midterm_solution - Math 340 Midterm Duration 80 nlnlutes...

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Unformatted text preview: Math 340 Midterm Feb 13, 2014 Duration: 80 nlnlutes Netn'le' L i 0 Studei'it Number: This exam shouid have 12 pages. No textbooks, calculators, or other aids are allowed. There are 5 questions in this exam, with a total worth of 60 points. Problem 3. (10 points) Use the two-phase si1‘1’1plex method to find. all mortintizev's of the followil'lg iinear program and the optimal value of its objective function. (Remember to check if it is in standard form. ) maximize 51:] + 22:2 + 2:3 subject to 2:] --i-- {132 + 3:3 2 1 23:1 -i— 2332 + 51:3 5 Ki :1: j , 2:2, {1:3 2 0 “RM, Voting Mm swam 40m: {Wt-cm }(i JCle'i‘KS gt 1 M, w \(3 ‘1 “I 4”“ $51M; M3 1‘3 Mcyatiue, I M Enfh‘afl . .. "" i” L. W‘Ch‘wm% l‘s 7"{35‘45—‘9m' to :- [*KtWL'Ksfifi ‘Phoi- {‘0 6111:3933; M: ~11: mm m We {a ‘3 x; : meat—let” h - W1 51.}: -X "i Va 310 “mg; KL! Lflaw) ‘ M W ; ml+Kc+K1W3’“’"‘1 w x "" K" 1 WWW»— %\ W) KO Math 340 Page 2 out of 12 OQH‘Mag £9!” PPM-rue; E WEE-O a?) 6W??:fi0.é, LP 33 Wfbm ‘Pkacg; 3.x walacmrc uaflvfiales: atom: (We! CUJCMdJflg. K53 2‘ X1, K3, 1A1 WW MMM it": I” Klfl‘zjs .{«‘33": Kg: J‘x3vr2’w’i 2,; M + 2%,"? 343 [A WW 61’ WI 543,951? Mad- WW. % s: Uwrzgsuéwwlxwg 2: PM; 4W. ‘39, M {Mh‘d «flea/fible W‘Cf’aw? 40! Nam: l [‘32 x\=1»xu«g+iw g_awaswfi KL:I~%¢”K5%JH } qumfifi Kg“?- 1 *Ez‘fiq )(n, W Kg 3 1 ~H’v3 “mil” 3 XS. Lew,“ “ExHKL *3“! E~SZwKI~K3~k2KL§ HLW 3 l J? ii K3 ” Si is (919*?Wi okfc’n‘ouxmd) 33:“: Km: 1 “K‘fiéxgjiww’ COW“ air 7‘2. 1‘5 0; Soy: E—SL‘WX‘ MKS, Xagtims\+}it 20 [molds fittm) x2;2»§%20 %.Ost5H 90‘ ’M 9,917 6? MIL ogrmwi Wuhan; Ls ‘ k ‘ a o iio,g~§)t,1+i,o]‘ 0£t q} [\3 Bil-(1th 340 Page 3 out. of 13 Problem 2 (10 points) rThe following optimal ('lielaicn'mry has been obtained when solving 2L linear progrzn'nn'nng problen'i, say (P), in stal'ldarcl l01‘11'i2 rt: 2 x l. - 23 1 — :1: ,1 {113 Z 2 "l- (1.5 1 + .’ l3 :1 z r 2 w 2:134 a) [3 ptsl Find the orighml linear }.n‘ogra.11'1 b) [l ptsl W llfll‘. is the optinml value> say 2*, of the ('}l_)j0Cl1lV(-3 function of (P)? o) [3 plus] State the dual (D) of (P) and find an optin‘ml solution for (D). l l ( d [1 pts] Illustl‘flfiK-I the strong duality theormn on this exannple ( ( ( [2 pts] Does (P) have an optimal solution 33* such that (n; m 1/2? If yes, find Such a solution. If no, explain why not. (0, CO!) (P) Wu; “lwo orig;qu uwfa‘olts -. KHK; {Ludo glatll, UMMWWS. 5 >09, ‘Zq TO “M W TM'HaJZ. alfafiwarf‘ji Meal *3 “Ml 95*: *9 l” W— W”: VM‘C‘WS‘ éo‘ we Pilaf : v.4 alas, KL \emw) M: lwlliflxl x33; 3+1X¢“Kl “‘3 —-~ “"“1; 3 an Jle; .__.W.W,....._-—- imjfiull cil‘ch‘OaM? 61/ C?) Math 34-0 Page 4 out of 13 U") “M ammo, 1,? cg: “Mm Wax 33M 32.. w ¢§¢ . _ \ . M Opfimfi'g 936“ ,{for (D) ‘. \L’J'O; flint? (Mu/TC (192%.) Oak/QM t; {2,1qu +19 f (WWW) as at X“ . O“ WK:- gth ffl’» '5 2 C ) S'l‘I‘ot/L} Mafifib? ‘FmdA‘d—Cd '4’} W Thy, @gfiIn/w-fl iaflotfoms ear C?) m %‘W k}? Le) mch wtzO} iL-tlht') x Oétéi} I $4 \ 9% «‘— WIL : is, §UCMQ sngn‘ .L "2, Math 3110 Page 5 out of 13 Problem 3 (10 points) Cunsider the LP: IVIa‘xil'nize 2:231 + 582 subject. to .132 S 2:17] —}— :32 S 3,931 — (1:2 5 1, and, 331,372 2 U. wWrite the slack variables for this linear 1,)1‘og1‘a‘1'3'1, and write the dim} linear pl'ogmm and dial 3331.014 val‘ialgles. I-Izwing done tlmt, check that; 51:1 2 2, 1:2 1 is an optimal solution by “Sing the (:011‘11)1(21‘1‘1011ézal'y slzl.c:i<1'less theorel'n. (rm MK 1b!»pr 3km: VMWOWWEMF (l) M' X” m :> m 2 «WW W (9e) m «2,5;3 ‘1‘ K2; = " “a Xv’fifil “if” B”%‘"‘(1 \Cugl XS” {M‘K‘ufi‘gl Mn mews $ch (b) 543‘ (flzfi ‘33 21 EH : @Lflfi‘fls "" 2” Lg?) fll‘l‘fllwtb 17/) ‘jgw fjfi‘flf‘fig”! ‘5‘;ij ‘33 30 w WMWMWWW WWWWWM} F...._.._.__.. ’v‘ -. N \ k: WWN Kfz ‘ 3 “3:305 M ’0' WM CS 1 “M WWMQX (MM % WW N50: {HO “ab {33:501.} KN“) 7") W "\‘w mom W): .M % ,,\ $5.2» ' as» +fl$ 2:; K do \fllg'B/13,$3 ,.. [2 (ma ‘3 O) .. ,5 0 _ / aha W“ Ihle: mummjofiw‘m W Y‘ 35’— 63) Nam afiwfez :22 /; (swam, s: \ m / a ,3; 0 a . Thad” “M; W gmg“ Sol \{3 a b Mréém/vmb VIN/COALme Eb mgflfid epfimafl“ Math 340 Page 6 011?. of 13 l‘dath 3110 ’age 7 out of 13 Problem 4 (10 + 2 points) Suppose we have a standard form prii’nal linear program maximize ch s11i:>jeet to AX g b, x > (l, with 5 original variables :12], . . . ,m5 and Il- inequality constraints (in addition to the nonw negativity constraints (11,: > 0). The primal (.)l_)jeetive is maximize 2.1;) + 33:2 ~|~ 4:125 and the dual objective is ininirnize W 11} i 1" 3/2 -— 219's +1901. This is all the in formation are have regarding the underlying optin‘iization probien‘i. A pos— sibly inaccurate linear mograii‘i solver 1')1‘0('luees the following (,zandidates for primal/ dual feasible solutirn‘m. Primal Solution Candidati-i Dual Solution Candidate a (“71> :3? e 11,, w 1 1. "(lil,;..i_;ifl___ __ (0,00, 1,0) (—1,,0,e,1) (1,0, 1.2, 0) (1,4,11,10) (1,0,1,1,(_J) (2, :1, o, :1) (0, 31,2, fg) (1,5,1,2) (a) [2 pts] Specify preeisely What we know about. the vectors b and c, and the matrix A (such as its ('lin‘iension, the values of its entries etc.) (b) [ii pts] For each pair of solution eandidaiac‘s, detern’rine whether they can possibiy be 1.)rin‘1al feasible and dual feasible, resl'ieetively, at the same time. State clearly your argument for each ease and cite the relevant theorems. (o) [1 pts] W hieh of these pairs could be prinlai/dual optimal? In each case, deterniii‘ie the corresponding eptin’ial value of the primal objm‘live funetimi. Again, provide an argument. ((1) [5 pts] Now, consider the folh'nving pair of candidates: Primal Solution Cai'ididate Dual Solution Candidate _u_ (:13;_,...,.’L’5) mmmm mm (291:- - - 2194) (1,0,1,1,1/2) (0, £1, 0, 1)) Find a linear program with the above prin‘ial and dual objective functions such that the given pair is primal / dual optimal for it. Math 3110 Page 8 out. of 12 Ka)§'q ?%QM~+‘3 waskmvimfi Q Q arffind waal4\gg :iDFvfis q¢5ngw 0W i\v\{oovx{1v. :4 CT: [1(3)OIO(L{J *3 LOTSi"1:\i"1I{]' (whomu) (‘5, mt final ~£em349m (Momquv rims?) 00) 3.554" 13th "2 '. Wining. 053‘ %» :3 2, Wk WWW WM WM *5. Sam/wad prme Asian} 436M :5“:- Math 340 O Page 9 out of 12 M h-c i0?“ an. " at?) . I ' K‘@X‘3 XL! XS :k )(EQG )(q {@[email protected]@wz q- "PIWE MATHEW ‘ $.31; , Enguk um ‘ M ma); +9 co. 6:19 C“) _..._-— a‘q‘OKL ( an?“+Q1LX1le3¥3*0«NXqTChSXS 5" f. ‘ T i < QMX|+an1i -- +075)“: -— l 4—- aqut—Mszht *Ql‘fxs “4 l + CM— “ - ’ at“ Kt ‘ - . O (as ewe-A) Md 7‘1. =0 (b3 Cowple Slack-MO?) ' rs cad—«4M4; M {‘N‘A‘H‘ W ord“- ( Quil “16“; +QWKV 1' 6hng S. “l 0.1.IXI “113 {3+ QufiO-l fats“ xg-s I 0.31XHQ33X3 Mann.» 4 “wk $«L Qqflfi +0193 x1. 1' am )(q ’caHS'YS é ‘- M :DQQL ’{fiaSE'o-‘Cw : Similar 1‘0 above. , GUM OMSUVM} '3;:‘dg.=\3‘:‘dfi:0 I W W jfiumjs—M) Gum +Q21‘fi; 1' Q3t‘33 i' aqn'flq =1 2. ’ cm“ =§ - 2 0mm + QZL3L+ 037233 + anflq 2 3 -:::—]> an Ll " 5‘ . :0 0mm ‘1 61:39:." 0133 35 NIH; ‘fiI-l =0 w W? G23 Ll ‘5‘2‘53’3‘59 a -LI :0 (1.14pm "r 01m Ebi amt ‘33 4(qu 3‘4 :0 that I“ 01531 ’r 97»? ‘12 4' 4:533 -i 6M5“ 3L1 :' L! Gus-L4 = ‘1 Math 3&0 Page 12 out. of" 12 Chfidrs f Q\U( T “gait; “é "\ .L 1;: / 2 "“11 [ am “film "ram + {:Qfi g “2‘ ORLAQHQJ‘fi‘QLH 4f 551d”? S _,_.___,A._m—- MW”. (Tm, m: aux: —~1 mm WWW-“:0 03‘ c. “‘2 0L3?) 703% #QES‘L—O @ Cm 21 ; Qua “WNW” MW (D g @ QWOWQA MW, «of: camp mg. Tm w+m "@612, So mgwwfio. Wfl/{fimaflz‘a W“; is a“ L? Wm“ M cam Wmd/mwl [>er 0-, am agh‘maé, Sgt/M}wa 12 Math 3-410 Page 10 out of 12 Problem 5 (10 points) (21) [3 pins] Let (P) be a. linear program in standard fin'i‘n. Define precisely what it 111031113 for (P) to he lll’liIJOiJllClCXl. (b) [3 ptS) Cam (P) be 1111lJ<nui<lo<l ii' any ioasiblo solution :1: [2:] . .. rail] of (P) satisfies Eng] 5 1000 for all j r: 3., . . . ,n. Justify your answer. (Q) [413125] Suppose, this time, that the feasible sot for (l’) is unhournlmL lo, for any number L > 0, there exists; at feasible Solution :1: with at least one coordinate larger than L. Does this mean (P) is inihounded'? If yes, give a proof. if no, (:02'1st1‘not a. bti):1n<lod (P) with an inilxnnu‘loCl feasible sot. (a) M ('9‘) be 7am on wax an; sit. Axel; , me. We 3% W C?) rs Whowa’ficfi I“; (of W M70 , W emit: a aimsbe swam KM mm (lam ECKM) : CTKM >M» GO) NO. M ErCTK be M objecrm a} (1’). TM I?! S {Cit-m1Hail-mili-wicnnxai g [000(('C.(+iC;|+w-‘l‘icni) 50: tour W M 7(ooo(tc.1+m+lci|), m is m «(ea/1:519 smrw 4w which (5% yM. LC) Hag, T5, a @Mfwuawgte-. “PM (mime gm (3 cm ‘i a 10%“): mm , Kiwi Law“ 945' X‘EO r5 elm mmmded. Gui 1w. “2:20 om mm #:o i é‘o fw— Fromm “is. not Mbflmded. ii} ...
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