Engineering Optimization: Theory and Practice, Fourth Edition

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.