This paper introduces the necessary and sufficient conditions that surrogatefunctions must satisfy to properly define frontiers of non-dominated solutionsin multi-objective optimization problems. These new conditions work directly onthe objective space, thus being 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. Toillustrate the practical consequences of adopting the proposed conditions, weuse Gaussian processes as surrogates endowed with monotonicity soft constraintsand with an adjustable degree of flexibility, and compare them to regularGaussian processes and to a frontier surrogate method in the literature that isthe closest to the method proposed in this paper. Results show that thenecessary and sufficient conditions proposed here are finely managed by theconstrained Gaussian process, guiding to high-quality surrogates capable ofsuitably synthesizing an approximation to the Pareto frontier in challenginginstances of multi-objective optimization, while an existing approach that doesnot take the theory proposed in consideration defines surrogates which greatlyviolate the conditions to describe a valid frontier.
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