Advertisement

Multiobjective Optimization and Phase Transitions

  • Luís F. SeoaneEmail author
  • Ricard Solé
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

Many complex systems obey to optimality conditions that are usually not simple. Conflicting traits often interact making a Multi Objective Optimization (MOO) approach necessary. Recent MOO research on complex systems report about the Pareto front (optimal designs implementing the best trade-off) in a qualitative manner. Meanwhile, research on traditional Simple Objective Optimization (SOO) often finds phase transitions and critical points. We summarize a robust framework that accounts for phase transitions located through SOO techniques and indicates what MOO features resolutely lead to phase transitions. These appear determined by the shape of the Pareto front, which at the same time is deeply related to the thermodynamic Gibbs surface. Indeed, thermodynamics can be written as an MOO from where its phase transitions can be parsimoniously derived; suggesting that the similarities between transitions in MOO-SOO and Statistical Mechanics go beyond mere coincidence.

Keywords

Multi Objective Optimization Pareto Front Order Transition Order Phase Transition Target Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work has been supported by an ERC Advanced Grant, the Botín Foundation, by Banco Santander through its Santander Universities Global Division and by the Santa Fe Institute. We thank CSL members for insightful discussion.

References

  1. 1.
    West, G.B., Brown, J.H., Enquist, B.J.: A general model for the structure and allometry of plant vascular systems. Nature 400, 664–667 (1999)ADSCrossRefGoogle Scholar
  2. 2.
    Pérez-Escudero, A., de Polavieja, G.G.: Optimally wired subnetwork determines neuroanatomy of Caenorhabditis elegans. Proc. Natl. Acad. Sci. USA 104(43), 17180–17185 (2007)ADSCrossRefGoogle Scholar
  3. 3.
    Shoval, O., Sheftel, H., Shinar, G., Hart, Y., Ramote, O., Mayo, A., Dekel, E., Kavanagh, K., Alon, U.: Evolutionary tradeoffs, Pareto optimality, and the geometry of phenotype space. Science 336, 1157–1160 (2012)ADSCrossRefGoogle Scholar
  4. 4.
    Schuetz, R., Zamboni, N., Zampieri, M., Heinemann, M., Sauer, U.: Multidimensional optimality of microbial metabolism. Science 336, 601–604 (2012)ADSCrossRefGoogle Scholar
  5. 5.
    Szekely, P., Sheftel, H., Mayo, A., Alon, U.: Evolutionary tradeoffs between economy and effectiveness in biological homeostasis systems. PLoS Comput. Biol. 9(8), e1003163 (2013)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Zipf, G.K.: Human Behavior and the Principle of Least Effort (1949)Google Scholar
  7. 7.
    Maritan, A., Rinaldo, A., Rigon, R., Giacometti, A., Rodrígued-Iturbe, I.: Scaling laws for river networks. Phys. Rev. E 53(2), 1510 (1996)ADSCrossRefGoogle Scholar
  8. 8.
    Ferrer i Cancho, R., Solé, R.V.: Least effort and the origins of scaling in human language. Proc. Natl. Acad. Sci. 100(3), 788–791 (2003)Google Scholar
  9. 9.
    Barthelemy, M.: Spatial networks. Phys. Rep. 499, 1–101 (2011)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Louf, R., Jensen, P., Barthelemy, M.: Emergence of hierarchy in cost-driven growth of spatial networks. Proc. Natl. Acad. Sci. 110(22), 8824–8829 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dawkins, R.: The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe Without Design. WW Norton & Company (1986)Google Scholar
  12. 12.
    Dennett, D.C.: Darwin’s Dangerous Idea. The Sciences (1995)Google Scholar
  13. 13.
    Seoane, L.F., Solé, R.: Phase transitions in Pareto optimal complex networks. Phys. Rev. E 92, 032807 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    Avena-Koenigsberger, A., Goñi, J., Solé, R., Sporns, O.: Network morphospace. J. Royal Soc. Interface 12(103), 20140881 (2015)CrossRefGoogle Scholar
  15. 15.
    Goñi, J., Avena-Koenigsberger, A., de Menizabal, N.V., van den Heuvel, M., Betzel, R., Sporns, O.: Exploring the morphospace of communication efficiency in complex networks. PLoS ONE 8, e58070 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    Priester, C., Schmitt, S., Peixoto, T.P.: Limits and trade-offs of topological network robustness. PLoS ONE 9(9), e108215 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Otero-Muras, I., Banga, J.R.: Multicriteria global optimization for biocircuit design. BMC Syst. Biol. 8, 113 (2014)CrossRefGoogle Scholar
  18. 18.
    Seoane, L.F., Solé, R.: A multiobjective optimization approach to statistical mechanics. http://arxiv.org/abs/1310.6372 (2013)
  19. 19.
    Gibbs, J.W.: A method of geometrical representation of the thermodynamic properties of substances by means of surfaces. Trans. Conn. Acad. 2, 382–404 (1873)zbMATHGoogle Scholar
  20. 20.
    Maxwell, J.C.: Theory of Heat. Longmans, Green, and Co., pp. 195–208 (1904)Google Scholar
  21. 21.
    Fonseca, C.M., Fleming, P.J.: An overview of evolutionary algorithms in multiobjective optimization. Evol. Comput. 3, 1–16 (1995)CrossRefGoogle Scholar
  22. 22.
    Dittes, F.M.: Optimization on rugged landscapes: a new general purpose Monte Carlo approach. Phys. Rev. Lett. 76(25), 4651–4655 (1996)ADSCrossRefGoogle Scholar
  23. 23.
    Zitzler, E.: Evolutionary algorithms for multiobjective optimization: methods and applications. A dissertation submitted to the Swiss Federal Institute of Technology Zurich for the degree of Doctor of Technical Sciences (1999)Google Scholar
  24. 24.
    Coello, C.A.: Evolutionary multi-objective optimization: a historical view of the field. IEEE Comput. Intell. M. 1(1), 28–36 (2006)CrossRefGoogle Scholar
  25. 25.
    Konak, A., Coit, D.W., Smith, A.E.: Multi-objective optimization using genetic algorithms: a tutorial. Reliab. Eng. Syst. Safe. 91(9), 992–1007 (2006)CrossRefGoogle Scholar
  26. 26.
    Solé, R., Seoane, L.F.: Ambiguity in language networks. Linguist. Rev. 32(1), 5–35 (2014)Google Scholar
  27. 27.
    Prokopenko, M., Ay, N., Obst, O., Polani, D.: Phase transitions in least-effort communications. J. Stat. Mech. 2010(11), P11025 (2010)CrossRefGoogle Scholar
  28. 28.
    Seoane, L.F., Solé, R.: Systems poised to criticality through Pareto selective forces. http://arxiv.org/abs/1510.08697 (2015)
  29. 29.
    Harte, J.: Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics. Oxford University Press (2011)Google Scholar
  30. 30.
    Mora, T., Bialek, W.: Are biological systems poised at criticality? J. Stat. Phys. 144(2), 268–302 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Touchette, H., Ellis, R.S., Turkington, B.: An introduction to the thermodynamic and macrostate levels of nonequivalent ensembles. Phys A. 2004(340), 138–146 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.ICREA-Complex Systems LabUniversitat Pompeu Fabra-PRBBBarcelonaSpain
  2. 2.Institut de Biologia Evolutiva, CSIC-UPFBarcelonaSpain
  3. 3.Santa Fe InstituteSanta FeUSA

Personalised recommendations