Problems 489Determine whether the solutionX=
√^0
√^2
2
is optimum by finding the values of the Lagrange multipliers.7.43 Determine whether the solution
X=
√^0
√^2
2
is optimum for the problem considered in Example 7.8 using a perturbation method with
xi= 0 .001,i= 1 , 2 ,3.7.44 The following results are obtained during the minimization of
f (X)= 9 − 8 x 1 − 6 x 2 − 4 x 3 + 2 x 12 + 2 x 22 +x^23 + 2 x 1 x 2 + 2 x 1 x 3
subject to
x 1 +x 2 + 2 x 3 ≤ 3
xi≥ 0 , i= 1 , 2 , 3
using the interior penalty function method:Starting point for
minimization of Unconstrained minimum
Value ofri φ (X, ri) ofφ (X, ri)=X∗i f (X∗i)=fi∗
1
0. 1
0. 1
0. 1
0. 8884
0. 7188
0. 7260
0.70720.01
0. 8884
0. 7188
0. 7260
1. 3313
0. 7539
0. 3710
0.15640.0001
1. 3313
0. 7539
0. 3710
1. 3478
0. 7720
0. 4293
0.1158Use an extrapolation technique to predict the optimum solution of the-problem using the
following relations:
(a) X(r) =A 0 +rA 1 ;f (r)=a 0 +ra 1
(b) X(r) =A 0 +r^1 /^2 A 1 ;f (r)=a 0 +r^1 /^2 a 1
Compare your results with the exact solutionX∗=
12
9
7
9
4
9
, fmin=^19