Engineering Optimization: Theory and Practice, Fourth Edition

796 Answers to Selected Problems

2.13positive semidefinite 2.15positive definite

2.17negative definite 2.19indefinite

2.21x∗ 1 = 0. 2 507 m, x∗ 2 = 5. 0879 × 10 −^3 m

2 .23a= 328 , b= − 376 2.26x∗= 72 , y∗= 12

2.27x∗= 001 2.28(a)minimum (b)minimum

(c)saddle point (d)none 2.30saddle point at (0, 0)

2.33dx 1 = rbitrarya , dx 2 = 0 2 .36radius= 2 r/ 3 , length=h/ 3

2.38length=(a^2 /^3 +b^2 /^3 )^3 /^2 2.40h∗=

(

4 V

π

) 1 / 3

, r∗=

h∗

2

2.41x∗ 1 =x 3 ∗= (S/ 3 )^1 /^2 , x∗ 2 = (S/ 12 )^1 /^2

2.43d∗=^16 { a( +b)−

√

a^2 − ab+b^2 } .47 2 200 mm×250 mm

2.50 X∗= { 4 , 2 , 2 } 2 .53 198 .43 ft× 113 .39 ft

2.55(a)fn∗ew= 51 π (b)fn∗ew 8 = 1 π 2.57(a)f∗= 1 / 3

(b)f∗= − 1 / 9 2 .61 X 2 is local minimum

2 .63(a)Kuhn–Tucker conditions satisfied

(b)λ 1 = 0. 4 ,λ 2 = 0. 2 ,λ 3 = 0 2 .65(a) S= { 1 ,− 3 } (b)none

2.67optimum 2.69x 1 ∗=^34 , x 2 ∗= 4169 2.73 convex 2.75none optimum

CHAPTER 3

3.3x 1 = 1 , x 2 = 2 , x 3 = 3 3 .5x 1 = 2 , x 2 = 4 ,x 3 = 6

3 .7x∗ 1 = 1 / 3 , x 2 ∗= 4 / 3 3 .9x∗ 1 = 225 , x 2 ∗= (^115)
3.12x∗= 3113 , y∗= 3112 3.15x∗= 5131 , y∗= 1131
3.17 all points on line joining (2, 10) and (7.4286, 15.4286)
3.18x∗= 01 , y∗= 81 3.20x∗= 9 / 7 , y∗= 04 / 7
3.23x∗= 6 , y∗= 1 3 .25x∗= 6 ,y∗= 0
3 .27x∗= 57 / 8 , y∗= 72 / 8 3.29x∗= 3 , y∗= − 2. 5
3 .31x∗= 4 , y∗= 0 3 .33unbounded 3.35x∗= 4 / 7 ,y∗= 03 / 7
3.37x∗= 63 / 7 , y∗= 51 / 7 3.39x∗= 61 / 5 , y∗= 1 / 5
3 .41infeasible 3.43unbounded
3.48x∗ 1 = 0003. 0 , x 2 ∗= 164. 7 , x∗ 3 = 2001. 0
3.50x∗ 1 ( arleyb )= 40 , x∗ 2 =x 3 ∗=x 4 ∗= 0 ,x 5 ∗( easedl )= 160