Statistical optimization of expensive multi-response black-box functions

Assume that a set of $P$ process parameters $p_i$, $i=1,\dots ,P$, determines the outcome of a set of $D$ descriptor variables $d_j$, $j=1,\dots ,D$, via an unknown functional relationship $\phi :\mathbf{p}\mapsto \mathbf{d},{\mathbb{R}}^P\to {\mathbb{R}}^D$, where $\mathbf{p}=(p_1,\dots ,p_P)$, $\mathbf{d}=(d_1,\dots ,d_D)$. It is desired to find appropriate values $\widehat{\mathbf{p}}=({\widehat{p}}_1,\dots ,{\widehat{p}}_P)$ for the process parameters such that the corresponding values of the descriptor variables $\phi (\widehat{\mathbf{p}})$ are close to a given target ${\mathbf{d}}^{\ast }=(d_1^{\ast },\dots ,d_D^{\ast })$, assuming that at least one exact solution exists. A sequential approach using dimension reduction techniques has been developed to achieve this. In a simulation study, results of the suggested approach and the algorithms NSGA-II, SMS-EMOA and MOEA/D are compared.