Active learning of Pareto fronts with disconnected feasible decision and objective spaces

A multi-objective optimization problem (MOP) is formulated as the joint minimization of m conflicting objective functions f1(x), . . . , fm(x) w.r.t a vector x of n decision variables. Typically, x ∈ Ω, where Ω ⊂ R is the feasible region, defined by a set of constraints on the decision variables. Objective vectors are images of decision vectors and can be written as z = f(x) = (f1(x), . . . , fm(x)), with z ∈ Φ, where Φ is the image of Ω, i.e., f : Ω → Φ ⊂ R. An objective vector z is said to dominate z, denoted as z ≻ z,if zk ≤ z k for all k and there exists at least one h such that zh < z ′ h . A point x is Pareto-optimal if there is no other x ∈ Ω such that f(x) dominates f(x). The set of Pareto-optimal solutions is called Pareto set (PS). The corresponding set of Pareto-optimal objective vectors is called Pareto front (PF). The Active Learning of Pareto fronts (ALP) algorithm [1] learns an analytical model of the Pareto front from a training set of approximated Pareto-optimal vectors. The training Pareto-optimal vectors are obtained by solving different scalarized instances of the original MOP. In order to minimize the computational effort (measured as number of evaluations of the MOP objective functions), informative training objective vectors are selected by applying active learning principles. The experimental results reported in [1] show that ALP outperforms the state-of-the-art MMEA and NSGA-II algorithms over widelyused continuous-optimization benchmarks, including a set of four well-known MOPS with disconnected Pareto front. However, the benchmarks considered in the experimental comparison have connected feasible decision and objective spaces. This paper highlights a possible generalization of ALP to tackle continuous MOPs where the feasible decision and objective spaces are disconnected. To validate the ALP extension, the formulation of a wellknown continuous MOP is modified to obtain disconnected feasible decision and objective spaces. We are not aware of established benchmark problems in the literature with this feature. Our contribution can also be considered a first attempt to fulfill this lack, in the spirit of simulating real-world optimization tasks.