Engineering Optimization: Theory and Practice, Fourth Edition

482 Nonlinear Programming III: Constrained Optimization Techniques

7.9 Minimizef (X)= 9 x 12 + 6 x 22 +x^23 − 18 x 1 − 12 x 2 − 6 x 3 − 8

subject to

x 1 + 2 x 2 +x 3 ≤ 4

xi≥ 0 , i= 1 , 2 , 3

Using the starting pointX 1 = { 0 , 0 , 0 }T, complete one step of sequential linear program-

ming method.

7.10 Complete one cycle of the sequential linear programming method for the truss of

Section 7.22.1 using the starting point,X 1 =

{ 1

1

}

.

7.11 A flywheel is a large mass that can store energy during coasting of an engine and feed

it back to the drive when required. A solid disk-type flywheel is to be designed for an

engine to store maximum possible energy with the following specifications: maximum

permissible weight=150 lb, maximum permissible diameter (d)=25 in., maximum

rotational speed=3000 rpm, maximum allowable stress (σmax)= 20 ,000 psi, unit weight

(γ )= 0 .283 lb/in^3 , and Poisson’s ratio (ν)=0.3. The energy stored in the flywheel is

given by^12 I ω^2 , whereIis the mass moment of inertia andωis the angular velocity, and

the maximum tangential and radial stresses developed in the flywheel are given by

σt=σr=

γ ( 3 +ν)ω^2 d^2

8 g

wheregis the acceleration due to gravity anddthe diameter of the flywheel. The distortion

energy theory of failure is to be used, which leads to the stress constraint

σt^2 +τr^2 −σtσr≤σmax^2

Considering the diameter (d)and the width (w)as design variables, formulate the opti-

mization problem. Starting from (d=15 in.,w=2 in.), complete one iteration of the

SLP method.

7.12 Derive the necessary conditions of optimality and find the solution for the following

problem:

Minimizef (X)= 5 x 1 x 2

subject to

25 −x 12 −x 22 ≥ 0

7.13 Consider the following problem:

Minimizef=(x 1 − 5 )^2 +(x 2 − 5 )^2

subject to

x 1 + 2 x 2 ≤ 15

1 ≤xi≤ 10 , i= 1 , 2

Derive the conditions to be satisfied at the pointX=

{ 1

7

}

by the search directionS=

{s 1

s 2

}

if it is to be a usable feasible direction.