  This paper introduces CLEO, a novel preference elicitation algorithm capable
of recommending complex objects in hybrid domains, characterized by both
discrete and continuous attributes and constraints defined over them. The
algorithm assumes minimal initial information, i.e., a set of catalog
attributes, and defines decisional features as logic formulae combining Boolean
and algebraic constraints over the attributes. The (unknown) utility of the
decision maker (DM) is modelled as a weighted combination of features. CLEO
iteratively alternates a preference elicitation step, where pairs of candidate
solutions are selected based on the current utility model, and a refinement
step where the utility is refined by incorporating the feedback received. The
elicitation step leverages a Max-SMT solver to return optimal hybrid solutions
according to the current utility model. The refinement step is implemented as
learning to rank, and a sparsifying norm is used to favour the selection of few
informative features in the combinatorial space of candidate decisional
features.
  CLEO is the first preference elicitation algorithm capable of dealing with
hybrid domains, thanks to the use of Max-SMT technology, while retaining
uncertainty in the DM utility and noisy feedback. Experimental results on
complex recommendation tasks show the ability of CLEO to quickly focus towards
optimal solutions, as well as its capacity to recover from suboptimal initial
choices. While no competitors exist in the hybrid setting, CLEO outperforms a
state-of-the-art Bayesian preference elicitation algorithm when applied to a
purely discrete task.
