Engineering Optimization: Theory and Practice, Fourth Edition

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.