7.20 Augmented Lagrange Multiplier Method 459
Figure 7.19 Numerical integration procedure.
whereris the number of discrete values ofθ, and θis the uniform spacing between
the discrete values so that
θ 1 =θl, θ 2 =θ 1 + θ,
θ 3 =θ 1 + 2 θ,... , θr=θ 1 + (r− 1 ) θ=θu
Ifgj( X,θ )cannot be expressed as a closed-form function ofX, the derivative∂gj/∂xi
occurring in Eq. (7.236) has to be evaluated by using some formof a finite-difference
formula.
Exterior Penalty Function Method
φ(X, rk) =f(X)+rk
∑m
j= 1
[∫θu
θl
〈gj( X,θ )〉^2 dθ
]
(7.238)
The method of evaluatingφ(X, rk) ill be similar to that of the interior penalty functionw
method.
7.20 Augmented Lagrange Multiplier Method
7.20.1 Equality-Constrained Problems
The augmented Lagrange multiplier (ALM) method combines the Lagrange multiplier
and the penalty function methods. Consider the following equality-constrained problem:
Minimizef (X) (7.239)
