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We investigate non-convex optimization problems in $BV(\Omega)$ withtwo-sided pointwise inequality constraints. We propose a regularization andpenalization method to numerically solve the problem. Under certain conditions,weak limit points of iterates are stationary for the original problem. Inaddition, we prove optimality conditions for the original problem that containLagrange multipliers to the inequality constraints. Numerical experimentsconfirm the theoretical findings.

A penalty scheme to solve constrained non-convex optimization problems in $BV(\Omega)$