Note that we do not consider the variables explicitly as optimization variables since they are prescribed by the Dirichlet condition (4).

*Equality constraints*
are obtained by applying the five-point-star to the
elliptic equation
in (2) in all points with
,

The derivative in the direction of the outward normal is approximated by the expression where

Then the discrete form of the Neumann boundary condition (3) leads to the

The control and state inequality constraints (5) and (6) yield the

Observe that the inequality constraints do not depend on the meshsize . Later on, this fact will require a scaling of the Lagrange multipliers. Finally, the discretized form of the cost function (1) is

Then the relations (2)-(8) define an NLP-problem of the form (1).

Associate Lagrange multipliers
,
and
with the equality constraints
(3) and (6)
resp. the inequality constraints (7) and (8).
Then the Lagrangian function for the above NLP-problem becomes:

The multipliers and satisfy complementarity conditions corresponding to (14):

Now we discuss the necessary conditions of optimality

for state and control variables assuming different combinations of indices . For

These equations contain multipliers for that do not appear in the Lagrangian (10). To make equations and definitions consistent, we put

This substitution corresponds to the Dirichlet condition (11). Relations (11) then reveal that the Lagrange multipliers satisfy the five-point-star difference equations for the adjoint equation in (9) if we use the following approximation for the Borel measure ,

where denotes a square centered at with area . Recall the decomposition (15) of the measure . If the singular part of the measure vanishes, i.e. then (13) yields the following appro-

ximation for the density ,

In case that the measure is a delta distribution, we obtain from (13) the relation

On the boundary part we get for indices assuming, e.g., :

Recalling (5) this represents the discrete version of the Neumann boundary condition (10).

Finally, necessary conditions with respect to the *control variables*
for indices
are determined by the
following two relations.
For
with, e.g.,
we get

This equation yields the discrete version of the optimality condition (12) for the control, if we use the identification

For indices with, e.g., we find

Observing and the approximation (5) of the normal derivative, the minimum condition (13) holds with the substitutions

2000-12-09