14.7 Sensitivity of Optimum Solution to Problem Parameters 753∂X
∂p=
∂x 1
∂p
..
.
∂xn
∂p
,
∂λ
∂p=
∂λ 1
∂p
..
.
∂λq
∂p
(14.74)
The following can be noted in Eqs. (14.69):
1.Equations (14.69) denote (n+q) simultaneous equations in terms of the
required sensitivity derivatives,∂xi/∂p (i= 1 , 2 ,... , n) and ∂λj/∂p (j=
1 , 2 ,... , q). BothX∗andλ∗are assumed to be known in Eqs. (14.69). Ifλ∗
are not computed during the optimization process, they can be computed using
Eq. (7.263).
2.Equations (14.69) can be solved only if the system is nonsingular. One of the
requirements for this is that the active constraints be independent.
3.Second derivatives off andgjare required in computing the elements of [P]
anda.
4.If sensitivity derivatives are required with respect to several problem parameters
p 1 , p 2 ,... ,only the vectorsaandbneed to be computed for each case and the
system of Eqs. (14.69) can be solved efficiently using the techniques of solving
simultaneous equations with different right-hand-side vectors.Once Eqs. (14.69) are solved, the sensitivity of optimum objective value with respect
topcan be computed as
df (X)
dp=
∂f (X)
∂p+
∑ni= 1∂f (X)
∂xi∂xi
∂p(14.75)
The changes in the optimum values ofxiand fnecessary to satisfy the Kuhn–Tucker
conditions due to a changepin the problem parameter can be estimated as
xi=∂xi
∂pp, f=df
dpp (14.76)The changes in the values of Lagrange multiplierλjdue to pcan be estimated as
λj=∂λj
∂pp (14.77)Equation (14.77) can be used to determine whether an originally active constraint
becomes inactive due to the change,p. Since the value ofλjis zero for an inactive
constraint, we have
λj+ λj=λj+∂λj
∂pp= 0 (14.78)from which the value ofpnecessary to make thejth constraint inactive can be
found as
p= −λj
∂λj/∂p, j∈J 1 (14.79)