Necessary and Sufficient Conditions for Surrogate Functions of Pareto Frontiers and Their Synthesis Using Gaussian Processes. (arXiv:1505.05063v1 [cs.AI])

This paper introduces the necessary and sufficient conditions that surrogate functions must satisfy to properly define frontiers of non-dominated solutions in multi-objective optimization problems. Given that this is the first time that those conditions are elicited, there is no reason to believe that the surrogates already proposed in the literature meet them. As a consequence, dominated solutions can be suggested by already proposed surrogates as valid candidates to represent the Pareto frontier. Conceptually speaking, the new conditions we are introducing work directly on the objective space, thus being agnostic on the evaluation methods. 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, an oversimplified model for the surrogate is shown to be capable of suggesting a valid frontier of non-dominated solutions, though not the expect one from the data provided. On the other hand, when applying Gaussian processes as surrogates endowed with monotonicity soft constraints and with an adjustable degree of flexibility, the necessary and sufficient conditions proposed here are finely managed by the multivariate distribution, guiding to high-quality surrogates capable of suitably synthesizing an approximation to the Pareto frontier in challenging instances of multi-objective optimization.

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