Biomedical Engineering Reference
In-Depth Information
Uzawa algorithm
An optimal control problem with
K
constraints is a problem of the form of problem
(
16
) where we add constraints
0
f
i
g
i
K
.
For such problems, direct methods are particularly suited and the discretised optimal
control problem can be solved by the Uzawa algorithm.
We denote
F
i
(
t
,
x
(
t
)
,
u
(
t
))
d
t
+
(
T
,
x
(
T
))
≤
0for
i
=
1
,...,
N
l
0
f
i
g
i
(
u
)=
∑
(
t
l
,
x
u
(
t
l
)
,
u
(
t
l
)) +
(
T
,
x
u
(
T
))
and we introduce the
=
Lagrangian
K
i
=
1
λ
F
0
i
F
i
(
,
λ
)=
(
)+
(
)
,
L
u
u
u
where
is a vector with one coordinate by state constraint called a Lagrange
multiplier. At a given iterate
λ
(
u
k
,
λ
k
)
,wesolve
u
k
+
1
=
arg min
u
L
(
u
,
λ
k
)
by a nonconstrained optimisation algorithm, as is the gradient algorithm, and then
we compute
i
k
i
k
F
i
λ
=
max
(
0
,
λ
+
α
(
u
k
+
1
))
, ∀
i
∈{
1
,...,
K
}
+
1
where
is an appropriate step size. If the constraint is an equality constraint instead
of an inequality constraint, we accept nonpositive values for
α
λ
and we do not
perform the maximum against 0.
Basdevant et al. used the Uzawa algorithm in [
15
] to solve the problem of
minimising the number of cancer cells while maintaining the number of healthy
cells over a tolerability threshold. They modelled the cell population dynamics and
the action of the drug by a set of coupled differential equations.
In [
27
], we solved the problem of minimising the asymptotic growth rate of the
cancer cell population while keeping the asymptotic growth rate of the healthy cell
population over a prescribed threshold; see a sketch of the method and of its results
below in Sect.
7
. We modelled the cell population dynamics by a McKendrick model
physiologically controlled by a circadian clock, considering a phase-dependent drug
acting on transitions. We firstly discretised the problem and then solved it by using
a Uzawa algorithm with augmented Lagrangian. That is to say, we replaced the
Lagrangian by
K
i
=
1
(
max
(
0
,
λ
1
2
c
F
0
i
cF
i
2
i
2
(
,
λ
)=
(
)+
+
(
))
−
(
λ
)
)
L
c
u
u
u
Compared to the classical Lagrangian, the augmented Lagrangian has better conver-
gence and stability properties for a small computational cost.
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