Optimization and Computational Fluid Dynamics - Department of ...

64 H.G. Bock and V. Schulz
min
z f(z) (3.9)
s.t. c(z)=0. (3.10)
We nevertheless keep in mind that all or some of the equality constraints
might later represent so-called active inequality constraints. Optimization
theory provides the following necessary conditions in terms of the Lagrangian
(L), which hold at an optimal solution to this problem, provided the functions
involved are sufficiently smooth
∂L(z,λ)/∂z =0, where L(z,λ):=f(z)+λ ⊤ c(z) . (3.11)
The principle of SQP methods starts at Newton’s method for the necessary
conditions (3.11) together with the constraints (3.10). This method iterates
over z and the adjoint variables λ in the form (zk+1 ,λk+1 )=(zk ,λk )+
(∆zk+1 ,∆λk+1 ), where the increments solve the linear system
� Hc ⊤ z
cz 0
�� ∆z
∆λ
� �
k −∇f − cz(z
=
k ) ⊤λk −c(zk �
. (3.12)
)
Here, H denotes the Hessian of the Lagrangian L with respect to z, i.e.,
H = Lzz(zk ,λk ) and subscripts denote respective derivatives.
Because the derivative generation needed to establish the matrix on the left
hand side of this equation often turns out to be prohibitively expensive, one
often uses approximations instead, i.e., G :≈ H and A :≈ cz (of course, these
approximations may also change from iteration to iteration). This substitution
will deteriorate the convergence behavior of this approximate Newton
method. However, � the�fixed points of the iteration are not changed as long
⊤ GA
as the matrix is nonsingular. If the approximations G and A are
A 0
sufficiently accurate, one may expect at least linear local convergence of the
iteration method. In the implementation, one will terminate the iteration as
soon as (∆z k+1 ,∆λ k+1 ) is sufficiently close to zero in a suitable norm. If an
estimate of the convergence rate is known an a priori estimate for the distance
to the limit point can be given.
In order to arrive at a generalization of this approach to inequality con-
strained problems, we first reformulate the system of equations
� �� � �
⊤
k GA ∆z −∇f − cz(z
=
A 0 ∆λ
k ) ⊤λk −c(zk �
)
(3.13)
in the form of an equivalent linear-quadratic problem. Setting ∆λ = λk+1 −
λk , we obtain the formulation
� ��
⊤ GA ∆z
A 0 λk+1 � �
k −∇f +(A− cz(z
=
k )) ⊤λk −c(zk �
.
)