Engineering Optimization: Theory and Practice, Fourth Edition

488 Nonlinear Programming III: Constrained Optimization Techniques

Figure 7.26 Two-bar truss subjected to a parametric load.

7.40 Solve the following optimization problem using the augmented Lagrange multiplier

method keepingrp=1 throughout the iterative process andλ(^1 )=0:

Minimizef=(x 1 − 1 )^2 +(x 2 − 2 )^2

subject to

−x 1 + 2 x 2 = 2

7.41 Consider the problem:

Minimizef=(x 1 − 1 )^2 +(x 2 − 5 )^2

subject to

x 1 +x 2 − 5 = 0

(a)Write the expression for the augmented Lagrange function withrp=1.

(b)Start withλ( 11 )=0 and perform two iterations.

(c)Findλ( 13 ).

7.42 Consider the optimization problem:

Minimizef=x 13 − 6 x 12 + 11 x 1 +x 3

subject to

x 12 +x 22 −x 32 ≤ 0 , 4 −x 12 −x 22 −x^23 ≤ 0 , x 3 ≤ 5 ,

xi≥ 0 , i= 1 , 2 , 3