Optimization and Computational Fluid Dynamics - Department of ...

3 Mathematical Aspects of CFD-based Optimization 69
3.2.3 Multiple Set-point Optimization
Often, it is not enough to compute an optimal solution for one specific problem
setting. Rather, one is interested in computing solutions, which optimize
the performance of the process averaged over a range of process parameters
or scenarios. Alternatively, one might want to optimize the worst case, which
leads to a min-max formulation. This goal leads naturally to robust optimization
or working range optimization, which we denote in the form of multiple
set-point optimization (cf. [8, 9]). There, instead of problem (3.18, 3.19) we
formulate the problem
min
y1,y2,y3,p ω1f1(y1,p)+ω2f2(y2,p)+ω3f3(y3,p) (3.29)
s.t. c1(y1,p) = 0 (3.30)
c2(y2,p) = 0 (3.31)
c3(y3,p)=0. (3.32)
For the sake of simplicity, we have restricted the formulation above to a problem
with three set-points coupled via the objective, which is a weighted sum
of all set-point objectives (weights: ω1,ω2,ω3), and via the free optimization
variables p, which are the same for all set-points. The generalization to more
than three set-points and to additional equality and inequality constraints is
obvious. The corresponding Lagrangian in our example is
L(y1,y2,y3,p,λ1,λ2,λ3)=
3�
ωifi(yi,p)+
i=1
3�
i=1
λ ⊤ i ci(yi,p) . (3.33)
The approximate reduced SQP method (3.22) applied to this case can be
writteninthefollowingform
⎡
⎢
⎣
0 0 0 0 A⊤ 1 0 0
0 0 0 0 0 A⊤ 2 0
0 0 0 0 0 0 A⊤ 3
0 0 0 B c⊤ 1,p c⊤2,p c⊤3,p A1 0 0 c1,p 0 0 0
0 A2 0 c2,p 0 0 0
0 0 A3 c3,p 0 0 0
⎤⎛
⎞ ⎛
∆y1 −L
⎥⎜∆y2⎟
⎜
⎥⎜
⎟ ⎜
⎥⎜∆y3⎟
⎜
⎥⎜
⎟ ⎜
⎥⎜
⎥⎜
∆p ⎟ = ⎜
⎥⎜
⎥⎜∆λ1⎟
⎜
⎟ ⎜
⎦⎝∆λ2⎠
⎝
∆λ3
⊤ y1
−L⊤ y2
−L⊤ y3
−L⊤ ⎞
⎟
p ⎟ . (3.34)
−c1
⎟
−c2
⎠
−c3
We notice that the linear sub-problems involving matrices A⊤ i are to be solved
independently, and therefore trivially in parallel. The information from all
these parallel adjoint problems is collected in the reduced gradient
g =
3�
i=1
ωif ⊤ p +
3�
i=1
c ⊤ p λi .