Page 252 - Numerical Methods for Chemical Engineering

Lagrangian methods for constrained optimization 241

(x) = 0 and S i is the set of inequality constraints,
S e is the set of equality constraints, c m∈S e
(x) ≥ 0. If µ is the vector of Lagrange multipliers that enforce the constraints, the
c m∈S i
constrained minimum satisfies

∇F − µ m ∇c m 0
m
= (5.114)
c(x) − s 0
Let us examine the Newton update to (5.114) from the current estimates of the constrained
[k]
minimum x [k] and the multipliers µ ,

2 [k]
T
(∇ L| x ) − C x [k] −∇F| x + C [k] µ [k]
T
[k]
[k]
= [k] (5.115)
[k]
C [k] 0 µ s − c
where
 T 
−− (∇c 1 ) −−
. 2 2 2
. (5.116)
 
.  ∇ L =∇ F − m µ m ∇ c m
C = 
−− (∇c n c ) T −−
Equation (5.115) is equivalent to the two coupled linear systems

[k] [k] T [k] [k] T [k]
∇ L [k] x − C µ =−∇F| x + (C ) µ
[k]
2
x
(5.117)
[k]
C x [k] = s − c [k]
[k] T
We can subtract (C ) µ [k] from the first of these linear systems to obtain
2 [k] [k] T [k+1]

(∇ L| x ) x − C µ =−∇F| x [k] (5.118)
[k]
[k]
where µ [k+1] = µ [k] + µ . Thus, (5.115) is equivalent to
T

2 [k]
[k]
(∇ L| x ) − C x −∇F| x [k]
[k]
= (5.119)
C [k] 0 µ [k+1] s − c [k]
Let us now consider the quadratic optimization problem
1
2
T
min p p (∇ L| x )p + ∇F| x · p
[k]
[k]
2
(5.120)
subject to C [k] p + c [k] − s = 0
The Lagrangian of this problem is
˜
1

2
T
L(p) = p (∇ L| x )p + ∇F| x · p − µ · C [k] p + c [k] − s (5.121)
[k]
[k]
2
˜
and the constrained minimum of (5.120) occurs at ∇L = 0,
2

(∇ L| x )p + ∇F| x − C [k] T µ = 0 (5.122)
[k]
[k]
which is equivalent to (5.118). Thus, we have the quadratic problem to compute x [k+1] =
x [k] + p,
1
2
T
min p p (∇ L| x )p + ∇F| x · p
[k]
[k]
2
subject to ∇c m | x · p + c [k] = 0 (5.123)
[k]
m m ∈ S e
∇c m | x · p + c [k] ≥ 0
[k]
m m ∈ S i