Biomedical Engineering Reference
In-Depth Information
variables
Z
Z
L
@Ǜ
i
D
@
@
@
D
1
f
i
C
1
Ǜ
i
;
u
1
C
2
.
ref
/:
(6.43)
Summarizing, the optimality system to be solved reads
8
<
K
X
u
C
b
r
u
C
u
C
u
3
D
Ǜ
i
f
i
in
State equations
:
i
D
1
u
D 0 on @
1
b
r
1
C
1
C 3
u
2
1
D
u
d in
1
D 0 on @
Adjoint equations
8
<
Z
1
1
Ǜ
i
D
1
f
i
i D 1;:::;K
Z
Optimality conditions
:
1
2
D
ref
C
1
u
(6.44)
This set of equations represents the so-called
Karush-Khun-Tucker (KKT) condi-
tions
[
69
].
In principle, this system provides the solution to the optimization problem in a
monolithic or “one-shot” fashion. In practice, the cases of interest when the system
can be solved directly are rare—in particular for nonlinear state problems, and we
need again to resort to iterative procedures.
Let a guess for ˛ and be given at the iteration j. Again, typically, we take
.0/
D
ref
. A reasonable iterative procedure reads as follows.
1.
Solve the state equations to compute
u
.j
C
1/
;
2.
Solve the adjoint problem to compute
.j
C
1/
1
and
.j
C
1/
2
.
3.
Update the control variables using the optimality conditions. In this example it is
natural to choose
Z
Z
1
1
1
2
Ǜ
.j
C
1/
i
.j
C
1
1
f
i
and
.j
C
1/
.j
C
1/
1
u
.j
C
1/
;
D
D
ref
C
until a convergence criterion is satisfied.
This procedure corresponds in fact to a fixed-step steepest descent method for
J
R
regarded as a function of the control variables. In fact, notice that the Lagrange
multiplier introduced here corresponds to introduced in the previous section.
With this perspective, Eq. (
6.38
) reads
D
J
R
DǛ
D 0
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