Necessary and Sufficient Conditions for Surrogate Functions of Pareto Frontiers and Their Synthesis Using Gaussian Processes

This paper introduces necessary and sufficient conditions that surrogate functions must satisfy to properly define frontiers of nondominated solutions in multiobjective optimization (MOO) problems. These new conditions work directly on the objective space, and thus are agnostic about how the solutions are evaluated. Therefore, real objectives or user-designed objectives' surrogates are allowed, opening the possibility of linking independent objective surrogates. To illustrate the practical consequences of adopting the proposed conditions, we use Gaussian processes (GPs) as surrogates endowed with monotonicity soft constraints and with an adjustable degree of flexibility, and compare them to regular GPs and to a frontier surrogate method in the literature that is the closest to the method proposed in this paper. Results show that the necessary and sufficient conditions proposed here are finely managed by the constrained GP, guiding to high-quality surrogates capable of suitably synthesizing an approximation to the Pareto frontier in challenging instances of MOO, while an existing approach that does not take the theory proposed in consideration defines surrogates which greatly violate the conditions to describe a valid frontier.