Rates of convergence for the continuum limit of nondominated sorting

Nondominated sorting is a discrete process that sorts points in Euclidean space according to the coordinatewise partial order, and is used to rank feasible solutions to multiobjective optimization problems. It was previously shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. We prove quantitative error estimates for the convergence of nondominated sorting to its continuum limit Hamilton-Jacobi equation. Our proof uses the maximum principle and viscosity solution machinery, along with new semiconvexity estimates for domains with corner singularities

High-order filtered schemes for the Hamilton-Jacobi continuum limit of nondominated sorting

We investigate high-order finite difference schemes for the Hamilton-Jacobi equation continuum limit of nondominated sorting. Nondominated sorting is an algorithm for sorting points in Euclidean space into layers by repeatedly removing minimal elements. It is widely used in multi-objective optimization, which finds applications in many scientific and engineering contexts, including machine learning. In this paper, we show how to construct filtered schemes, which combine high order possibly unstable schemes with first order monotone schemes in a way that guarantees stability and convergence while enjoying the additional accuracy of the higher order scheme in regions where the solution is smooth. We prove that our filtered schemes are stable and converge to the viscosity solution of the Hamilton-Jacobi equation, and we provide numerical simulations to investigate the rate of convergence of the new schemes

Hamilton-Jacobi scaling limits of Pareto peeling in 2D

Pareto hull peeling is a discrete algorithm, generalizing convex hull peeling, for sorting points in Euclidean space. We prove that Pareto peeling of a random point set in two dimensions has a scaling limit described by a first-order Hamilton-Jacobi equation and give an explicit formula for the limiting Hamiltonian, which is both non-coercive and non-convex. This contrasts with convex peeling, which converges to curvature flow. The proof involves direct geometric manipulations in the same spirit as Calder (2016).Comment: 50 pages, 18 figures; v2 improves exposition and extends main theorem to cover any norm in R^

Hamilton-Jacobi Equations for Sorting and Percolation Problems.

In this dissertation we prove continuum limits for some sorting and percolation problems that are important in mathematical, scientific, and engineering contexts. The first problem we study is non-dominated sorting, which is a fundamental combinatorial problem in multi-objective optimization. The sorting can be viewed as arranging points in Euclidean space into fronts according to a partial order. We show that these fronts converge almost surely to the level sets of a function that satisfies a Hamilton-Jacobi equation in the viscosity sense. Of course, multi-objective optimization is ubiquitous in scientific and engineering contexts, and, as it turns out, non-dominated sorting is also equivalent to the longest chain problem, which has a long history in probability and combinatorics. We present a fast numerical scheme for solving this Hamilton-Jacobi equation and prove convergence and various properties of the scheme. We then show how to use the scheme to design a fast approximate non-dominated sorting algorithm and we demonstrate the algorithm on synthetic data as well as a large-scale real-world dataset. The second problem we study is directed last passage percolation (DLPP), which is a stochastic growth model with applications in directed polymer growth, queuing systems, and stochastic particle systems. DLPP is closely related to the longest chain problem, and by using similar techniques we prove that a DLPP model with macroscopic and discontinuous weights has a continuum limit that corresponds to solving a Hamilton-Jacobi equation. We further prove convergence of a numerical scheme for this Hamilton-Jacobi equation and present an algorithm based on dynamic programming for finding the asymptotic shapes of maximal directed paths.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/108848/1/jcalder_1.pd