efficient model order reduction methods for parameter optimization constrained by coercive variational state equations using the [*first optimize, then discretize*]{} approach. The methods are based on successive enrichment of the underlying reduced order models within the framework of Trust-Region optimization. Optimization problems constrained by partial differential equations (PDEs) arise in many fields of application in engineering and across all sciences. Examples of such problems include optimal (material) design or optimal control of processes and inverse problems, where parameters of a PDE model are unknown and need to be estimated from measurements. The numerical solution of such problems is very challenging as the underlying PDEs have to be solved repeatedly within outer optimization algorithms and the dimension of the parameters that need to be optimized might be very high or even infinite dimensional. PDE constrained optimization problems have been of interest for many decades. Classically, the underlying PDE (forward problem) is approximated by a high dimensional full order model (FOM) that results from discretization, e.g., by the Finite Element or Finite Volume method. Hence, the complexity of the optimization problem directly depends on the numbers of degrees of freedom (DOF) of the FOM. Mesh adaptivity has been advised to minimize the 