Random Pareto front surfaces
This work provides a framework for uncertainty quantification and decision-making in multi-objective optimization, applicable in fields like design of experiments and extreme value theory, though it is incremental as it builds on existing parameterization ideas.
The authors tackled the problem of representing and analyzing Pareto front surfaces in multi-objective optimization by proposing a polar coordinate parameterization using a scalar-valued length function, which enabled rigorous derivation of statistics like expectation, covariance, and quantiles for stochastic Pareto front surfaces.
The goal of multi-objective optimisation is to identify the Pareto front surface which is the set obtained by connecting the best trade-off points. Typically this surface is computed by evaluating the objectives at different points and then interpolating between the subset of the best evaluated trade-off points. In this work, we propose to parameterise the Pareto front surface using polar coordinates. More precisely, we show that any Pareto front surface can be equivalently represented using a scalar-valued length function which returns the projected length along any positive radial direction. We then use this representation in order to rigorously develop the theory and applications of stochastic Pareto front surfaces. In particular, we derive many Pareto front surface statistics of interest such as the expectation, covariance and quantiles. We then discuss how these can be used in practice within a design of experiments setting, where the goal is to both infer and use the Pareto front surface distribution in order to make effective decisions. Our framework allows for clear uncertainty quantification and we also develop advanced visualisation techniques for this purpose. Finally we discuss the applicability of our ideas within multivariate extreme value theory and illustrate our methodology in a variety of numerical examples, including a case study with a real-world air pollution data set.