The optimization variable in (4.1) is taken as the vector

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

Then the discrete form of the Neumann boundary condition in (2.2) leads to the

The control and state inequality constraints (2.3) and (2.4) yield the

Observe that these inequality constraints do not depend on the meshsize . Later on, this fact will require a scaling of the Lagrange multipliers. The discretized form of the cost function (2.1) is

Then the relations (4.3)-(4.8) define an NLP-problem of the form (4.1). The Lagrangian function for this NLP-problem becomes

The Lagrange multipliers , resp.

are associated with the equality constraints (4.3) and (4.5), the inequality constraints (4.6), resp. the inequality constraints (4.7). The multipliers and satisfy complementarity conditions corresponding to (2.10):

The discussion of the necessary conditions of optimality

will be performed for different combinations of indices . For indices we obtain the relations

Hence, the Lagrange multipliers satisfy the five-point-star difference equations for the adjoint equation in (2.7) if we make the following identification for the Borel measure . Let denote a square centered at with area . Then we have the approximation

Recall the decomposition (2.11) of the measure . If the singular part of the measure vanishes, i.e. then (4.10) yields the approximation for the density :

In case that the measure is a delta distribution, we obtain from (4.10) the relation

For indices on the boundary we get, e.g. for :

This is just the discrete version of the Neumann boundary condition (2.8) if we identify

where is a line segment on of length centered at . For the special decomposition this leads to the identifications

Similar relations hold for other indices . Finally, necessary conditions with respect to the control variables are determined, e.g., for indices by

This is the discrete version of the optimality condition (2.9) for the control, if we use again the identification .

2002-11-25