optimization

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

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

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document
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
Ask a homework question - tutors are online