752 Practical Aspects of Optimization
14.7.1 Sensitivity Equations Using Kuhn–Tucker Conditions
The Kuhn–Tucker conditions satisfied at the constrained optimum designX∗are given
by [see Eqs. (2.73) and (2.74)]∂f (X)
∂xi+
∑
j∈J 1λj∂gj(X)
∂xi= 0 , i= 1 , 2 ,... , n (14.64)gj(X)= 0 , j∈J 1 (14.65)
λj> 0 , j∈J 1 (14.66)whereJ 1 is the set of active constraints and Eqs. (14.64) to (14.66) ar e valid with
X=X∗andλj=λ∗j. When a problem parameter changes by a small amount, we
assume that Eqs. (14.64) to (14.66) remain valid. Treatingf, gj, andX, λjas functions
of a typical problem parameterp, differentiation of Eqs. (14.64) and (14.65) with
respect topleads to∑nk= 1
∂
(^2) f (X)
∂xi∂xk
+
∑
j∈J 1λj∂^2 gj(X)
∂xi∂xk
∂xk
∂p+
∑
j∈J 1∂λj
∂p∂gj(X)
∂xi+
∂^2 f (X)
∂xi∂p+
∑
j∈J 1λj∂^2 gj(X)
∂xi∂p= 0 , i= 1 , 2 ,... , n (14.67)∂gj(X)
∂p+
∑ni= 1∂gj(X)
∂xi∂xi
∂p= 0 , j∈J 1 (14.68)Equations (14.67) and (14.68) can be expressed in matrix form as[
[P]n×n [Q]n×q
[Q]Tq×n [0]q×q]
∂X
∂pn× 1
∂λ
∂pq× 1
+
{
an× 1
bq× 1}
=
{
(^0) n× 1
(^0) q× 1
}
(14.69)
whereqdenotes the number of active constraints and the elements of the matrices and
vectors in Eq. (14.69) are given byPik=∂^2 f (X)
∂xi∂xk+
∑
j∈J 1λj∂^2 gj(X)
∂xi∂xk(14.70)
Qij=∂gj(X)
∂xi, j∈J 1 (14.71)ai=∂^2 f (X)
∂xi∂p+
∑
j∈J 1λj∂gj(X)
∂xi∂p(14.72)
bj=∂gj(X)
∂p, j∈J 1 (14.73)