  In the present paper we consider minimization based formulations of inverse
problems $(x,\Phi)\in\mbox{argmin}\{\mathcal{J}(x,\Phi;y)\colon(x,\Phi)\in
M_{ad}(y) \}$ for the specific but highly relevant case that the admissible set
$M_{ad}^\delta(y^\delta)$ is defined by pointwise bounds, which is the case,
e.g., if $L^\infty$ constraints on the parameter are imposed in the sense of
Ivanov regularization, and the $L^\infty$ noise level in the observations is
prescribed in the sense of Morozov regularization. As application examples for
this setting we consider three coefficient identification problems in elliptic
boundary value problems. Discretization of $(x,\Phi)$ with piecewise constant
and piecewise linear finite elements, respectively, leads to finite dimensional
nonlinear box constrained minimization problems that can numerically be solved
via Gauss-Newton type SQP methods. In our computational experiments we revisit
the suggested application examples. In order to speed up the computations and
obtain exact numerical solutions we use recently developed active set methods
for solving strictly convex quadratic programs with box constraints as
subroutines within our Gauss-Newton type SQP approach.
