Environmental Engineering Reference
In-Depth Information
would satisfy the (global) variational problem (
2.27
)-(
2.29
). These parameters mod-
ulate the intensity of the basic release rates to fulfil the requirements for the nutrient
concentrations in all oil-contaminated zones
ʩ
i
. Such correction on the basic dis-
charge rates is needed because the nutrient discharged in one zone could reach the
other zones during the time interval
(
0
,
T
)
due to the processes of advection and
diffusion.
Substituting Eq. (
2.65
) in the variational problem (
2.27
)-(
2.29
) we obtain a
quadratic programming problem whose solution determines the optimal parameters
ʳ
i
, and hence, the optimal discharge rates at points
r
i
,
i
=
1
,...,
N
:
N
1
2
p
j
ʳ
2
j
minimize
m
(ʳ
1
,...,ʳ
N
)
=
(2.66)
j
=
1
N
subject to:
c
i
−
ʱ
i
≤
a
ij
ʳ
j
≤
c
i
+
ʲ
i
,
i
=
1
,...,
N
(2.67)
j
=
1
ʳ
j
≥
0
,
j
=
1
,...,
N
(2.68)
=
0
2
dt
and
a
ij
=
0
where
p
j
Q
j
(
Q
j
(
r
j
,
[
t
)
]
t
)
g
i
(
t
)
dt
,
i
,
j
=
1
,...,
N
.
The solution of the quadratic programming problem (
2.66
)-(
2.68
) exists because
the corresponding feasible space is a compact set in
N
and the objective function
m
is a continuous function of several real variables [
17
]. Besides, such a solution
is unique because
m
is also a strictly convex function and the feasibility space is a
convex set in
R
N
[
5
]. It is assumed here that the feasibility space is a non-empty set
due to the introduction of suitable (large enough) parameters
R
ʲ
i
.
The quadratic programming problem (
2.66
)-(
2.68
) can be solved using the
quad-
prog
routine of MATLAB as soon as the adjoint functions are determined. Regard to
this routine, we point out that, when the only constraints of the problem are the upper
and lower bounds of variables, i.e., no linear inequalities or equalities are specified,
the default
quadprog
algorithm is the large-scale method. Moreover, if the prob-
lem has only linear equalities, i.e., no upper and lower bounds or linear inequalities
are specified, the default quadprog algorithm is also the large-scale method. This
method is a subspace trust-region method based on the interior-reflective Newton
method described in Coleman and Li [
6
]. Each iteration involves the approximate
solution of a large linear system using the preconditioned conjugate gradient method
(PCG). Otherwise, medium-scale optimization is used, and
quadprog
uses an ac-
tive set method, which is also a projection method, similar to that described in Gill
et al. [
12
]. It finds an initial feasible solution by solving a linear programming prob-
lem [
25
,
51
]. Due to the structure of quadratic programming problem (
2.66
)-(
2.68
),
the second method of
quadprog
routine is applied in the examples.
ʱ
i
and
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