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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## 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.

### Page1 / 3

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online