  We investigate non-convex optimization problems in $BV(\Omega)$ with
two-sided pointwise inequality constraints. We propose a regularization and
penalization method to numerically solve the problem. Under certain conditions,
weak limit points of iterates are stationary for the original problem. In
addition, we prove optimality conditions for the original problem that contain
Lagrange multipliers to the inequality constraints. Numerical experiments
confirm the theoretical findings.
