2.4 Multivariable Optimization with Equality Constraints 81Necessary Conditions for a General Problem. The procedure indicated above can
be generalized to the case of a problem innvariables withmconstraints. In this case,
each constraint equationgj( X)= 0 ,j= 1 , 2 ,... , m, gives rise to a linear equation in
the variationsdxi, i= 1 , 2 ,... , n. Thus there will be in allmlinear equations inn
variations. Hence anymvariations can be expressed in terms of the remainingn−m
variations. These expressions can be used to express the differential of the objective
function,df, in terms of then−mindependent variations. By letting the coefficients
of the independent variations vanish in the equationdf=0, one obtains the necessary
conditions for the constrained optimum of the given function. These conditions can be
expressed as [2.6]
J
(
f, g 1 , g 2 ,... , gm
xk, x 1 , x 2 , x 3 ,... , xm)
=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
∂f
∂xk∂f
∂x 1∂f
∂x 2· · ·
∂f
∂xm
∂g 1
∂xk∂g 1
∂x 1∂g 1
∂x 2· · ·
∂g 1
∂xm
∂g 2
∂xk∂g 2
∂x 1∂g 2
∂x 2· · ·
∂g 2
∂xm
..
.
∂gm
∂xk∂gm
∂x 1∂gm
∂x 2· · ·
∂gm
∂xm∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
= 0 (2.26)
wherek=m+ 1 , m+ 2 ,... , n. It is to be noted that the variations of the firstmvari-
ables(dx 1 , dx 2 ,... , dxm) ave been expressed in terms of the variations of the remain-h
ingn−mvariables(dxm+ 1 , dxm+ 2 ,... , dxn) n deriving Eqs. (2.26). This implies thati
the following relation is satisfied:
J
(
g 1 , g 2 ,... , gm
x 1 , x 2 ,... , xm)
= 0 (2.27)
Then−mequations given by Eqs. (2.26) represent the necessary conditions for the
extremum off(X) under themequality constraints,gj( X)= 0 ,j= 1 , 2 ,... , m.
Example 2.8
Minimizef (Y)=^12 (y 12 +y^22 +y 32 +y 42 ) (E 1 )subjectto
g 1 (Y)=y 1 + 2 y 2 + 3 y 3 + 5 y 4 − 01 = 0 (E 2 )g 2 (Y)=y 1 + 2 y 2 + 5 y 3 + 6 y 4 − 51 = 0 (E 3 )SOLUTION This problem can be solved by applying the necessary conditions given
by Eqs. (2.26). Sincen=4 andm=2, we have to select two variables as independent
variables. First we show that any arbitrary set of variables cannot be chosen as indepen-
dent variables since the remaining (dependent) variables have to satisfy the condition
of Eq. (2.27).