2.4 Multivariable Optimization with Equality Constraints 91
or
db=d
̃
g=
∑n
i= 1
∂
̃
g
∂xi
dxi (2.51)
Equation (2.49) can be rewritten as
∂f
∂xi
+λ
∂g
∂xi
=
∂f
∂xi
−λ
∂
̃
g
∂xi
= 0 (2.52)
or
∂
̃
g
∂xi
=
∂f /∂xi
λ
, i= 1 , 2 ,... , n (2.53)
Substituting Eq. (2.53) into Eq. (2.51), we obtain
db=
∑n
i= 1
1
λ
∂f
∂xi
dxi=
df
λ
(2.54)
since
df=
∑n
i= 1
∂f
∂xi
dxi (2.55)
Equation (2.54) gives
λ=
df
db
or λ∗=
df∗
db
(2.56)
or
df∗=λ∗db (2.57)
Thusλ∗denotes the sensitivity (or rate of change) offwith respect tobor the marginal
or incremental change inf∗with respect tobatx∗. In other words,λ∗indicates how
tightly the constraint is binding at the optimum point. Depending on the value ofλ∗
(positive, negative, or zero), the following physical meaning can be attributed toλ∗:
1 .λ∗> 0. In this case, a unit decrease inbis positively valued since one gets a
smaller minimum value of the objective functionf. In fact, the decrease inf∗
will be exactly equal toλ∗since df=λ∗( − 1 )=−λ∗< 0 .Henceλ∗may be
interpreted as the marginal gain (further reduction) inf∗due to the tightening
of the constraint. On the other hand, ifbis increased by 1 unit,f will also
increase to a new optimum level, with the amount of increase inf∗being
determined by the magnitude ofλ∗since df=λ∗( + 1 )> 0. In this case,λ∗
may be thought of as the marginal cost (increase) inf∗due to the relaxation
of the constraint.
2.λ∗<. Here a unit increase in 0 b is positively valued. This means that it
decreases the optimum value off. In this case the marginal gain (reduction)
inf∗due to a relaxation of the constraint by 1 unit is determined bythe value
ofλ∗as df∗=λ∗( + 1 )<0. Ifbis decreased by 1 unit, the marginal cost
(increase) inf∗by the tightening of the constraint isdf∗=λ∗( − 1 )> 0 since,
in this case, the minimum value of the objective function increases.