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Neumann boundary conditions

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

Equality constraints are obtained by applying the five-point-star to the elliptic equation in (2.2) in all points with :
 (4.3)

The derivative in the direction of the outward normal is approximated by the expression where
 (4.4)

Then the discrete form of the Neumann boundary condition in (2.2) leads to the equality constraints
 (4.5)

The control and state inequality constraints (2.3) and (2.4) yield the inequality constraints
 (4.6) (4.7)

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
 (4.8)

Then the relations (4.3)-(4.8) define an NLP-problem of the form (4.1). The Lagrangian function for this NLP-problem becomes
 (4.9)

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
 (4.10)

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 :
 (4.11)

In case that the measure is a delta distribution, we obtain from (4.10) the relation
 (4.12)

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
 (4.13)

where is a line segment on of length centered at . For the special decomposition this leads to the identifications
 (4.14)

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 .

Next: Dirichlet boundary conditions Up: Discretization and optimization techniques Previous: Discretization and optimization techniques
Hans D. Mittelmann
2002-11-25