2.4 Multivariable Optimization with Equality Constraints 87
Necessary Conditions for a General Problem. The equations derived above can be
extended to the case of a general problem withnvariables andmequality constraints:
Minimizef (X) (2.39)
subject to
gj( X)= 0 , j= 1 , 2 ,... , m
The Lagrange function,L, in this case is defined by introducing one Lagrange multiplier
λjfor each constraintgj( asX)
L(x 1 , x 2 ,... , xn, λ 1 , λ 2 ,... , λm)
=f(X)+λ 1 g 1 (X)+λ 2 g 2 ( X)+· · · +λmgm(X) (2.40)
By treatingLas a function of then+munknowns,x 1 , x 2 ,... , xn, λ 1 , λ 2 ,... , λm,
the necessary conditions for the extremum ofL, which also correspond to the solution
of the original problem stated in Eq. (2.39), are given by
∂L
∂xi
=
∂f
∂xi
+
∑m
j= 1
λj
∂gj
∂xi
= 0 , i= 1 , 2 ,... , n (2.41)
∂L
∂λj
=gj(X)= 0 , j= 1 , 2 ,... , m (2.42)
Equations (2.41) and (2.42) representn+mequations in terms of then+munknowns,
xiandλj. The solution of Eqs. (2.41) and (2.42) gives
X∗=
x∗ 1
x∗ 2
..
.
x∗n
and λ∗=
λ∗ 1
λ∗ 2
..
.
λ∗m
The vectorX∗corresponds to the relative constrained minimum off(X)(sufficient
conditions are to be verified) while the vectorλ∗provides the sensitivity information,
asdiscussed in the next subsection.
Sufficiency Conditions for a General Problem. A sufficient condition forf (X)to
have a constrained relative minimum atX∗is given by the following theorem.
Theorem 2.6 Sufficient Condition A sufficient condition forf (X)to have a relative
minimum atX∗is that the quadratic,Q,defined by
Q=
∑n
i= 1
∑n
j= 1
∂^2 L
∂xi∂xj
dxidxj (2.43)
evaluated atX=X∗must be positive definite for all values ofdXfor which the
constraints are satisfied.