Problems 487Starting point for Unconstrained
Value of minimization of minimum of
k rk φ (X, rk) φ (X, rk)=X∗k f (X∗k)=fk∗
1 1 (−0.4597,−5.0) (0.2361,−0.5) 0.1295
2 10 (0.2361,−0.5) (0.8322,−0.05) 2.0001
Estimate the optimum solution,X∗andf∗, using a suitable extrapolation technique.7.35 The results obtained in an exterior penalty function method of solution for the optimization
problem stated in Problem 7.15 are given below:
r 1 = 0. 01 , X∗ 1 ={
− 0. 80975
− 50. 0}
, φ 1 ∗= − 24. 9650 , f 1 ∗= − 49. 9977r 2 = 1. 0 , X∗ 2 ={
0. 23607
− 0. 5}
, φ∗ 2 = 0. 9631 , f 2 ∗= 0. 1295Estimate the optimum design vector and optimum objective function using an extrapola-
tion method.7.36 The following results have been obtained during an exterior penalty function approach:
r 1 = 10 −^10 , X∗ 1 ={
0. 66
28. 6}r 2 = 10 −^9 , X∗ 2 ={
1. 57
18. 7}Find the optimum solution,X∗, using an extrapolation technique.7.37 The results obtained in a sequential unconstrained minimization technique (using an exte-
rior penalty function approach) from the starting pointX 1 =
{ 6. 0
30. 0}
arer 1 = 10 −^10 , X∗ 1 ={
0. 66
28. 6}
; r 2 = 10 −^9 , X∗ 2 ={
1. 57
18. 7}r 3 = 10 −^8 , X∗ 3 ={
1. 86
18. 8}Estimate the optimum solution using a suitable extrapolation technique.7.38 The two-bar truss shown in Fig. 7.26 is acted on by a varying load whose magnitude
is given byP (θ )=P 0 cos 2θ; 0 ◦≤θ≤ 360 ◦. The bars have a tubular section with
mean diameterdand wall thicknesst. UsingP 0 = 50 ,000 lb,σyield= 30 ,000 psi, and
E= 30 × 106 psi, formulate the problem as a parametric optimization problem for min-
imum volume design subject to buckling and yielding constraints. Assume the bars to be
pin connected for the purpose of buckling analysis. Indicate the procedure that can be
used for a graphical solution of the problem.
7.39 Minimizef (X)=(x 1 − 1 )^2 +(x 2 − 2 )^2
subject to
x 1 + 2 x 2 − 2 = 0
using the augmented Lagrange multiplier method with a fixed value ofrp=1. Use a
maximum of three iterations.