08midsol

# 08midsol - 1. Given the ﬁnear programming problem...

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Unformatted text preview: 1. Given the ﬁnear programming problem minimize z = w3x1 — :32 subject \$0 3:; + 322 ~§~ .733 m 2 4.1233 “3'” \$2 + 1‘4 3 4 \$12\$23\$3)x4 Z 0 £5 {:3 If? E (:1) Write the dual linear programming problem. {a a gig; 5‘;% 3 (‘0) Given that m" m (2/3A/3,0,0)T is an optimai mint-ions to the given mini» mizatian problem, use compiementazy Siackness to solve the dual problem‘ {ﬁg} gyfjgiéx {A} 23§§-§~ 3mg 3% 5% wréi-Ejg i “‘3 x2 + SE “’3 g; v; f G g _ ‘3 g; 3 a; {Egg if ” ‘L m w §gﬂga” ‘57 L»: -%« grmi 2. Let A be: an m x 71 matrix and b an m x 1 mattrix. Prove ‘zhat éhe set SE{IEIALE“—*bgq;20} REE? X Méririfé if §€%é5/ gafég _"M;_§ﬂ we Ema/E 333g 2;, Ski; Z"; #5 Egg: kip £5.39 it» i :: EL? M73: x a A, égmwwg—a) 3. Salve the foiiowmg linear pmgraxrmﬁng problem by the simplex method, starting with the basis {1:3, x4}, showing your work. minimize z 2 W331 + 2392 subjectto \$1+932+cc3 = 2 :51 +334 2 \$13323\$33\$4 2 0 5%” émii 93% \$115: 2" X3 3 “’1' )5; H ER \$1.; : 25”}; é g ’ ' w» m 2:? Ear}: 5c; ﬁég‘2e g A; 25; quf";{§39 22\$? ﬂ@/{ﬁ/% ; “ ("w ‘ Qi‘ai'ii {gigxg‘i w;f§y{?\ gdgm.\q(2’LW\ A _m. 3 {ER >5! “:2 \$1 §g i); if} 5 at “i *r 5*“ ﬁg? 4. Lei;- S m {2:Am m 67\$ _>_ D} and Eat f 2 (égﬁg, . . . \$71)? be a vector in 8. Suppose that p 2 (£011,102, , . . 1pm)1r is a nonzero vector such that A3) a 0 and pi = 0 if @1- : 0. Prove that both p and —p are feasible directions at E. if g; 5/; 5 ~ j“:- 5A5>w f 5. Let S be the set of vectors x = (\$1, \$2}? that satisfy the inequalities 2 3 0 2h “1‘” 332 3V 3:1 + 3552 IV IV \$11112 (a) Show that d =2 (1,1)? is a direcéion of unhoundedness for S, (b) Write :17 m (2,1)T as the sum of a scalar muitiple of d : (1,1)T and a. convex combination of extreme points of S. ...
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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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08midsol - 1. Given the ﬁnear programming problem...

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