We exploit RB methods for the efficient solution of parameter-dependent PDE-constrained optimization problems. According to the optimization strategy adopted, these problems feature a very large size, in case a monolithic strategy is chosen, or huge computational cost due to the need of solving a parametrized PDE many times, when preferring an iterative optimization method.We concentrate on
optimization problems, where control variables are described in terms of a vector of parameters, and
optimal control problems, in which the parameters affect instead the state system and the control variables are functions to be determined. We propose efficient RB strategies to speedup the solution of these problems, pursuing either a
(i) state reduction
in the former case, or
state and control reduction
in the latter.