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Unformatted text preview: 1. Find all the local mieimizers of the function
f($13$2)=($1—§$2)2 + 333 W 31‘2 and demonstrate that they satisfy the sufﬁcieet conditions presented in the
class. 2. Show that (—2/11,6/11, —2/11)T is a local minimizer for the problem Minimize flml = “5131: $2; $3) = If? + 117% + 33%
subject to 2331+ $2+m3
~$1+3$2—$3 = 2 by demonstrating that (~2/11,6/11, ~2/11)T satisﬁes the sufﬁcient condim
tions presented in the course. 3. Using Newton’s method to ﬁnd the stationary points of
ﬁx) 2 f($1,:z:2) = 23‘;1 —~ 2291 — ﬁmg m $3 we start with 539 2 (mi /2, 1/2)? Calculate E1 in fraction form. 4. Show that (1,1)T is a local minimizer for the problem Minimize f($) = ﬁn, m2) = 21:? + 2x% — $§$2 subject to
33: “1“ 332 Z 2
33:2 2 331
$2 :3 2 by demonstrating that (1, If satisfies the sufﬁcient conditions presented in
the course.
5. Consider the canonical form linear programming problem
Minimize z m m ch
subject to Am 2 b, x 2 0 and its dual canonical form problem Maximize w = = My
subject to ATy g c, y 2 8. Prove that if x and y are feasible solutions to the given problem and its
duel, respeetjoreiyg tiles w(y) g You should assume say properites of
matrices that you need. Be sure to indicate how anti where you make use of
the facts that m 3 6 and y a O. 6. Show that (wt/W3, 131/6, “fl/6):? is a locai minimizer for the probiem Minimize f(3?) m f(z1,:tg,x3) m 3331 i— 332 + 323
subject to _ 223422245333 = 0
$§+$g+$§ = 1 by demonstrating that (1/2/3, 1/1/6, 41/6)?" satisﬁes the sufficient condi—
tions presented in the course. 7. Consider the problem Minimize ﬂxl, 932, $3) ﬂ :13?5 ~— 231532 + 13%
subject to I71 W$3 = 1
233142132 = 2 (a) Fizﬁ a basis null Space matrix Z for the constraint matrix A of this probiem.
(b) For at m (1,—1,1)T, ﬁnd 1) and A such that won) a Z2; + ATA 8. Show that (0, —1)T is a 10031 minimizer for the problem Minimize ﬁre) == f(:1:1,:c2) 2 225513 + $2
subject to 2 $2 3:3
IV iV by demonstrating that (0, 1)T satisﬁes the sufﬁcient conditions presenteé
in the course. 9. Consider the following linear programming problem. Minimize z = mxl —} 3532 + 334 .g. 335
subject to
$1+2$2+§5€3+x4+m5 : 1
“2xl ‘i— 3172 + 3:3 "i" $4 = 1 $1, $2, $3,334,$5 2 0 At a certain step in soiving this problem by the simplex method1 the basis
is {232, $4}. Determirie the next basis according to the rules of the simplex
method; showing your work in detail. 10. Prove that if 23* is a vector such that V f (33*) z 0 and ng is positive
deﬁnite, then 37* is a local minimizer for f ...
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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.
 Spring '10
 Brown

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