2.4 Multivariable Optimization with Equality Constraints 85
difficult task and may be prohibitive for problems with more than three constraints.
Thus the method of constrained variation, although it appears to be simple in theory, is
very difficult to apply since the necessary conditions themselves involve evaluation of
determinants of orderm+1. This is the reason that the method of Lagrange multipliers,
discussed in the following section, is more commonly used to solve a multivariable
optimization problem with equality constraints.
2.4.3 Solution by the Method of Lagrange Multipliers
The basic features of the Lagrange multiplier method is given initially for a simple
problem of two variables with one constraint. The extension of the method to a general
problem ofnvariables withmconstraints is given later.
Problem with Two Variables and One Constraint. Consider the problem
Minimizef (x 1 , x 2 ) (2.31)
subject to
g(x 1 , x 2 )= 0
For this problem, the necessary condition for the existence of an extreme point at
X=X∗was found in Section 2.4.2 to be
(
∂f
∂x 1
−
∂f/∂x 2
∂g/∂x 2
∂g
∂x 1
)∣
∣
∣
∣
(x 1 ∗, x∗ 2 )
= 0 (2.32)
By defining a quantityλ, called theLagrange multiplier, as
λ= −
(
∂f/∂x 2
∂g/∂x 2
)∣
∣
∣
∣
(x∗ 1 , x∗ 2 )
(2.33)
Equation (2.32) can be expressed as
(
∂f
∂x 1
+λ
∂g
∂x 1
)∣
∣
∣
∣
(x∗ 1 , x 2 ∗)
= 0 (2.34)
and Eq. (2.33) can be written as
(
∂f
∂x 2
+λ
∂g
∂x 2
)∣
∣
∣
∣
(x∗ 1 , x 2 ∗)
= 0 (2.35)
In addition, the constraint equation has to be satisfied at the extreme point, that is,
g(x 1 , x 2 )|(x∗ 1 ,x 2 ∗)= 0 (2.36)
Thus Eqs. (2.34) to (2.36) represent the necessary conditions for the point(x∗ 1 , x∗ 2 ) ot
be an extreme point.
Notice that the partial derivative(∂g/∂x 2 )|(x 1 ∗, x∗ 2 )has to be nonzero to be able
todefineλby Eq. (2.33). This is because the variationdx 2 was expressed in terms
of dx 1 in the derivation of Eq. (2.32) [see Eq. (2.23)]. On the other hand, if we
