optimization - ECON 5 000 Part IV: Optimization Part IV:...

Info iconThis preview shows pages 1–39. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 16
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 18
Background image of page 19

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 20
Background image of page 21

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 22
Background image of page 23

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 24
Background image of page 25

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 26
Background image of page 27

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 28
Background image of page 29

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 30
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECON 5 000 Part IV: Optimization Part IV: Introduction to Optimization Optimization an 12‘: In order to introduce the language of optimization theory and to illustrate some of the difficulties that arise, consider in detail the one variable case. Let Df ch be the domain of f:D——>R1 Defn.: Suppose f is defined at x0 and that xoeDf. x0 is a critical point of f if fx(xo) = 0 or does not exist. Assume xoeDf. _Defn: f has a local maximum at x0 if f changes from an increasing to a decreasing function at x0. Defn: f has a local minimum at x0 if f changes from a decreasing to an increasing function at x0. Defn: f has a global maximum at x0 if f(x0)2f(x) V xeDf. Defn: f has a global minimum at x0 if f(xo)$f(x) \fxeDf. Result: If f has a local maximum or minimum at xoeDf and if Df is an open interval then x0 is a critical point of f. Result: If f has a local maximum or minimum at XOEDf and if Df is a closed interval [a,b] then x0 is a critical point of f or xo=a or xo=b. First Derivative Test: Suppose f(x) is differentiable in an open interval about x0 but not necessarily including x0. Let 5 be small positive number such that x0+5 and xo—S are both in this open interval. Then 9 If fx(x0+5)<0 and fx(xo—5)>0, f has a local maximum at x0. 6 If fx(x0+8)>0 and fx(x0—5)<0, f has a local minimum at x0. ECON 5000 Part IV: Optimization Second Derivative Test: Suppose fn(xo) exists and thm)=0. if fxx(xo) < 0, f has a local maximum at x0. if flm(xo) > O, f has a local minimum at xm if fmAxo) = 0, f may have a local maximum, a local minimum or neither at xm Defn: suppose that fmAxU) exists in an open neighbourhood of XO possibly not including x0. Then if 5 is a small number such that that xg+8 and xo—B are both in this open interval then x0 is a point of inflection of f if for all 3 sign(fxx(xo+3)) = - sign(fxx(xo-3) ) . That is, at x0, fgx(xg) = O or fxx does not exist but fxx changes sign at xm Result: if flm(xo) = O and n is the first integer such that dnf/dxn¢0 then 0 if n is an odd number, x0 is a point of inflection of f. 0 if n is an even number and d“f/dx“<0, then x0 is a local maximum of f. 9 if n is an even number and d“f/dx“>0, then x0 is a local minimum of f. Extreme value Theorem: If f is continuous on the (finite) closed interval [a,b] then f takes on a maximum and a minimum on [a,b]. The only candidates for theSe maxima and minima are the critical points of f in [a,b] and a and b. jg. _ 1mm "@524er _¢/¢m__¢égm____ ML*a‘W%fiL0éfi_M_fiMLL “64.5146: 4sz ._,_ __#Mm_wm_220._/w_1é£_ipt_w_xm_y_ fmaz’igtza) .5575) éféq ___ hxveg: LMLTJWQCM _-Z4LA__2¢fl,-JZ,WM_ 17:4,;— __fiCA-I_~__C - ,-__ _____w ML. ngégiwfw,égw4¢_}§eg_fn __. __ LEI- “-2232 .5 F_omatm a? _ __/3’C'1M—‘{ L 3&4 _._A__ ,, _._,7LM A; ___C; ,WA.__ZMAL.E(?FM)FM_L . __ ___M*_z.*¢__r=gg2flofi4_m _fl-_d "M fwdamwg fig ___#-_fi __ ,w , '___-_L7LLM;C_- :._z_,:j’c§d.w. _ _V#___ [/1 L: _Hg@/_fi;nacw__aw_. A 20¢) p ‘ ___.__V._ 35(ka "9H (so. .JLILJX—JE C - "flaw", __ éyWM 14w 21 A; I. 4A,} _— Oz —-—- :1 .L 5(1) W 7 L — flit) Hr. . “M N20,; % - ) /LMMI74¢_,1, x c—__C¢ gm: fuck,qu Z. _ Q—fi; ;Z:’/N jaw. f .— __ _ , .Lfiat____L _ _ __._A- .____ _ __Mgf_&awfl_~fl£M_wL—7&n;é£ggg:weia_w 7. __ _ __.@M.AL{«_1€CCJM. ,7 ___@:_E; A 6 __W_r _M_ZZM "am 99445 Zfifihfiég) : A. 24: [email protected]fiflwL%_figgugflfldfiaxfi;_ 'afig) 75 . 1‘0 fifl-gn/ZTV—s _______:.i _m_4cfi%,4 Mfaam_ __M_._J& a, MHZ/C. afivaCZEE‘ Zégubwu: M42 ' / -l ___f# _€ZL.M - 9 Wu: b @h%l_flifflm’ [email protected]__ 1‘- . ._5_,¢ —Lf____;.— [M5- W - L 4: E51"? " /o , Ego —- _z_____(§;_r_z,__ -_._..__. ‘ . -7 ..*__u_.. A__ .._ _- #KflH/md‘: / 4;, ._ k _ _“_M_M~C _‘gc7x:é_¢w9 A; 3” __ ._.._.!_éky_-_#¢ F .H___2§7—?C zéj‘é _) 7?: _.. . _-_.___,. 2‘14 x'yg, —_— 1, ,L, W 44; X Xa; _ .,_._,-Zffi4/~%JC»J __fl_ ; Eiwiéflgéiiéflé TZKM__C?(7:) ‘. ‘ x_._flwa_d%¢/_&XZJM ____ér_€__€_w_«_3a ..;___z€:___ Lat—Q _._;:c/L : =1 w Fflggz‘z/ ,afl fl}; m.-. [-714 I LflafL-/%M'mw:m, W? M. 74.591.44 ___._.._.__%__Q12<)__fl’ 26.x; zmwévm _M W’xil fl,_,_;_____;_:‘; _~ w 44:74,; m M; ,7 ‘_i;.:': ~ _fi _n__,zla_:u{:l«é Ha. _ fl. 4». _L..,,-__.. _ ‘ ' "_ "_'_M Q: wl;___ T- fiflfi;_ AVE-ma ELL g .,. "__14) éfm.’ 7 paw: fa I :7g-JM w W) ._ 16-95% fig xvi—4.450. a? 11.. _ 14;? 21’s /,éx mi.” czm+‘wa___. ‘ d D; ‘ {x 6AM] (311x) egg-g Fm *2 (if 0 Ar. 74ng‘ 22:5; m _J_.i'/r~.44“'2._'. rgdvhnv,.¢"‘¢% OJZL- .szzz... “(M M Aw. a. ’ ;-A_#____..._ ._ __1__L __7C )_7C _'£/";,_,Q§_M/Mt;:__ A»: @5130, _é/UL_ AflKQ' _:___._,£v.<— 41AM_6£__ _ _ - 4.x- ._,¢._ _ .. .. _ ._ ' _ wéwo-«Cy it“. _ $512..“ .2 c.4154. 07 AlgD=/¢! /‘L_¥Mpmm7.JuL/DWLMQ.PLKM flu”; 1;: me z’usamfguuvgraonrM C1)fi'D«éA-L¢M fat... HAILLWA (37 D <22” 5 cw MW 71:0) .1:- mm“. A... r’LM we}; u x’e 7) MM x2." -. “Lag. (fusing (we‘wzm: flaw) (<1) 21- Did/4"" .3 7;... (4.: AW) 7%.. 71.“ n “W2; (£5. hum); 14-- xéC031)>, 4-4- m QZMM—_ Ii flu» m. n. 71;. it; “Lt. Am AWL-brig... cmww 1.} fl..sz 4):). %,Agm4n Dru—a7 Cow-1‘14;- MIL—m)“. C39 ITL faatn-aflgr’égflr WWWLfik PA 6.4M flu- ZMPL (W 2:3? #214»— . (4) 13¢ rLUA’dgm/‘jf q,“ Edna—L- %‘M rig/3;, cw /‘¢ 7% (mam wryam—mm; fl-NZGL—m (I) j} flcmxmla [7" a “MM? 71‘ CM, rt’x #5 .44 7% jg”: ar—n'. aSiW a4 {a 9.qu ‘CLWL. mfréauI-fi wdfl‘rw fl 0) I? -)l? Ac. a'kw Cmfifimfi WC: - 7:12! 74; ff” "1 WW” ""“”""’“‘ flfw MAM-r4. ZFJxIW/y‘fio. .27 *Xébjfradflé‘)?“ "fa. ,5}; MAME-:21; m a, 9mm, ,‘1 :‘x'eb Miti- fiaqyfi?) =0 71;. ,! AMA. E ‘ g :nfimiii one x". A M/‘M 1 All-X 72...“. 13;. Ida-4...; ‘6 F renal! XeD,X5E,k’J 2%") > Rx). (é) xFéor-IQWM/zfizfmtfi *fibflfixghjffigh mjAWAflJ.“ x+ (N(e,x‘)) Mfg: Away) CD) fig“??- ‘CQO. HAM/m4...— {-J 4.01.4.1 mad/m1»... (‘0 x’ea’) p, ,. Zena—é “yaw mmghg FMJ 74. 15w) fr chi-L71. I‘L trad.“ g 149M147,” ,5.._:7 7’- x". 67W ) w: “my: ,Zx. 74 04%;; zu¢amy {.4 Cw}; .; ml? )2!) Mai. 2%.; m 5,11; ,4: f‘.’ 71”, (a) $1 x’ ,5. a, 51% 74, F “.1 mm. (£21.24?) CHM”; Mr A Quads» win [x 0.1% Jami: Mm__ (b) fnx'flwfim fly. (Eda-1 5mm; 7’4”;- X’xivafiflul.‘ fiféfig:fié. (my fl,‘ 675—6: S'uf ii it“... M 00%49 lira-175129.. [fife-[EIL-"g LL. ma. MNLWL’? Wk TDZJ XI" 54. (L..ng M 7"- 7%... Wm- W {(36) Sag/L; 1’: x69. 2% F 3) : - :a L‘:r,..Jn. (cc) x c (we 5; > FCC [WW-w"; 1.4}!an J /7..J_,,/ .[gc alt—lag firm-Lt;_ Pmi [\d NCE,X') A; d... filnv-X‘ {U :‘UCE ,XP) .4..— D, 1g 6 Zn Ac. a... Mé.‘+rM-y mac-1cm- M—«k 12?} 330A“. emf“ cad— I‘Lfi J-fl elVCfi,x’)_ fun-U- KF ’3 7w [7%) 22’ Mae-.4... 1‘4. («—4. 7’11: (n 7%») a Féxnm) - 3'41, 914:4. I TC?! [fl‘hfiML-re 6W) [7?- Z‘j/L'J’ EMMA“) Q) 150:" +541) = PM) + S'fiTgMFW) +09) Eff-0U. .—--§ (3 d4 {—30 g ‘ Suéus-fi (:2) “LA 0: ML: (3) o 2 5417—31»! #0”) 4- C(S) A; J70 5...! an)o “Ad... 0 3 «LT #(X') ?{$Cv‘-—1-wv-- le 7Cw1" X? I'M-HT Mfl Ac 3,44... (71:. Fl J fang“... ,znhéaéu. a. m (,mAEZ... M Aff- a/vflWmW flww,w N74; 4;... MAng Wu. 75 I} Ito-(a— mHI-Q-w— . FflLv‘fi’CO I {he—LE $14441»; /(x)=xL A... ruthru-L... x’ =0 64 Ilka) =0- #6,.) fix) :7;3 A“. [79:0 Aux x"=a A; W Lszmmwn‘a-ua (a!) 5L Foc’r 4.1L.“ 7&3’ a 1- ' ,9; )7 “Ma. ,4: _ . LiEW-r) - sLOL‘JvAi mum... A 74% gwfl W74 W (“Ax Me. two-uh? uufim’ M J W7 05%; dual-«Fr M ML... AH’?’ fhmr; Aux n . 47 WW, 7%».51/ 74m “(E/f: (i) Lmz area,» (1-way; (’ #779054 1E5 fiwé. (haw; M if Xxé D n WP 7Ze_/ fly. 7%. May: jVaJ‘F-(k’p—‘a w M, A; mam, U a. mum = fixvdfrwér) +£-¢~"#¢~F<x*M 7‘45 2. WM egg-f") 0 ac S‘L-Qa H <6 i dun-4.6.. (214%, for) ; fora—M) A] 0.5.49 bx f“: £00m 1:771; Masha-M7 5mm) <?./} ACME»: a: ngj Mai-094, 1* (It) a ,- D-‘u'raL,’ L711 911,. {.7 (Ito 4.1 fjof‘ Me an: {aw-L '- a 2 _. 4" ALMMHM 1 LS'oc) 72%, m Ag. 14 a H mm Mim- Ma iv"; 7/1 9W ,1:- mm WWG 5m: Wm"; 5“ EM £77m flfif aJT‘ X+5D ) '30 L1 Kr a. fawn-nu... g ]£_ firm—ffi] flow/2H Mow-47 7:96,... An ! MJW'7; 4AM x”. QM O) mid-twin. ' ' 2417‘; [mad—u— M 12 9-19- X 61’) 2 GK.) Que.) (CI) #9» F0") Aim-4 (WA a... (ti) ?—n4{(x') :9 AL. u. mama-7 mil..- mud/41.; Cd“) 534—415(3’) =0 a—J HM. 1£(k’).r~ /a§va-£ (W5) TL. x’AA-KnJ' ‘ 03f. gnu. Cayman—«(7 7m. mast-Hg, a rep , 7f 4...“ (awe MW“. at x” $41"!ch ‘ ' X’rfiv a544,“! é!) a.th M (3) {gdcmcw‘ mad M“ x “Hm; 75L {- omb. Ava”, x’pW; U q 1‘0 3d ML cwzqcm 4;”. mm»;— [Baa-v.4- 1% .997 slau- F 5,. fuflLMm flu. Aflwufléuh—L 77 u TMC-Fv-fllfinvk 2,414. skL-L; r4? 7%. [#4212 abrnfi-a-g’ g Xi’- (7£::V._,X‘r)V Xréb [Za— 6:... {u Orb—1.4.1 (was. A14 J LI;— MwaCa; on. A7 pagan-1...? (‘L jw f 34 _ 5+ 1‘65 _ __ 3; 256.2? > 1;; n _ _ *_1___ _____ ECP [U5 MWCMM , _ WW 7pm) _ 7 5f jaw-uh): 0L _ m d _. W ,MZQfC/éfl (Aflxfla; [MM/QM. Zz‘fl/owwf» a {ma _ M. ‘mwycm 74 )3-W’ ' 3t 2 KW (UV . Fm g; : JCIXHZD, ’4‘)? 711.2752. '1' - 24%. _/__5fl_2_]4.r,,fl;/érz$.‘7 5, 7’4”? 5 K, /L‘—- I, ( M“ éfi fl '_ 7. '4___¢.;¢wa%¢/%, " 44' , __ -” ‘ flj_.,[email protected]_.- hmzxzvsawvi - a : . M _ 74;, Wfi___:&flé_%m€_lé7flh ._f __ k W k, m _ a, 72? W W ’ 4 1&0) flflé 7 flan/J 6.»; fix@-;) ,4 #A‘AM #W 0’7”/)_2< Q70 I!‘ [email protected] MW ‘ 7 WééfioW/Q ,ufiALZ/M f4 QJ'M W mic )6/7‘5’ A; zip/LL L‘JJ‘fi W mm 6», JUM Ma a CH. M 74.7. Aim. Faiths/«14,. m M A» __ 7L K’é‘f gm. 7 7.51; cm 74: 722; J'.-/ ’4 MW 2 NEW? 7L 4% mung. «9T , , ' 9? __ fifflvafimmfla , a saw-,2; 4’7.Lc~flh~§/L I r. Fags ax 906'; o H - f: ‘72“.fi‘f/MMW7 W' _ - W- e lvéh _ 5% fix) . “2”” I777 — V F" , i a M__.-fzc . _ W _ .__;gf‘$ QVWV 7, ' ‘ ' -7/ . 72,73; A (M can/KW Zt’l. 7L M flaw we; dig? mu 2;, .40 741‘ M. Wm) 2;, W; on") , 7. _,,, Qkh—O ,. it"? __ _ , 7 _ r*fi;[:w_ (nave: M 166-541) X, 65,1 _ __ , 224‘ __ _ £4; af'C’??‘:7f‘:‘“‘:T.':_f_fl , 2!»; = 43m.) flea”) ’) fifixh '5, 7i _ figs/um 3 u 3 (4(2)) 7.. «r34 MIC/t {( 4. 7X“ 7 __ __01_fl{:n1f__3'bt—r_ __ ___--- _.. _ __ ___ _ ___ ____ _ _ ,_ _.___, _-___..- 77+.-.__‘_.__..._.. . . _T H_. .. __,._. __ - _.v.__..‘ _.___ _-___ _-_._@_::(__.£/;._¢Jw___Mflmfléw. _ x: f _a 74f ' __Dw‘Tv-v‘__ :_V-__¢L¢<1~_/2____g_ _ ‘ _ («alga-v I q 1‘ - I ' - I” ' ’ ' ' ‘ " L _ 7_.1__-__) , ___—40.496» ._£W__vfi.__¢c___f%éjrww: _ M 53:" .L f___.__-______'_ f: 7753) = Eifiilflic —;L can)" ” ‘ ":;—f ____ _m__LL:._:0 _C-f, n__ __ ___p“ , ___ __F Hfifl, _ fa 64k.on «.4 chhm' fifmawé.‘ “.4 Farm; but oil-90w —un-t-" f—Aié— . ZZ- flw..._ a; W. A M,.;,,;_j_z,“_;-cn.tw M ,An-éffir’ 7!“ W'Céfl-cw haiku-A, 72441.9»— C/t-yrm cfiffim> anAMWM by [-4.4% k‘cc‘1f’ ...n-\ 3 ‘4‘ WC“ K71; P “‘4 IQ") xezfl 94-4}? It: 560:) ta ( ;{) M— Z“ 20 {1-1, 91 fl 5:: ; 9_r-_ x,_ :0 no u C 37:; 3h gt. 39 , flt’gt :c L,“ H 76 _ . Za "I" ’1 J 1:. 3° t.=!,. m 77*— I‘Kw act's“: e@*,x’) flap/Luc- C mu 946de :4 3’4; fa «ML- Fa -/’/m;c..: #4.: m gm mm; wwéufmg ,4. Q/‘ué‘i/f (a) ram A mu. .1: #94 #4.: ago!) :10 ‘37:; ~ (3) 935 0,!) >5 ,6“ W MM awn/4rd. x; m... ’01.; _ I a , fit. a. I"'\ Q n C a f a. Hob-L. f.»- - 1:” - (9 CC) 143)? 0 fidfljurud H Ito.) J; k ‘L 335 05 X' Q!) 1th) ,rla W @fiugf {£943,611 fumi— CW ((4—- 15K ,Lcmm, r’ ,L a. 36444 “6%) ((0%. (LL-7' flat-4. own/t. Fk-M 9!. e K" 5"“ fiiQQfJL' {:I’MM XLILO Lin-3" fl w» Wfiiwmemxh -1 7"“ V flaw-5&5) h- ‘jJO‘Jffl’c 20. -S’.{. fix,rfixu. -.£ j-T7o 3!;an- 7" x, ;z,_ , a ‘l g: 1:11.. a" I‘flxI—chp ,zz_gm éo 30-)1‘xl_fl,rpcl:o x, _ afl £0 ,1!) Ina-mow» =° <53 I-mw-mnw ;‘”flLf-r’.x.—m~)=° ac! ; a )(i’) 2:17 o :1 3° Wm <53) 4 MAWufiqw 74:008., .lezh :21" - Lg) 1 :o {L}. Ari-La?) mick—.0 M ,f«_ (n .40) 1:5}; act—.9 $4wa A4“ 1;,xL2c-a- mu— ‘w‘lwhé. AIL 9745—;— can») :@)a)o) {#J': (14.; Ami} M,h W‘firh MALI rhmo'm'u. («Ha 1rd. Hem-— a—J 0.1.; 61494.1. 71L Swami eff-J1; kc; (3mm -— .r. .r: ,1 7 2H. 3?. 13,3; I . . r _ f-l—s-Jz-gm (In) m 2.500“) -. L1: 21. »o 2’9 " 2m. r 7 w z ,— ---- V— #___H_ E fl E mersw?“ _ gay ..- ‘ A _,_ , rfi.#.___\ .# ) A- ____\—a __ , ;,__fifi_._.,—-_-:;, .‘ _ W.” ____.¥ 7. ,_ 7.k__ . —* J” 71* W W 49 24a W M WVM éffx’ra, 4) “ha WW N_r___>»777 3711/ r f‘ I: :1 , ‘_ f. _. lflfifio AM} :41. f4; _ _‘ m A}? :2 £1 _ _ ,, m fl ._.-—- Lui— ofiaa 2a.. *“m_,_.:_2:€;__2£224MG ,. , 2:45 9x,_u_’a_x¢ u; _ ...
View Full Document

This note was uploaded on 11/20/2009 for the course ECON 5000 taught by Professor Smith during the Summer '09 term at York University.

Page1 / 39

optimization - ECON 5 000 Part IV: Optimization Part IV:...

This preview shows document pages 1 - 39. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online