Optimization and Computational Fluid Dynamics - Department of ...

3 Mathematical Aspects of CFD-based Optimization 65
If g is symmetric and positive definite on the null space of A,thisisequivalent
to the linear quadratic problem
min 1/2∆z ⊤ G∆z+ � ∇f k − (A − cz(z k )) ⊤ λ k�⊤ ∆z (3.14)
s.t. A∆z + c(z k ) = 0 (3.15)
where the adjoint variable for Eq. (3.15) is λ k+1 . In this way, accurate evaluations
of the matrices cz or H can be avoided. Nevertheless, the matrix cz
appears in the objective of the quadratic program (QP), however, only in the
form of a matrix-vector product with λ k , which can be efficiently realized
such as in the adjoint mode of automatic differentiation [17].
This formulation is generalizable to inequality constraints, cf. [6, 7], e.g.,
of the form c(z) ≥ 0 in (3.10), which leads to linear-quadratic sub-problems
of the form
min 1/2∆z ⊤ G∆z+ � ∇f k − (A − cz(z k )) ⊤ λ k�⊤ ∆z (3.16)
s.t. A∆z + c(z k ) ≥ 0 . (3.17)
The adjoint variables of Eq. (3.17) converge to the adjoint variables of the inequality
constraints at the solution. Therefore, they provide a proper decision
criterion within an active-set strategy. This formulation of SQP methods allowing
for approximations of derivatives, forms also the basis for the real-time
investigations presented later (see also [14]).
3.2.2 Modular SQP Methods
Here, we discuss again the separability framework, i.e., problems of the type
(3.4-3.6) – first without inequality constraints
min f(y,p) (3.18)
y,p
s.t. c(y,p)=0. (3.19)
The essential feature is the nonsingularity of cy, which means that we can
look at the problem as an unconstrained problem of the form
min
p f(y(p),p) . (3.20)
When applied to the necessary optimality condition ∇pf(y(p),p)=0,Newton’s
method, or its variants, yield good local convergence properties. Every
iteration consists of two steps
(1) solve B∆p = −∇pf(y(p k ),p k )=0,whereB ≈∇p 2 f(y(p k ),p k )
(2) update p k+1 = p k + τ · ∆p, whereτ is an appropriate step length