Page 248 - Numerical Methods for Chemical Engineering

Lagrangian methods for constrained optimization 237

where v is tangent to all equality constraint curves and all active inequality constraint
curves,
= 0 | = 0 (5.85)
v · ∇g i | x min v · ∇h j∈S A x min
Thus, we can move a differential distance in v and still satisfy all of the active constraints,

= 0
g i (x min + εv) ≈ g i (x min ) + εv · ∇g i | x min

= 0 (5.86)
h j∈S A (x min + εv) ≈ h j∈S A (x min ) + εv · ∇h j∈S A x min
Now, using (5.84) and (5.85), the change in cost function associated with this move is

2
= ε|v| (5.87)
F(x min + εv) − F(x min ) ≈ εv · ∇F| x min
Thus, to have a constrained minimum, v = 0, so that

= n e + n i (5.88)
∇F| x min λ i ∇g i | x min κ j ∇h j | x min
i=1 j=1
j∈S A (x min )
In (5.87) we considered only movement along the active constraint surfaces, but now we
require as well that F(x) cannot decrease whenever we move anywhere into the interior of
the feasible region. That is, we must not be able to decrease F(x) by moving away from
x min in any direction p that is tangent to each equality constraint curve and that points in a
direction where each active h j (x) is nondecreasing,

= 0 ≥ 0 (5.89)

p · ∇g i | x min p · ∇h j∈S A x min
The change in F(x) during the move x min → x min + εp must be nonnegative,
F(x min + εp) − F(x min )
0 ≤ ≈ p · ∇F| x min (5.90)
ε

Substituting from (5.88),
 
F(x min + εp) − F(x min ) n e n i
 
λ i ∇g i | + κ j ∇h j (5.91)
ε = p ·  x min x min 
i=1 j=1
j∈S A (x min )
= 0, only the second term in the square bracket contributes,
As p · ∇g i | x min
F(x min + εp) − F(x min ) n i
0 ≤ = κ j p · ∇h j | x min (5.92)
ε
j=1
j∈S A (x min )
Let us now write

|
p = n i c m ∇h m | x min + u u · ∇h m∈S A x min = 0 (5.93)
m=1
m∈S A (x min )