7.20 Augmented Lagrange Multiplier Method 463
7.20.3 Mixed Equality–Inequality-Constrained Problems
Consider the following general optimization problem:
Minimizef (X) (7.255)
subject to
gj( X)≤ 0 , j= 1 , 2 ,... , m (7.256)
hj( X)= 0 , j= 1 , 2 ,... , p (7.257)
This problem can be solved by combining the procedures of the two preceding sections.
The augmented Lagrangian function, in this case, is defined as
A(X,λ, rk) =f(X)+
∑m
j= 1
λjαj+
∑p
j= 1
λm+jhj(X)
+rk
∑m
j= 1
α^2 j+rk
∑p
j= 1
h^2 j(X) (7.258)
whereαjis given by Eq. (7.253). The solution of the problem stated in E qs. (7.255)
to (7.257) can be found by minimizing the functionA, defined by Eq. (7.258), as in
the case of equality-constrained problems using the update formula
λ(k+^1 )=λ(k)j + 2 rkmax
{
gj( X),−
λ(k)j
2 rk
}
, j= 1 , 2 ,... , m (7.259)
λ(km++j^1 )=λ(k)m+j+ 2 rkhj( X), j= 1 , 2 ,... , p (7.260)
The ALM method has several advantages. As stated earlier, the value ofrkneed not
be increased to infinity for convergence. The starting design vector,X(^1 ), need not
be feasible. Finally, it is possible to achievegj( X)= 0 andhj( X)= 0 precisely and
the nonzero values of the Lagrange multipliers (λj= ) identify the active contraints 0
automatically.
Example 7.12
Minimizef (X)= 6 x 12 + 4 x 1 x 2 + 3 x 22 (E 1 )
subjectto
h(X)=x 1 +x 2 − 5 = 0 (E 2 )
using the ALM method.
SOLUTION The augmented Lagrangian function can be constructed as
A(X, λ, rk)= 6 x^21 + 4 x 1 x 2 + 3 x^22 + λ(x 1 +x 2 − 5 )
+rk(x 1 +x 2 − 5 )^2 (E 3 )
