Engineering Optimization: Theory and Practice, Fourth Edition

7.5 Sequential Linear Programming 391

Figure 7.7 Linearization of constraint aboute.

Example 7.1

Minimizef (x 1 , x 2 )=x 1 −x 2

subject to
g 1 (x 1 , x 2 )= 3 x^21 − 2 x 1 x 2 +x 22 − 1 ≤ 0

using the cutting plane method. Take the convergence limit in step 5 asε= 0 .02.
Note:This example was originally given by Kelly [7.4]. Since the constraint
boundary represents an ellipse, the problem is a convex programming problem. From
graphical representation, the optimum solution of the problem can be identified as
x 1 ∗= , 0 x 2 ∗= , and 1 fmin= − 1.

SOLUTION

Steps 1, 2, 3: Although we can start the solution from any initial pointX 1 , to avoid

the possible unbounded solution, we first take the bounds onx 1 andx 2

as− 2 ≤x 1 ≤ and 2 − 2 ≤x 2 ≤ and solve the following LP problem: 2

Minimizef=x 1 −x 2