Page 249 - Numerical Methods for Chemical Engineering

238 5 Numerical optimization

Then, (5.92) becomes

0 ≤ n i κ j n i c m (∇h m | x min · ∇h j | x min ) (5.94)
j=1 m=1
j∈S A (x min ) m∈S A (x min )
Defining the real-symmetric matrix with elements
(5.95)
mj = ∇h m | x min · ∇h j | x min = jm
the requirement (5.90) becomes
0 ≤ κ (A) · ( c) κ (A) = [κ j∈S A ] c = [c m∈S A ] (5.96)

If the active inequality constraint gradients are linearly independent, is positive-definite.
Now, by moving in the direction p, each active inequality constraint function must also be
nondecreasing,
(x min )
h j∈S A (x min + εp) − h j∈S A
0 ≤ = p · ∇h j∈S A x min
|
ε
= n i c m jm = ( c) j (5.97)
m=1
m∈S A (x min )
Thus, for the active inequality constraint functions to be nondecreasing we must have
( c) j ≥ 0, and (5.96) then requires that for the cost function to be also nondecreasing in
this direction, each active inequality constraint multiplier must be nonnegative,
≥ 0 (5.98)
κ j∈S A
What about the multipliers for the inactive inequality constraints?
Let us rewrite (5.88) as

= n e + n i (5.99)
∇F| x min λ i ∇g i | x min κ j ∇h j | x min
i=1 j=1
≥ 0 for the active
where κ j /∈S A = 0 for the inactive constraints with h j (x min ) > 0 and κ j∈S A
constraints with h j (x min ) = 0. We avoid treating the inactive and active constraints sepa-
rately if we require all constraints to satisfy
κ j ≥ 0 κ j h j = 0 (5.100)

The latter of these is known as the complementarity condition, and requires that whenever
a constraint is inactive with h j > 0, its Lagrange multiplier is zero, so that it does not
influence the local search for a minimum.
Combiningtheseresults,weidentifythefirst-orderoptimalityconditionsthataresatisfied
at a constrained minimum x min , known as the Karush–Kuhn–Tucker (KKT) conditions.We
define the Lagrangian as

L(x; λ, κ) = F(x) − n e λ i g i (x) − n i κ j h j (x) (5.101)
i=1 j=1