Engineering Optimization: Theory and Practice, Fourth Edition

7.5 Sequential Linear Programming 389

Figure 7.5 Graphical representation of the problem stated by Eq. (7.21).

subject to
c≤x≤d (7.22)

The optimum solution of this approximating LP problem can be seen to bex∗=c.
Next, we linearize the constraintg(x)about pointcand add it to the previous constraint
set. Thus the new LP problem becomes

Minimizef (x)=c 1 x (7.23a)

subject to
c≤x≤d (7.23b)

g(c)+

dg

dx

(c)(x−c)≤ 0 (7.23c)

The feasible region ofx, according to the constraints (7. 23 b)and( 7. 23 c), is given by
e≤x≤d(Fig. 7.6). The optimum solution of the approximating LP problem given
by Eqs. (7.23) can be seen to bex∗= e.Next, we linearize the constraintg(x)≤ 0
about the current solutionx∗= eand add it to the previous constraint set to obtain the
next approximating LP problem as

Minimizef (x)=c 1 x (7.24a)

subject to
c≤x≤d (7.24b)