Optimization and Computational Fluid Dynamics - Department of ...

66 H.G. Bock and V. Schulz
The approximation B of the Hessian of f is often performed by Quasi-Newton
update formulas as discussed in [29]. Step (1) of this algorithm involves two
costly operations which, however, can be performed in a highly modular
way. First, y(p k ) has to be computed, which means basically a full nonlinear
solution of the flow problem abbreviated by equation (3.19). Furthermore,
the corresponding gradient, ∇pf(y(p k ),p k ), has to be determined. One of the
most efficient methods for this purpose is the adjoint method which means
the solution of the linear adjoint problem, since for y k = y(p k )oneobtains
∇pf(y(p k ),p k )=fp(y k ,p k ) ⊤ + cp(y k ,p k ) ⊤ λ
where λ solves the adjoint problem
cy(y k ,p k ) ⊤ λ = −fy(y k ,p k ) ⊤ .
Assuming that the CFD-model (3.19) is solved by Newton’s method, one
might wonder whether it may be enough to perform only one Newton step per
optimization iteration in the algorithm above. This results in the algorithmic
steps
⎡ ⎤⎛
⎞ ⎛ ⎞
(1) solve ⎣
0 0 c ⊤ x
0 Bc ⊤ p
cx cp 0
⎦⎝
∆y
∆p⎠
= ⎝
λ
−f ⊤ y
−f ⊤ p
−c
(2) update (y k+1 ,p k+1 )=(y k ,p k )+τ · (∆y k+1 ,∆p k+1 )
⎠
This algorithm is called a reduced SQP algorithm. The local convergence can
be again of quadratic, super-linear or linear type [26, 34], depending on how
well B approximates the so-called reduced Hessian of the Lagrangian
0 Bc ⊤ p
cx cp 0
B ≈Lpp −Lpyc −1
y cp − (Lpyc −1
y cp) ⊤ + c ⊤ p c −⊤
y Lyyc −1
y cp .
The vector λ produced in each step of the reduced SQP algorithm converges
to the adjoint variable vector of problem (3.18, 3.19) at the solution. This
iteration again can be written in the form of a Newton-type method
⎡
0 0 c
⎣
⊤ ⎤⎛
x
⎦⎝
∆y
⎞ ⎛
−L
∆p⎠
= ⎝
∆λ
⊤ ⎞ ⎛
y y
⎠, ⎝
k+1
⎞ ⎛
y
⎠ = ⎝
k
⎞ ⎛
⎠ + τ · ⎝ ∆y
⎞
∆p⎠
. (3.21)
∆λ
−L ⊤ p
−c
p k+1
λ k+1
This iteration can be generalized to inexact linear solves with an approximate
matrix A ≈ cx such that the approximate reduced SQP iteration reads as
⎡
0 0 A
⎣
⊤
⎤⎛
⎦⎝
∆y
⎞ ⎛
−L
∆p⎠
= ⎝
∆λ
⊤ ⎞ ⎛
y
⎠ , ⎝ yk+1
⎞ ⎛
⎠ = ⎝ yk
⎞ ⎛
⎠ + τ · ⎝ ∆y
⎞
∆p⎠
. (3.22)
∆λ
0 Bc ⊤ p
Acp 0
−L ⊤ p
−c
p k+1
λ k+1
It is shown in [23] that in this case, the use of an approximation of the
consistently reduced Hessian, i.e.,
p k
λ k
p k
λ k