Engineering Optimization: Theory and Practice, Fourth Edition

116 Classical Optimization Techniques

Determine whether the following search direction is usable, feasible, or both at the design

vectorX=

{ 5

1

}

:

S=

{

0

1

}

, S=

{

− 1

1

}

, S=

{

1

0

}

, S=

{

− 1

2

}

2.67 Consider the following problem:

Minimizef=x^31 − 6 x 12 + 11 x 1 +x 3

subject to

x^21 +x^22 −x 32 ≤ 0

4 −x^21 −x^22 −x 32 ≤ 0

xi≥ 0 , i= 1 , 2 , 3 , x 3 ≤ 5

Determine whether the following vector represents an optimum solution:

X=







0

√

2

√

2







2.68 Minimizef=x 12 + 2 x 22 + 3 x 32

subject to the constraints

g 1 =x 1 −x 2 − 2 x 3 ≤ 12

g 2 =x 1 + 2 x 2 − 3 x 3 ≤ 8

using Kuhn–Tucker conditions.

2.69 Minimizef (x 1 , x 2 )=(x 1 − 1 )^2 +(x 2 − 5 )^2

subject to

−x^21 +x 2 ≤ 4

−(x 1 − 2 )^2 +x 2 ≤ 3

by (a) the graphical method and (b) Kuhn–Tucker conditions.

2.70 Maximizef= 8 x 1 + 4 x 2 +x 1 x 2 −x 12 −x^22

subject to

2 x 1 + 3 x 2 ≤ 24

− 5 x 1 + 12 x 2 ≤ 24

x 2 ≤ 5

by applying Kuhn–Tucker conditions.