  Real-world problems often involve the optimization of several objectives
under multiple constraints. An example is the hyper-parameter tuning problem of
machine learning algorithms. In particular, the minimization of the estimation
of the generalization error of a deep neural network and at the same time the
minimization of its prediction time. We may also consider as a constraint that
the deep neural network must be implemented in a chip with an area below some
size. Here, both the objectives and the constraint are black boxes, i.e.,
functions whose analytical expressions are unknown and are expensive to
evaluate. Bayesian optimization (BO) methodologies have given state-of-the-art
results for the optimization of black-boxes. Nevertheless, most BO methods are
sequential and evaluate the objectives and the constraints at just one input
location, iteratively. Sometimes, however, we may have resources to evaluate
several configurations in parallel. Notwithstanding, no parallel BO method has
been proposed to deal with the optimization of multiple objectives under
several constraints. If the expensive evaluations can be carried out in
parallel (as when a cluster of computers is available), sequential evaluations
result in a waste of resources. This article introduces PPESMOC, Parallel
Predictive Entropy Search for Multi-objective Bayesian Optimization with
Constraints, an information-based batch method for the simultaneous
optimization of multiple expensive-to-evaluate black-box functions under the
presence of several constraints. Iteratively, PPESMOC selects a batch of input
locations at which to evaluate the black-boxes so as to maximally reduce the
entropy of the Pareto set of the optimization problem. We present empirical
evidence in the form of synthetic, benchmark and real-world experiments that
illustrate the effectiveness of PPESMOC.
