7.8 Rosen’s Gradient Projection Method 407
Figure 7.9 Situation whenSi=0 and someλjare negative.
and this vector will be a nonzero vector in view of the new computations we have
made. The new approximationXi+ 1 is found as usual by using Eq. (7.68). At the
new pointXi+ 1 , a new constraint may become active (in Fig. 7.9, the constrai ntg 3
becomes active at the new pointXi+ 1 ) In such a case, the new active constraint.
has to be added to the set of active constraints to find the new projection matrix
atXi+ 1.
We shall now consider the computational details for computing the step lengthλi
in Eq. (7.68).
7.8.1 Determination of Step Length
The step lengthλiin Eq. (7.68) may be taken as the minimizing step lengthλ∗ialong
the directionSi, that is,
f(Xi+λ∗iSi) =min
λ
f (Xi+λSi) (7.74)
However, this minimizing step lengthλ∗i may give the point
Xi+ 1 =Xi+λ∗iSi
