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| Convex relaxation of binary-continuous optimization problems and their numerical solution by semi-smooth Newton methods are discussed. The proposed framework involves $L^0$-type penalties that pointwise are zero on the admissible set and one otherwise. Such penalties can be used to promote controls that are sparse or that take values only from a given discrete set (called ``multi-bang'' controls) but are non-convex and lack weak lower-semicontinuity, application of Fenchel duality yields a formal primal-dual optimality system that admits a unique solution. Under appropriate conditions, it is possible to derive a generalized multi-bang principle, i.e., to prove that this solution is optimal and almost everywhere takes on values only from the admissible set. This is illustrated by numerical examples. |
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