08finsol - 1 Using Newton’s methoé ’50 film the stationary points of f(1171,$2 $133.3 — 222 235"if 4 starting Wiih in(1,1 calculate

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Unformatted text preview: 1. Using Newton’s methoé ’50 film}: the. stationary points of f(1171_,$2) :: $133.3 — 222%.; + 235] "if 4., starting Wiih in : (1,1)? calculate both 331 a-Eld V f (f; ), in fractioa form. an}: 2? m “g 1% gs; mg 3 g g» [j wgim: 5 Q “a; ‘3'"; f X? gig g E 34%; (If; “82;; igémfi: $- $15" '3: if- w g “'5 i H 2 Will“ M"? :3 ii” : wgfiéwékw; all §$2§ggf§2§mi 53? My "a “:2 m a; - 3x s g} sag“; {gig g V”? "g; 53"? iflgfiwng {Vii/M]; 23% yé WW “ _ a: , ’01 i big“ i} gigging“ {*"E’ EEC??? 4? E‘“ 5 m “:Ww 3"“:mg 3% S E“ E; i} a: {figg’g 5M5 M gas ‘7 {a J 5 “3; “a a a 2. Given the probiem Minimize fiat}? 33271?) x $1 subject to 2?] {L 322 + 3:3 m {3 2 2 2 3:] + 3:2 + 9:3 x 1 figsk‘giga) "Write the Lagrangian function £($1,$251£53,A1,/\2) of this probiem as a function g of five variabies. (b) Show that (Wx/a/Bixfé/fiai/Bial/x/B) is a stationary point. of the La— 7 grangian function. :75: im%¢*%§}(c) Show that (wx/{mi/SE V6761 is a regular point. ; ,za’xr . 7— _‘ _W;V 5‘; Qfl-kéw.§ gig? Egfiiflgfi/Afg R}; V: W/iggfi§?€£g?g§}mgw§{x“ “$22; “in” 3 g Wm , , E E E i m W Mm Emfiiwfifigkg ii”§ gait; ' . ‘ i ME", i “it £51 3;}? w g a. M g w s W g; f" g3 - aa— aa 1‘ v 2 it I if), “WM * fl z i: ‘f 2 i i ’ if»; a ngz ii,” } ff; 7 ii; iii 53:; g Emfifgg Egg 1??? i i 5" i if , 1 ad? Emgfi is: a a? s; 3. Prove the “Weak Duality Theorem: If 3: is a feasible solution to €116 linear programw ming problem: Minimize z x CTLI: Siibjeci; to A3? : b; :1: Z 0 and y is a feasible soiution to the (iuai probiem: Maximize fl) : {fry subjeci io Arty g C, then 5%: S 0T2: «w? "m: 5% ‘3‘ if: i; <3 o :31 a: W}; a}; £1” ., g," \ i v?" 53?: g: : $i§fi girwfoxwfii bgffhikme/Z: or g3 in W g a: to} 4: {SE 32 T: » (3T1 , x2) that minimizes the function 4. Solve the problem: Find .1“. f<£13i7$2) = 33:: W ~ 27} + 2$2 41 Showng your work. ] (31 0, ml)?" in the form :r: = p + q where p is in the 111111 space of A and q is in the range of AT. 11-1 [204 Am 5. Given the mairix express 1‘ ..§z%s§§§§§§s& QE: w L 6. Show that 2? 94);" is a. local minimum for the problem Minimize f(:51,:2:2) = W334 + 5172 subject. to SEQ-+2 1'2 323} ,2 3/1 1/\ IV by demonstrating tbsp: (3 / 2, 9 / 4F satisfies the sufficient conditions yresented in the course. é ? g a ‘ 3:? WE} z V i:‘ mi Z; > infigfii‘ZaU/S L22") 5” in, m x a} a , w 33:: a; 5x {a}; :: ngfiggwgiwgzwgwg fa 5AM; ? 1/; gm “5%”? ‘2“ 1:3? “7%?! } figwgfggggfig? Ev? jig g? \fé“ {XS }% j I “A I K : Q i. {WA& 5993 “’5”??? g a 3 7. Given the iinear programming problem Minimize 2 m —331 + 2162 + $3 sub‘éeet to ‘2. 2 G {3 I1+ $2 $1+4IE2 —— x3 EV IV W “’37; ‘i— 333 3:12:5127133 1V ; égéfi (a) Show that 35‘ I (0, Z, 2)T ie a feasible solution to the probiem and identify which [MD constraints are active and which are inactive at that point. r ifw Show that p x (2, —1, ~3)T is a. feasible direction at 36 x (8,1,2)? {4’ { mg Determifie the maximal step length a > 8 such that $+ap {emains feasible, Where i“ is“ jWE w x and p are as iii part (13). fig: Wig; (1) Find ail the feasibie diEections p :: (plypgypBF a}; g; m (9:132)? 5m gm e3§§ei>fli Mm $+%{§}w&:£ mm 3‘ (y. ' "‘ ~ ‘. ‘j__g{;3{,f‘vxfei{/r we wxfse Maggi»? figge flag/h"? ENE. W. 8. Find all the 20031 minimizere for the problem Minimize f(nt'1,mg,$3) m ~221332373 Subject to 2$1+$2+$3 = and. demonetraie that they satisfy the SUf‘ECiGBt conditions presented in the class. 3“"?{éfiiig Q; E ,2,” m 92; E W _ k i i“ K; 14;» g j fix, g‘if'? U {Si} N gig; 953%. _ ‘ mvcgljjg‘: “Kgxggfém “5% zigggefig . t g “a My w“ Qfigfigxz$aygt§ 35"" Xéz’kgé'xé é A‘ g s ’1“ /§”§W / “’5. {EL gg’ZC} 55/1“ I}??? “12% 9351:5313 Wig; “32' g; #2:} 5 5 5;; :g g 5:; e. 5; gfixé‘wg‘} (43m; £%§L¥%€Qj i532; £5; 3}; f ; zg g “W giE-Fii} {5;} t M e i g ‘ {$9 3 5:; :1 MI}? 5R1 gngégfiegggfl §~::; ggtwfi e. "a g b?) gm; ¥§§2§w§flgxx§ " 12”"? «2,3 5”“: IX? “Ki? ‘7: Egg 3i2€935WXg i3 '2”; E a she}??? x 52‘"% Hat; ix 3 ;‘ E; g? if? E M ;~ M e5}! V: g 3% §;’i/gg*§ wng 553;? g f Q i : f, g WW E; M £2}; E if: {5% é ‘2 jg { 9. For the problem: Minimize fix) subject :0 Act: 2 b and Z a basis mil} space matrix for A, prove that. if ZTVf = (37 then there exists a vector A)k (the vector of Lagrange multipliers) such that Vf m ATM. YOE may assume any other theorems ihat you need for the proof. :W‘hw} 5-: 1 > f“- * g a“. §/ 5, if _ Mam-£6”, {gig 33‘ Maj-=4} 5&9: ; éhhii '1“ 1} {gig} :3 gay}; agflfgé A4} 10. F ind all the local minimizers for ézhe problem Minimize flack 3:2) 2 ~£~ 2:135 + x1332 w 4:51 + 2:52 ~i» 1 subject to $1 2 32 (1) in the foliowing cases. g} Al} Ehree constraints are active (b) All three constraints are in&ctive. (c) Constraint (1) is active and consiraints (2) and (3) are iaactive. {Eg- ‘“" g&&fl£’¥;\i+ Egg" 5“??? “QXXH “Wk: 5%? m Egg} 2 . - »‘ 0 fl vg? £2i“3A”;23;;:2ng My? ¢%g%}awfig§g:§ j E; ii}; {a} r Mm :3: 3%} V-thgé$g~ ;€”%§;+&5?}§ 7% “gxg”‘” Q i {as}? j I? J" ’2 f :2? g "WW 3 (75‘ - ’ $1.355 :M}; V JL 5%} i 53 if A g} 7%; i we «if {5; “X.pr gait/)3 31”} m “ “3 §3%W%W§EA § 1 é§%3w%.} giwg fr W2} gggfrg “nil; 1g; Ems: E ...
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This note was uploaded on 01/18/2011 for the course MATH Math 164 taught by Professor Brown during the Spring '10 term at UCLA.

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08finsol - 1 Using Newton’s methoé ’50 film the stationary points of f(1171,$2 $133.3 — 222 235"if 4 starting Wiih in(1,1 calculate

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