A continuum limit for non-dominated sorting

Non-dominated sorting is an important combinatorial problem in multi-objective optimization, which is ubiquitous in many fields of science and engineering. In this paper, we overview the results of some recent work by the authors on a continuum limit for non-dominated sorting. In particular, we have discovered that in the (random) large sample size limit, the non-dominated fronts converge almost surely to the level sets of a function that satisfies a Hamilton-Jacobi partial differential equation (PDE). We show how this PDE can be used to design a fast, potentially sublinear, approximate non-dominated sorting algorithm, and we show the results of applying the algorithm to real data from an anomaly detection problem.