An adaptation of Response Surface Methodology (RSM) when the covariate is of high or infinite dimensional is proposed, providing a tool for black-box optimization in this context. We combine dimension reduction techniques with classical multivariate Design of Experiments (DoE). We propose a method to generate experimental designs and extend usual properties (orthogonality, rotatability,...) of multivariate designs to general high or infinite dimensional contexts. Different dimension reduction basis are considered (including data-driven basis). The methodology is illustrated on simulated functional data and we discuss the choice of the different parameters, in particular the dimension of the approximation space. The method is finally applied to a problem of nuclear safety.