervl u( ) 5 Optiiztion Methos for Uniol Functions Intervl eliintion ethos Golen section etho Fioncci serch Interpoltion ethos Qurtic interpoltion Cic interpoltion Newton etho Secnt etho 6 3
4 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Intervl Eliintion Methos Assuing uniol intervl t the -th itertion, we cn lwys eliinte sintervl y evluting the function t 2 interior points u u u u Reucing the intervl If u If u u u the iniu lies in [, the iniu lies in [, ] ] 1, 1 1, 1 7 Golen Section Metho The interior points re syetriclly selecte The previous interior points re re-use t the next itertion The se reltive reuction is use t ech itertion [ (1 2)] ( 3 5) / 2 Since 0 < < 0.5, ( 3 5) /
5 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Golen Section Metho Greek Geoeters The golen rtio or golen proportion (wikipei.org) 9 Golen Section Algorith * = GolenSection(u,, ) u: ;,, * egin = 0 ; en ; ; ( 3 5) / 2 ( ) ; (1 )( ) u u( ) ; u u( ) repet until StoppingCriteri if u u 1 ; ; (1 )( ) u u ; u u( ) else 1 ; 1 en = + 1 en * ( ; ( ) 1 1 u u( ) ; u )/ 2 1 u 10 5
6 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Fioncci Metho The interior points re syetriclly selecte The previous interior points re re-use t the next itertion A ifferent reltive reuction is use t ech itertion (1 ) Fioncci Metho (cont) A sequence of nuers tht stisfy is the following 1 1 F F N N 1 FN F N F 1 F N 1 N 2 where F k is the k-th Fioncci nuer. The Fioncci sequence is Fk 1 Fk Fk 1 with F 1 0, F0 1 1, 2, 3, 5, 8,13, 21, 12 6
7 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Golen Section Metho vs Fioncci Metho The Fioncci etho yiels lrger intervl reuctions thn the Golen Section etho (higher rte of convergence) For very lrge nuer of itertions (N lrge), oth ethos chieve lost the se uncertinty intervl The Golen Section etho is preferre ecuse it oes not require to efine N in vnce 13 Qurtic Interpoltion Metho At the -th itertion it lso ssues n uniol intervl [, It fins n initil interior point,, such tht It fits qurtic polynoil to the function u() over the three previous points t ech itertion The iniu of the qurtic polynoil, n 2 of the 3 previous points re use for successive interpoltions Convergence is gurntee u( ) u( ) n u( ) u( ) ] 14 7
8 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Qurtic Interpoltion Illustrtion 15 Qurtic Interpoltion Illustrtion (cont) 16 8
9 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Qurtic Interpoltion Illustrtion (cont) u( ) Qurtic Interpoltion Illustrtion (cont)
10 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Qurtic Interpoltion Illustrtion (cont) Qurtic Interpoltion Illustrtion (cont)
11 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Qurtic Interpoltion Forul At the -th itertion, let c u u() u u() u c u(c) The iniizer of the qurtic,, is clculte using ( c ) u ( c ) ( ) uc 2 ( c) u ( c ) u ( ) u c (Bnler, 1997) 21 Qurtic Interpoltion Metho Next Points The next points re otine using If u n u u n u 1 u u u u 1 then then then then ,,,, ,,,, c c (Bnler, 1997)
12 Uniiensionl Serch Methos Dr. José Ernesto Rys-Sánchez Ferury 11, 2015 Qurtic Interpoltion Metho Next Points The next points re otine using If u n u u n u 1 u u u u 1 then then then then ,,,, ,,,, c c (Bnler, 1997) Aville Con in Mtl The stnr version of Mtl hs the following con for iniizing sclr uniiensionl functions: x = finn(fun,x1,x2) returns sclr x tht is locl iniizer in the intervl x1 x x2 of the sclr uniiensionl function whose ne is in string vrile fun Mtl eploys n lgorith se on the Golen Section n the qurtic interpoltion ethos; the etho is very efficient 24 12
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