  This paper introduces a multi-level (m-lev) mechanism into Evolution
Strategies (ESs) in order to address a class of global optimization problems
that could benefit from fine discretization of their decision variables. Such
problems arise in engineering and scientific applications, which possess a
multi-resolution control nature, and thus may be formulated either by means of
low-resolution variants (providing coarser approximations with presumably lower
accuracy for the general problem) or by high-resolution controls. A particular
scientific application concerns practical Quantum Control (QC) problems, whose
targeted optimal controls may be discretized to increasingly higher resolution,
which in turn carries the potential to obtain better control yields. However,
state-of-the-art derivative-free optimization heuristics for high-resolution
formulations nominally call for an impractically large number of objective
function calls. Therefore, an effective algorithmic treatment for such problems
is needed. We introduce a framework with an automated scheme to facilitate
guided-search over increasingly finer levels of control resolution for the
optimization problem, whose on-the-fly learned parameters require careful
adaptation. We instantiate the proposed m-lev self-adaptive ES framework by two
specific strategies, namely the classical elitist single-child (1+1)-ES and the
non-elitist multi-child derandomized $(\mu_W,\lambda)$-sep-CMA-ES. We first
show that the approach is suitable by simulation-based optimization of QC
systems which were heretofore viewed as too complex to address. We also present
a laboratory proof-of-concept for the proposed approach on a basic experimental
QC system objective.
