solutions practice final ex problems

solutions practice final ex problems - 1. (30 points total)...

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Unformatted text preview: 1. (30 points total) Consider the linear prografinfing problem maximize 53:1 + 0n + 0333 + 1.0% at In + 1322 + 133 + 13:4 3 4 ‘— 111'31 + 2332 + 3173 + 31'4 S 1 8:31 — 6592 + 7333 + 2374 S 2 l 9131 + 10232 — 11.33 + 45174 S $1201$22Q$32013420 (a) (5 points) Write down the linear programming dual of (Let yi be the dual variable associated with the ith Iess—than—or—equal—to constraint of M'xrx LI" \‘l‘i K 13‘” 2‘ .-a:€”¥fi“'gt%kf .. 139;. Nb? v~\\6L%%E_ " “b 1' ...\7<\ elf), +3‘figt 3% H in 31in w em. mm WW ““ “i *3“ :32wa (6%” “(70+ 2:“?«2 ‘3“‘NijthQ (“w Mom “\Wcza A‘Qirtwfig 2 (c) (5 points) Explain why the vectors as": (4, —-1,0,0)T and y = (0,0,D,1) are n__ot ,Cornplementary for this pair of problems (P) and ‘rs 5‘ ‘r I a% XE «KBLfEFb easel QK\+[oxs_ull><3+LlKég-=tu> - l '— fun- 5;, w a a K17“: We “as lawsgy raises X-Léfio Gui lgfi‘l-Lbi—‘(ogg ngfleeQ (d) (15 points) You need to check whether the feasible solution for (P) given by 331=l,:32=l,1;3=D,:t4=0 is optimal. Use your answers to (a) and (b) to determine this; do not use the simplex method. State any results that you are using. On . (“fiery 1c. finewfl_ E amstQLwMfifj as flfiaiflia I 2(th Eb} “fivkafib 2. (30 points total) We are solving a linear programming problem of the form maximize c2: subject to Ar S b :L' 2 0, by the revised simplex method with explicit basis inverse The matrix A is 4 X 5. The objective function coefficients of the five decision variables are: CI = 40102 : 5163 z 50,64 = —20,cs = 32. The columns of coefficients Aj of the five decision variableskand the right—hand-side vector b are: A1 A2 A3 A4 14.5 b 1 0 2 —1 2 50 ' 1 2 l O 0 20 IO 1 9 0 0 205 O 0 1.5 l —1 5 Slack variables 2:5,m7,3:3,$g were added, and after two pivots of the revised simplex method the vector of ordered basic indices is B = (5, 1, 8, 9) and 0.5 —0.5 0 0 15 11_ 0 1 0 0 —+ 20 AB — 0 —10 1 0 b‘ 5 0.5 —0.5 0 1 20 Perform one iteration of the revised simplex method using the explicit A; Verify that your calculation of y is correct by checking that 53- = 0 for each of the basic indices j. Use the maximum Ej rule for column selection. Make sure to show how you updated 5 and A31. B : {3,13%a\ a gag: (arena) {:33 3; (3111.10) 0) s3 3. (12 points total) We are solving a large sparse linear programming maximization problem by the revised simplex method. For the next calculation we require the inner product yAg. (Vile have calculated 3; already.) ' x (a) (3 pants) WhiCh of the 3 major Steps 0f thé Simplex iteration are we in? Explain in ‘i a few words. _ I J ' ‘ ‘m mfiu» Q eluww flick/b71993, 3A3 15 cbWai claim ' C3 =2 513‘" 5 it: 4 (b) (9 points) The arrays VAL( ), ROI/W ) and FIRSTNZ( ) begin with 15 16 7 8 9 10 11 -21 13 41 -32 16\ 18 52 81 110 112 6 24 17 3 4 5 62 31 4.5 15 16 15 21 i: 1 2 VAL(i): ID —3 ROWU): 2 4 8 FIRSTNZ(I'): 1 7 12 1 -13 125 143 149\915 968 Explain 'as specifically as possible how yA3 would be computed. How many arith- - metic operations are performed in this calculation? I 4. 128 points total) Consider a. general linear programing problem (P) of the form: max ca," (P) 3.75. AI 5 b x 2 D where A is an m x n matrix. (a) (3 points) Write down the duel linear programming problem (b) (4 points) State in full detail the Weak Duality Theorem as it applies tO' such pairs (P) and (D)_ A , , mm A: ’ Wong \ l? ’Eiu M: l)? gwflufu 1) WWW; ‘\ {‘3 9;]; v . Q3 fl; 8 (c) (9 points) Prove the Weak Duality Theorem as it applies to such pairs (P) and “ M: ii? \ $5 fight: @2ng k Dr {1le fl” 5‘19 r‘\ i?” 'm “$3 1513‘“ my) “LEA; Fax: A »~ LC; ’\”:>»-%9¢w $3 :5“ P (d) (5 points) Suppose that our (P) and (D) are both feasible. State precisely the Strong Duality result for (P) and - I mm'tw x . \r\ M W _ can a». Cl?) it” Q” A @flic‘ilx 9 (e) (7 points) We discussed how the simplex method can be used to provide a construc- ative proof of the strong duality result above. In a few lines give an overview of that argument. \ C t) :93 ab am; )wffirg j “flew {Kr “SAM Madge worked max/am) CQVAUD deed/hugger; Menu: ...
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solutions practice final ex problems - 1. (30 points total)...

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