  This paper proposes a method for solving optimization problems in which the
decision-maker cannot evaluate the objective function, but rather can only
express a preference such as "this is better than that" between two candidate
decision vectors. The algorithm described in this paper aims at reaching the
global optimizer by iteratively proposing the decision maker a new comparison
to make, based on actively learning a surrogate of the latent (unknown and
perhaps unquantifiable) objective function from past sampled decision vectors
and pairwise preferences. The surrogate is fit by means of radial basis
functions, under the constraint of satisfying, if possible, the preferences
expressed by the decision maker on existing samples. The surrogate is used to
propose a new sample of the decision vector for comparison with the current
best candidate based on two possible criteria: minimize a combination of the
surrogate and an inverse weighting distance function to balance between
exploitation of the surrogate and exploration of the decision space, or
maximize a function related to the probability that the new candidate will be
preferred. Compared to active preference learning based on Bayesian
optimization, we show that our approach is superior in that, within the same
number of comparisons, it approaches the global optimum more closely and is
computationally lighter. MATLAB and a Python implementations of the algorithms
described in the paper are available at
http://cse.lab.imtlucca.it/~bemporad/idwgopt.
