Engineering Optimization: Theory and Practice, Fourth Edition

92 Classical Optimization Techniques

3.λ∗=. In this case, any incremental change in 0 bhas absolutely no effect on the

optimum value offand hence the constraint will not be binding. This means

that the optimization off subject tog=0 leads to the same optimum point

X∗as with the unconstrained optimization off.

In economics and operations research, Lagrange multipliers are known asshadow prices

of the constraints since they indicate the changes in optimal value of the objective

function per unit change in the right-hand side of the equality constraints.

Example 2.11 Find the maximum of the functionf (X)= 2 x 1 +x 2 + 0 subject to 1

g(X)=x 1 + 2 x 22 = using the Lagrange multiplier method. Also find the effect of 3

changing the right-hand side of the constraint on the optimum value off.

SOLUTION The Lagrange function is given by

L(X, λ)= 2 x 1 +x 2 + 01 +λ( 3 −x 1 − 2 x^22 ) (E 1 )

The necessary conditions for the solution of the problem are

∂L

∂x 1

= 2 −λ= 0

∂L

∂x 2

= 1 − 4 λx 2 = 0

∂L

∂λ

= 3 −x 1 − 2 x^22 = 0

(E 2 )

The solution of Eqs.(E 2 ) si

X∗=

{

x 1 ∗

x 2 ∗

}

=

{

2. 97

0. 13

}

λ∗= 2. 0

(E 3 )

The application of the sufficiency condition of Eq. (2.44) yields

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

L 11 − z L 12 g 11

L 21 L 22 − zg 12

g 11 g 12 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

= 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

−z 0 − 1

0 − 4 λ−z− 4 x 2

− 1 − 4 x 2 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

−z 0 − 1

0 − 8 −z − 0. 52

− 1 − 0 .52 0

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

= 0

0. 2704 z+ 8 +z= 0

z= − 6. 2972

HenceX∗will be a maximum off withf∗= f(X∗) = 16 .07.