466 Nonlinear Programming III: Constrained Optimization Techniques
test for the satisfaction of these conditions before taking a pointX as optimum.
Equations (2.73) can be written as∑j∈j 1λj∂gj
∂xi= −
∂f
∂xi, i= 1 , 2 ,... , n (7.261)whereJ 1 indicates the set of active constraints at the pointX.Ifgj 1 (X)=gj 2 (X)=
·· · =gjp( X)= 0 , Eqs. (7.261) can be expressed asG
n×p
λ
p× 1= F
n× 1(7.262)
whereG=
∂gj 1
∂x 1∂gj 2
∂x 1· · ·
∂gjp
∂x 1
∂gj 1
∂x 2∂gj 2
∂x 2· · ·
∂gjp
∂x 2
..
.
∂gj 1
∂xn∂gj 2
∂xn· · ·
∂gjp
∂xn X
λ=
λj 1
λj 2
..
.
λjp
and F=
−
∂f
∂x 1−∂f
∂x 2
..
.−
∂f
∂xn
XFrom Eqs. (7.262) we can obtain an expression forλasλ=(GTG)−^1 GTF (7.263)If all the components ofλ, given by Eq. (7.263) are positive, the Kuhn–Tucker
conditions will be satisfied. A major difficulty in applying Eq. (7.263) arises from the
fact that it is very difficult to ascertain which constraints are active at the pointX.
Since no constraint will have exactly the value of 0.0 at the pointXwhile working
on the computer, we have to take a constraintgjto be active whenever it satisifes the
relation|gj( X)|≤ε (7.264)whereεis a small number on the order of 10−^2 to 10−^6. Notice that Eq. (7.264)
assumesthat the constraints were originally normalized.