  This paper provides a theoretical and numerical investigation of a penalty
decomposition scheme for the solution of optimization problems with geometric
constraints. In particular, we consider some situations where parts of the
constraints are nonconvex and complicated, like cardinality constraints,
disjunctive programs, or matrix problems involving rank constraints. By a
variable duplication and decomposition strategy, the method presented here
explicitly handles these difficult constraints, thus generating iterates which
are feasible with respect to them, while the remaining (standard and
supposingly simple) constraints are tackled by sequential penalization. Inexact
optimization steps are proven sufficient for the esulting algorithm to work, so
that it is employable even with difficult objective functions. The current work
is therefore a significant generalization of existing papers on penalty
decomposition methods. On the other hand, it is related to some recent
publications which use an augmented Lagrangian idea to solve optimization
problems with geometric constraints. Compared to these methods, the
decomposition idea is shown to be numerically superior since it allows much
more freedom in the choice of the subproblem solver, and since the number of
certain (possibly expensive) projection steps is significantly less. Extensive
numerical results on several highly complicated classes of optimization
problems in vector and matrix paces indicate that the current method is indeed
very efficient to solve these problems.
