Engineering Optimization: Theory and Practice, Fourth Edition

Problems 481

7.4 Consider the tubular column described in Example 1.1. Starting from the design vector
(d= 8 .0 cm,t= 0 .4 cm), complete two steps of reflection, expansion, and/or contraction
of the complex method.

7.5 Consider the problem:

Minimizef (X)=x 1 −x 2

subject to

3 x 12 − 2 x 1 x 2 +x 22 − 1 ≤ 0

(a)Generate the approximating LP problem at the vector,X 1 =

{− 2

2

}

.

(b)Solve the approximating LP problem using graphical method and find whether the

resulting solution is feasible to the original problem.

7.6 Approximate the following optimization problem as(a)a quadratic programming problem,
and(b)a linear programming problem atX=

{ 1

− 2

}

.

Minimizef (X)= 2 x 13 + 15 x^22 − 8 x 1 x 2 + 15

subject to

x 12 +x 1 x 2 + 1 = 0

4 x 1 −x^22 ≤ 4

7.7 The problem of minimum volume design subject to stress constraints of the three-bar
truss shown in Fig. 7.21 can be stated as follows:

Minimizef (X)= 282. 8 x 1 + 100. 0 x 2

subject to

σ 1 −σ 0 =

20 (x 2 +

√

2 x 1 )

2 x 1 x 2 +

√

2 x 12

− 20 ≤ 0

−σ 3 −σ 0 =

20 x 2

2 x 1 x 2 +

√

2 x^21

− 20 ≤ 0

0 ≤xi≤ 0. 3 , i= 1 , 2

whereσiis the stress induced in memberi,σ 0 =20 the permissible stress,x 1 the area

of cross section of members 1 and 3, andx 2 the area of cross section of member 2.

Approximate the problem as a LP problem at (x 1 = 1 , x 2 =1).

7.8 Minimizef (X)=x 12 +x 22 −^6 x 1 −^8 x 2 +^10

subject to

4 x^21 +x^22 ≤ 16

3 x 1 + 5 x 2 ≤ 15

xi≥ 0 , i= 1 , 2

with the starting pointX 1 =

{ 1

1

}

. Using the cutting plane method, complete one step of
the process.