Biomedical Engineering Reference
In-Depth Information
that is exactly what we want to obtain when we are looking for a minimum of
J
R
.
As a matter of fact, the iterative algorithm introduced in the previous section to
minimize
J
R
is an iterative algorithm to solve the KKT conditions.
Sequential Quadratic Programming Algorithm
In contrast with what is done in the unconstrained approach considered so far, con-
strained algorithms try to compute the solution to the minimization problem through
the convergence of the state and parameter variables (
u
.j/
;Ǜ
.j/
/ simultaneously.
This approach can be very effective in presence of nonlinear constraints, as the
constraints need not to be solved at each iteration. In this section we consider one
of these methods, the sequential quadratic programming (SQP) method [
12
]which
consists of iteratively approximating the original problem with a quadratic problem
subject to linear constraints. Such quadratic problem is then solved using quadratic
programming (QP) algorithms. Assume that the problem is already discretized, and
let the vector
x
.j/
include both the state (
u
.j/
) and the parameter (˛
.j/
) vectors
u
.j/
˛
.j/
:
x
.j/
D
The Lagrangian functional of the problem
.
x
/>is
approximated at iteration j with the paraboloid tangent to the Lagrangian in
x
.j/
,
i.e.,
L
.
x
;/ D
J
R
.
x
/ < ;
F
L
x
;
.j/
L
1
2
ı
.j/;T
.
x
.j/
;
.j/
/ C L
.j/;T
x
ı
.j/
x
H
.j/
ı
.j
x
;
C
x
LJ
LJ
LJ
LJ
x
.j/
, ı
.j/
LJ
LJ
LJ
LJ
x
.j/
@
2
@
@
x
L
@
x
2
where L
.j/
x
.j/
,andH
.j/
D
D
x
D
is the
Hessian
x
x
matrix. Such approximation of the Lagrangian is minimized w.r.t. ı
.j/
x
, subject to
F
the linearization of the constraint
.
x
/ D 0
x
.j/
C F
.j/;T
x
ı
.j/
x
F
D 0:
(6.45)
LJ
LJ
LJ
LJ
x
.j/
. Exploiting the fact that F
.j/;
x
ı
.j/
@
@
x
where the matrix F
.j/
x
D
is constant w.r.t.
x
ı
.j/
x
because of (
6.45
), one can reformulate the quadratic programming problem as
1
2
ı
.j/;T
D argmin J
.j/;
x
ı
.j/
ı
.j/
x
H
.j/
ı
.j/
x
C
x
x
(6.46)
D
F
x
.j/
;
s.t. F
.j/;
x
ı
.j/
x
LJ
LJ
LJ
LJ
x
.j/
.Thevalue
x
.j
C
1/
is obtained as
@
J
R
@
x
where the column vector J
.j/
D
x
C ı
.j
x
, where the step length 2 .0;1 is chosen using a line
x
.j
C
1/
x
.j/
D
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