  Photoionization with attosecond pulses populates hole states in the photoion.
Superpositions of hole states represent ideal candidates for time-dependent
spectroscopy, for example via pump-probe studies. The challenge consists in
identifying pulses that create coherent superpositions of hole states while
satisfying practical constraints. Here, we employ quantum optimal control to
maximize the degree of coherence between these hole states. To this end, we
introduce a derivative-free optimization method with Sequential PArametrization
update (SPA-optimization). We demonstrate the versatility and computational
efficiency of SPA-optimization for photoionization in argon by maximizing the
coherence between the $3s$ and $3p_0$ hole states using shaped attosecond
pulses. We show that it is possible to maximize the hole coherence while
simultaneously prescribing the ratio of the final hole state populations.
