sfinsol - w 1 Find all the local minimizerg of the function...

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Unformatted text preview: w 1. Find all the local minimizerg of the function 1 f(2;1, .752) : (5131 W :2- .L2 )2 + 13:3 w- 3:52 aria. demonstrate that they satisfy the sufficient condétéens presentefi in the 63388. 2:29 :3 5:} fin Hi. 6, \E '3 m 222 2‘ 22:4 2}; '3'“ ’ § “ i’ ; . 4 7 W} 5’ ‘ w Mi: ”‘- 2‘?! WM 52 , k j} , "v“: M” ”’2: (“.4 24 1 g X g; Q . (":1 w W ‘53 9% ,2 3) 5p f“ gig 313% 53/?“ m 2; w s g = in , E f i ; s/ 3 Q 5 V i - m ‘ .5 é 259,1} {2;} V j _ é gig/Q i L <3 5 3 sir" f ; _»~ r ”3 _ E g x: r g Q a 5s {Mg M g ; r1 “’5. fl»? E °“ g2 a“ 3 g, j 5%afl;{ M ”3 w? 2:52 {)4- § 3? , i 5 WW”? (:3; In ,7 g« \i g g 3 4 A f ”“5 5 T ‘5 ' 23f 2 2‘5 2g; 2’2 52 WE; “if? 2: L ' U“ § 5 kg? W -w i {7 éfi ‘2? W' i’ 32 {A f g ~ g V“- 5 -' £2222 {3? J 7,1}2 {ff/kg“, f; ; J g ‘ 5 - W5. g 5 £33 fig 53 {imgéfi Ema ,5 u ‘3, Show that (—2/11) 6/11, —2/11)T is a local minimize: for the problem Minimize flat) 2 f{:£1,x2, 3:3) 2 mg ~i~ $3 + $3 subject. to 0 2 2x1+ (£2 +$3 ”£1 "E‘ 3332 ~ $3 I! H by demonstrating that (—2/ 11,6/11, ——2 / 11V satisfies the sufficient conditions pre— Sented m the cauz‘se. EMA : (-i +~é_-.§:.. To j§>+égféf> (1:)“u—i§\_:;\ ....._.: ' '22] I; f / f9} Vim: 9% xii/31% 3 *2] P; “:[O "1 o. o - 0 02 a] Pufla’x‘fle Jflcfim’éw I0 a a 0 gTvflf’Q) Z :1 Wm, sing Newton’s method to find the stationary points of 3 2 m2$1W$'12’I22“1/' (”1/2, 1/2)T. Caicuiate i; in fraction foam. we start with 5130 if,“ m sfféax 3.35.53; aiw may , w izfx 3% Q 1;? _ 2D/ "\m Q. ‘55:. «f. M Q 3 % $321.3: Wm wig 3:} V yam. rfiiiiw k 3% 3w Hing n3» 5: VA? 3H? J ,VAz W W H? W ________ 5i, réfigéx §§§§§ Z M m M Q 5 r3551» M w M; fffff 13, Etc] 3!“ E W i Airfk. .11.. Ex Q 5, «35:53 1 W 4.? wz w; W \335} i xw z? i fl is A "5 4,23 U? «JV W? M. 1‘95} _______ gigv Nam is”; 7. x33; M 1.541% 335% EM ” «Em \EE;; 5 .W a}: 3 VA 3.1. 313 a“; Eigéiéig ,2? . 3%. m. l S? 33% ®§;§,§E,E}J h; a n V ,,,,,,,,, giggly 2 aaaaaaaaaaaaa 92/ a w? W fir 4,!va Exam; M ,,,,,,,,,,,,,,,,,,,, 21.3.» W Mzeézziiuieié 3R w §§§§§§§ 511V m w 4. Show thai (1, 1)? is a. local minimizer for {he probiem Minimize fix) 2 f(a:1,1:2) m :5? + 23:; -— 371322 subjec: to {fivjffimi 371+372‘ Z 2 is}: "X; “if“ figmtfi“ cagii’24a x2 2 31 {a} :2 ”fig’+ X; fw’ygémgé, :17!) < 2 ” ‘ ‘ '- : d W 353:2} 7-7 “Kg“! “3% {:5 <2 by demonstrating that (3. NT saiisfies the sufiicient conditions presented in the course. r; g ”m ,1; f? m 5 if f. g m 5” x 504% M 322 ‘ f; H 5W; ; g kg Tow r 3 g ” ‘ ; .4 o a 1 i _ g Hémwgg g is? ”’5 i 23 g“.— “e ‘15} r“? ~» ~ é-‘s y - J“; M, {:3 a W E ‘ g f ”5 g S E M E2“; £12," m g g 5 ”J g}: 2 A x f g a a, § } r?” ‘ "i E m ,3 w; fifitra; {MAE/51;} / fix , f , fig}; “"5 7'93 “5""? A w {’3 if] g; if E” g? f E - E g ’~ , e“ * s 3‘1; m » g g :g fiovxw‘fliéxiww S «m {g g i f/ g W; i 5. Cousider the canonical form linear programming probiem Minimize z : z($) :2“ CT}? subject to Am 2 b, a: 3 0 and its dual canonicai form problem Maximize w : 10(1)) : bTy su‘bject t0 .4713; g (:7 y 2 0. Prove that if x aad y are feasfioie soiutéons to tile given pmbiem and its {Ema}, re— spectively, then u{y) 5 3(1‘) You Should assume any properites of matrices that you need. Be sure to indicate how and Where you make use of the facts that :1: 2 {3 and 3! Z 0. 5"? k? ' a M f x A . {f g. 1/ 6}]- is a local minimizer for the problem 31:1 + :52 + :33 1/6, f($1;132afl73) subject to 6. Show that («~— 2/3: Minimize f(:r:) Mg} «2;; E 7M §§§§§§§ 3% «Kiss 2: M. «fa ”xx/flaw ZJ/ m5 am; .3 . Wy/{w «xii: “(252% A x E? En, {a a“ X»? a a. a 2,. a; f g Q”... w ,W, : as» Q 6... 5 i; 29. I . s w, m g 1: 79 : Ca ,. fa ,fié if: , \ 3:5: I... 7 m X5 \xfla by; «V g U WM, 2%.: wéfifiia _ is m a. fi 5% 3;; a, R ,, . m. G a m p 2, i % £5 X .9, 0 six“, E E K? M iv , . 53 57?: fl 2 O m {.51 RE? a Q \1 View, . 3m; 5, as w. x. a T; g, w. m mg: a, a E : E , ,,,,,,, $§ 77777 a E .4 a. g u fig? VA. Q my Q & cum/M if fwm 55 Wésiiwxw .v C3 : m2.” (mi. \1} EV M i m 4‘ l m grezfgiiiL VM: 9; gig & 7 J 7 g . X g 2: gig 5 E 3. it ,3 35.3%? m m m w .X 4552.312; X 3. g, Q 9 1A, w 3% . 2 ffék A, (M3 Q 72 a 2m. 3 3 23 mV} 1 g: ,3}! .M VW 9/”; fwgfg Jami x x ,6; :5 23% E 3 ,3 E r s an 5 f6 5% «w + + 1 w r x. Z X X f: 1?}. 5;: Q ,1“ :p 5% m. 2% z i 2, a. a? 3., a? a a. , Mg x, + + mm: M33; W R aw \zfi rm”, w m 3.5 {m K 1 3. 1... 5% Q g E, 1 a; 2 y, g 4.“, m r. x . .x E. W . if: I . a f 3 w .5. A? m,“ 3 3 ii A...» U a g 6 i W; y, kw a. M. W w. W g Q 3 5.. 5 WW a... g s g, g g 3...... E v a. a m .9 : ~ if . «(a \2! 3: a , k ,,,,,,,,,, 55;; h : xiv M HM .w . f A}; mi! Q 5%.. i, fi§e m . 3? /§ 4;. M M m 3 ,,,,,, A, W ,,,,, -. Mo % sfizhv 3 z 2? 7} a1 v W x352 a W: a, 23?» r m s 21!; . fins... a . a. .n zzzzz m M M VAN: .pz. ............ w 13.: 3k . {WW ”m c 7% ;;;;;;; g5 fl \ w , 5.. mm; X .H w ‘ .1 ,9 {b 1w . 2a ”A. I. imp“ g m. m t H _ ”xv/f; WWW; WML W O m fig \if... 13,; «:3 MW» § r!» {em w 7m i A E... 5. “ a m m d K i f. X as}, 3 m i g i n m g X 5 a} m. 2... x g 5%; . ,1 Q If; 2/... $13 :5, 5%? J :% 7., PL “”53. 122mg ”Xv waif; rim ”H35“? em 7. Consiqier the problem ‘ : n I 'r ‘ W 1 :3 , 2 Minimize f(3:1,232; 3:3) —— .131 —~ $3932 +~ 1:3 subject :0 (a) Find a basis 111111 space matrix Z for the constraint matrix A of this probiem. {1, m1, if“, find 3) am} A Each that (b) F0: :2: WM) = 22; + ATA fig: MW; 3 m.“ «in. a, A? 3; fifl Téaégegissifiiz Wig in Au, m AV .3 5 3p” .y ¥é5i§§z§§§f§z§zm a: :5 a. M 3 rigiifi; L w: .1332 3.! grafts; M iii} ii 3% may négfféis: m; £35... :2 s- 1m E» «5% Tigsggé Taggéaséj 3m 3: é : «a g: er§§§s @ 3a 3m, fiéfl 5 + 5. 4 . aim; a}; . Egg .2; am 3% as ERA. / 4: 1w 35. W 73, x195!) a? w M 25:; .3 3% a 3.. {hi Mi fir K/ 2 «am. 3 fizx e? 32 3U «13A: fl m" 3.5% 33A“: M ET 316 rji r M w m m m, W m W if; "yaw mg 2% £3 Ma {2 1 3w “2% WK {Mi REM, «\i/ «fan, 3%“ + Q 4p Asa, g + ,3 may“ 3% g sifl W \afl; W a”. f % I 3 RA; , MM My 1%,, Ni V? $ Q w E x . : r, _ W 3.. 9i \1... . _\ 9% M 6 gm; A: rye: a}? §§§§§ §&\\. 5.: iiaéik 8= Shaw that (0, AUT is a locai minimize}: for the probiem Minimize fix) : f(:{7;;:{52) 2 2:11:512 + :52 subject to . .5 5 f? 5 5}» wax% ‘E2 W 1V 7‘ ‘ w. __.£ 5. m Y ,, a ‘ , M mgtwé {$1311 ‘5: 52 ‘24:; v5 2:”? g: 535 by demonstrating that (07 W1)?“ satisfies the sufficient conditions presenteci in the course. {,2 Q; ”‘3 ‘2 1 f ,. g1: a g 5’“ :3: "f g g M (’1‘ 5/55 w A7 ) ‘ m ‘2 532/ “gm m w A 5 5 X. ‘5“ 5 f 3, ‘ 5 .33 3% it??? g5 5“ w”; €25 5 5 ‘9‘ 5 """ ' ’5 w w / gm“ If 2%,}; 1 :1 E m E 5 1k 25 ”f5” (3 I}. 3 K g E: 5w} 3;; ‘1 _ ”X 5V"f§\55>§5fljy5 gag 5 :3 32353553”; "5’53 g g M f 5w 1 5 w i w 5’“ 5 5 ‘ 5 l9“ 5 N j i. 55 j i, ‘ W m , fig :55 i / g "1“” f ””5 5*?” .7 L: ' .t 55 ”"35 ‘ 5 2’ 5 5 . , , _ a if“ 5:;wa : M; n . 2: g g; a a: 3 “'5 j 55’ 55 5555; :55" [5555» v 55 4:53; -.; 55 9 ”E < b: i I = f \5 EM k w”! 5’ y 5 a a g: 5:} f {:3 {N r- «and M; i; “é, : E 512’ K5 ‘ 5. l 3:: M\ MM 25 m “\f f \ W; 9. Consider the foliowing linear programming problem. Minimize z 2 ~37); + 3:233 -§« 334 + $5 subject to 1:; we 2% + 3173 + :54 m2" 375 :~. 1 —2I1+3I2+ 373—933;; I 1 I13$21$31$41$5 Z 0 At a certain step in solving this problem by the simpiex method? the basis is {$2,114}. Determine the next basis according to the rakes of the simplex method: showing your work in deiail. 5 5 - em mi" WW in wi§{'~??§§ ekg+a>2g€gnfié§2€3‘Xg’jfibif 5?; up? “33" 3“? M ,. y A? 3% 2o 2%? :3.» ”2i; M3 fa, W 13?”. M :2 s” 2:? 3;: £15- ; 3 A .5’: 3g {3“ f I 2ft" m w, e; £2 {7: Q 3%,» f i m “75“ g" ” a; e 10. Prove that K135 is a vector such that Vf($*) x G and Vaffim) is positive definite, then an is a local minimizer for fix). g“ (M; 5” 5 if i Q X rbgjéj’} Fig 3: j/ g; 2:: 5, V? hmfiwi 3%; ” {MW 5% in “M ”1W g i W”? 5 E E «if ”a 3M g $— 1»? 33W“? 3‘2 ‘33 If A; 13 5 i z 5" {5%} r ii {Kg} v?» 3"? v Mg} a»; h g {7 //2 g? ‘ f: :3: 359% \j éwik’xémg 394*“ a f i; E? 7:31;”; E my; Eggxgfl 3‘? V j; {igéflw ii \z , i x « % M g: «Q? E iM J” ; in "‘1 g“? \/&é(§)g2> wav {z w?” - > em £3“ng : g: 5E > “% $53? if %<§}f Er g, A: {E A L’ “ f w aw g r? {x MM “25%., MA” Q w: Ly? PR? 3‘} g; gri} {SW/{Xi {w £5? 3%th EW :fiii‘y’” 7;; {XE} a; ...
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