Engineering Optimization: Theory and Practice, Fourth Edition

Problems 115

2.63 Consider the following optimization problem:

Maximizef= −x 1 −x 2

subject to

x 12 +x 2 ≥ 2

4 ≤x 1 + 3 x 2

x 1 +x 24 ≤ 30

(a)Find whether the design vectorX= { 1 , 1 }Tsatisfies the Kuhn–Tucker conditions for

a constrained optimum.

(b)What are the values of the Lagrange multipliers at the given design vector?

2.64 Consider the following problem:

Maximizef (X)=x 12 +x 22 +x^23

subject to

x 1 +x 2 +x 3 ≥ 5

2 −x 2 x 3 ≤ 0

x 1 ≥ 0 , x 2 ≥ 0 , x 3 ≥ 2

Determine whether the Kuhn–Tucker conditions are satisfied at the following points:

X 1 =







3

2

3

2

2







, X 2 =







4

3

2

3

3







, X 3 =







2

1

2







2.65 Find a usable and feasible directionSat (a)X 1 = {− 1 , 5 }Tand (b)X 2 = { 2 , 3 }for the
following problem:

Minimizef (X)=(x 1 − 1 )^2 +(x 2 − 5 )^2

subject to

g 1 (X)= −x 12 +x 2 − 4 ≤ 0

g 2 (X)= −(x 1 − 2 )^2 +x 2 − 3 ≤ 0

2.66 Consider the following problem:

Maximizef=x^21 −x 2

subject to

26 ≥x 12 +x^22

x 1 +x 2 ≥ 6

x 1 ≥ 0