Skip to main content
SpringerLink
Log in

Hamilton–Jacobi scaling limits of Pareto peeling in 2D

  • Original Paper
  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

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 (Nonlinear Anal 141:88–108, 2016).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

Data availability

Not applicable.

Code Availability

Code to produce the figures in this article is included in the arXiv upload and on Github at https://github.com/nitromannitol/2d_pareto_peeling.

References

  1. Aldous, D., Diaconis, P.: Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Relat. Fields 103(2), 199–213 (1995)

    Article  MathSciNet  Google Scholar 

  2. Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston (1997) With appendices by Maurizio Falcone and Pierpaolo Soravia

  3. Bellettini, G.: Anisotropic and crystalline mean curvature flow, A sampler of Riemann–Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, pp. 49–82. Cambridge University Press, Cambridge (2004)

  4. Barles, G., Souganidis, P.E.: A new approach to front propagation problems: theory and applications. Arch. Ration. Mech. Anal. 141(3), 237–296 (1998)

    Article  MathSciNet  Google Scholar 

  5. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  6. Calder, J.: Minicourse: Partial Differential Equations for Data Peeling

  7. Calder, J.: A direct verification argument for the Hamilton–Jacobi equation continuum limit of nondominated sorting. Nonlinear Anal. 141, 88–108 (2016)

    Article  MathSciNet  Google Scholar 

  8. Calder, J.: Numerical schemes and rates of convergence for the Hamilton–Jacobi equation continuum limit of nondominated sorting. Numer. Math. 137(4), 819–856 (2017)

    Article  MathSciNet  Google Scholar 

  9. Cook, B., Calder, J.: Rates of convergence for the continuum limit of nondominated sorting. SIAM J. Math. Anal. 54(1), 872–911 (2022)

    Article  MathSciNet  Google Scholar 

  10. Calder, J., Esedo\(\bar{{\rm g}}\)lu, S., Hero, A.O.: A continuum limit for non-dominated sorting. In: Information Theory and Applications Workshop (ITA), vol. 2014, pp. 1–7. IEEE (2014)

  11. Calder, J., Esedoglu, S., Hero, A.O.: A Hamilton–Jacobi equation for the continuum limit of nondominated sorting. SIAM J. Math. Anal. 46(1), 603–638 (2014)

    Article  MathSciNet  Google Scholar 

  12. Calder, J., Esedoḡlu, S., Hero, A.O.: A PDE-based approach to nondominated sorting. SIAM J. Numer. Anal. 53(1), 82–104 (2015)

    Article  MathSciNet  Google Scholar 

  13. Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)

    Article  MathSciNet  Google Scholar 

  14. Corne, D.W., Knowles, J.D., Oates, M.J.: The Pareto envelope-based selection algorithm for multiobjective optimization. In: International Conference on Parallel Problem Solving from Nature, pp. 839–848. , Springer (2000)

  15. Calder, J., Smart, C.K.: The limit shape of convex hull peeling. Duke Math. J. 169(11), 2079–2124 (2020)

    Article  MathSciNet  Google Scholar 

  16. Dalal, K.: Counting the onion. Rand. Struct. Algorithms 24(2), 155–165 (2004)

    Article  MathSciNet  Google Scholar 

  17. Durier, R., Michelot, C.: Geometrical properties of the Fermat–Weber problem. Eur. J. Oper. Res. 20(3), 332–343 (1985)

    Article  MathSciNet  Google Scholar 

  18. Durier, R., Michelot, C.: Sets of efficient points in a normed space. J. Math. Anal. Appl. 117(2), 506–528 (1986)

    Article  MathSciNet  Google Scholar 

  19. Durier, R., Michelot, C.: On the set of optimal points to the Weber problem: further results. Transp. Sci. 28(2), 141–149 (1994)

    Article  MathSciNet  Google Scholar 

  20. Durier, R.: Sets of efficiency in a normed space and inner product. In: Recent Advances and Historical Development of Vector Optimization, pp. 114–128. Springer (1987)

  21. Durier, R.: On Pareto optima, the Fermat–Weber problem, and polyhedral gauges. Math. Program. 47(1) (Ser. A), 65–79 (1990)

  22. Hammersley, J.M.: A few seedlings of research. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. I: Theory of statistics, pp. 345–394 (1972)

  23. Kingman, J.F.C.: Poisson Processes, Oxford Studies in Probability, vol. 3, The Clarendon Press, Oxford University Press, New York, Oxford Science Publications (1993)

  24. Kuhn, H.W.: On a pair of dual nonlinear programs. Nonlinear Programming (NATO Summer School, Menton, 1964). North-Holland. Amsterdam, pp. 37–54 (1967)

  25. Kuhn, H.W.: A note on Fermat’s problem. Math. Program. 4, 98–107 (1973)

    Article  MathSciNet  Google Scholar 

  26. Lions, P.-L.: Generalized solutions of Hamilton–Jacobi equations, Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston, London (1982)

  27. Laporte, G., Nickel, S., Saldanha-da Gama, F.: Introduction to Location Science, Location Science, pp. 1–21. Springer (2019)

  28. Luc, D.T.: Structure of the efficient point set. Proc. Am. Math. Soc. 95(3), 433–440 (1985)

    Article  MathSciNet  Google Scholar 

  29. Luc, D.T.: Generalized convexity in vector optimization, Handbook of generalized convexity and generalized monotonicity. Nonconvex Optim. Appl., vol. 76, pp. 195–236. Springer, New York (2005)

  30. Morfe, P.S., Souganidis, P.E.: Comparison principles for second-order elliptic/parabolic equations with discontinuities in the gradient compatible with finsler norms. J. Funct. Anal. 285(4), 109983 (2023)

    Article  MathSciNet  Google Scholar 

  31. Marsden, J.E., Tromba, A.: Vector Calculus. Macmillan (2003)

  32. Ndiaye, M., Michelot, C.: A geometrical construction of the set of strictly efficient points in the polyhedral norm case. In: Proceedings of the 9th Meeting of the EURO Working Group on Locational Analysis (Birmingham, 1996), no. 11, pp. 89–99 (1997)

  33. Ndiaye, M., Michelot, C.: Efficiency in constrained continuous location. Eur. J. Oper. Res. 104(2), 288–298 (1998)

    Article  Google Scholar 

  34. Nouioua, K.: Enveloppes de Pareto et réseaux de Manhattan, Ph.D. thesis. L’Université de la Méditerranée (2005)

  35. Pelegrin, B., Fernandez, F.R.: Determination of efficient points in multiple-objective location problems, vol. 35, Multiple criteria decision making, pp. 697–705 (1988)

  36. Pelegrin, B., Fernandez, F.R.: Determination of efficient solutions for point-objective locational decision problems, vol. 18, Facility location analysis: theory and applications (Namur, 1987), pp. 93–102 (1989)

  37. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (2015)

    Google Scholar 

  38. Schey, H.M.: Div, grad curl, and all that: an informal text on vector calculus, 3rd edn. (1997)

  39. Schneider, R.: Convex bodies: the Brunn–Minkowski theory, expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge (2014)

  40. Smith, H.K., Laporte, G., Harper, P.R.: Locational analysis: highlights of growth to maturity. J. Oper. Res. Soc. 60(1), S140–S148 (2009)

    Article  Google Scholar 

  41. Tran, H.V.: Hamilton–Jacobi Equations-Theory and Applications, Graduate Studies in Mathematics, vol. 213, p. 2021. American Mathematical Society, Providence (2021)

    Book  Google Scholar 

  42. Thisse, J.-F., Ward, J.E., Wendell, R.E.: Some properties of location problems with block and round norms. Oper. Res. 32(6), 1309–1327 (1984)

    Article  MathSciNet  Google Scholar 

  43. Ward, J.E., Wendell, R.E.: Using block norms for location modeling. Oper. Res. 33(5), 1074–1090 (1985)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Jeff Calder for helpful suggestions and encouragement. A.B. thanks Charles K. Smart for many inspiring discussions. P.S.M. gratefully acknowledges his thesis advisor, P.E. Souganidis, for introducing him to viscosity solutions and homogenization and for unwavering support these past few years. A.B. was partially supported by Charles K. Smart’s NSF Grant DMS-2137909 and NSF Grant DMS- 2202715. P.S.M. was partially supported by P.E. Souganidis’s NSF Grants DMS-1600129 and DMS-1900599 and NSF Grant DMS-2202715.

Funding

A.B. was partially supported by Charles K. Smart’s NSF Grant DMS-2137909 and NSF grant DMS- 2202715. P.S.M. was partially supported by P.E. Souganidis’s NSF Grants DMS-1600129 and DMS-1900599 and NSF Grant DMS- 2202715.

Author information

Authors and Affiliations

  1. University of Chicago, Chicago, IL, 60637, USA

    Ahmed Bou-Rabee & Peter S. Morfe

Authors
  1. Ahmed Bou-Rabee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter S. Morfe
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors contributed equally to all aspects of the manuscript.

Corresponding author

Correspondence to Ahmed Bou-Rabee.

Ethics declarations

Conflict of interest

Not applicable.

Ethics approval

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bou-Rabee, A., Morfe, P.S. Hamilton–Jacobi scaling limits of Pareto peeling in 2D. Probab. Theory Relat. Fields 188, 235–307 (2024). https://doi.org/10.1007/s00440-023-01234-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-023-01234-4

Keywords

Mathematics Subject Classification

Search

Navigation