sfin - 1. Find all the local mieimizers of the function

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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 _ 2234-2224-5333 = 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.

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sfin - 1. Find all the local mieimizers of the function

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