Neighboring Local Optimal Solutions and Its Applications

  • Hsiao-Dong Chiang
  • Tao Wang


The number of neighboring local optimal solutions is an important index for assessing the complexity of nonlinear systems and the computational complexity of numerical methods for nonlinear optimization. Sperner’s lemma provides an effective tool for this quantitative study. It has been shown that, in general there are at least 2n local-optimal solutions neighboring to any given one, for a class of nonlinear optimization problems. Furthermore, if a collection of neighboring local-optimal solutions retains the local-independence, then each solution must have at least n(n + 1) neighboring local-optimal solutions instead. The local-independence has been justified for the planar case at the end.


Nonlinear optimization Local optimal solution Lower bound 



The presented work was partially supported by the CERT through the National Energy Technology Laboratory Cooperative Agreement No. DE-FC26-09NT43321, and partially supported by the National Science Foundation, USA, under Award #1225682.


  1. 1.
    Sperner, E.: Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 6(1), 265–272 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bagemihl, F.: An extension of Sperner’s lemma, with applications to closed-set coverings and fixed points. Fundam. Math. 40(1), 3–12 (1953)MathSciNetGoogle Scholar
  3. 3.
    Kuhn, H.W.: Some combinatorial lemmas in topology. IBM J. Res. Dev. 4(5), 518–524 (1960)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cohen, D.I.: On the Sperner lemma. J. Comb. Theory 2(4), 585–587 (1967)CrossRefzbMATHGoogle Scholar
  5. 5.
    de Longueville, M.: A Course in Topological Combinatorics, pp. 5–6. Springer, New York (2012)Google Scholar
  6. 6.
    Monsky, P.: On dividing a square into triangles. Am. Math. Mon. 77(2), 161–164 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Naber, G.L.: Topological Methods in Euclidean Spaces. Courier Dover Publications, Mineola (2000)zbMATHGoogle Scholar
  8. 8.
    Kuratowski, K.: Topology, vol. I. PWN-Polish Scientific Publishers/Academic, Warsaw/New York (1966)Google Scholar
  9. 9.
    Chiang, H.D., Chu, C.C.: A systematic search method for obtaining multiple local optimal solutions of nonlinear programming problems. IEEE Trans. Circuits Syst. 43(2), 99–106 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lee, J., Chiang, H.D.: A dynamical trajectory-based methodology for systematically computing multiple optimal solutions of nonlinear programming problems. IEEE Trans. Automat. Control 49(6), 888–899 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chiang, H.D., Lee, J.: Trust-tech paradigm for computing high-quality optimal solutions: method and theory. In: Lee, K.Y., El-Sharkawi, M.A. (eds.) Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems, pp. 209–233. Wiley, Hoboken (2007)Google Scholar
  12. 12.
    Wang, T., Chiang, H.D.: Neighboring stable equilibrium points in spatially periodic nonlinear dynamical systems: theory and applications (submitted)Google Scholar
  13. 13.
    Hirsch, M.W.: Differential Topology. Springer, New York (1976)CrossRefzbMATHGoogle Scholar
  14. 14.
    Chiang, H.D., Fekih-Ahmed, L.:Quasi-stability regions of nonlinear dynamical systems: theory. IEEE Trans. Circuits Syst. 43(8), 627–635 (1996)Google Scholar
  15. 15.
    Aiello, G., Alfonzetti, S., Borzì, G., Saleron, N.: Computing spatially-periodic electric fields by charge iteration. IEEE Trans. Magn. 34(5), 2501–2504 (1998)CrossRefGoogle Scholar
  16. 16.
    Fardad, M., Jovanović, M.R., Bamieh, B.: Frequency analysis and norms of distributed spatially periodic systems. IEEE Trans. Autom. Control 53(10), 2266–2279 (2008)CrossRefGoogle Scholar
  17. 17.
    Rokhlenko, A., Lebowitz, J.L.: Modeling electron flow produced by a three-dimensional spatially periodic field emitter. J. Appl. Phys. 108, 123301 (2010)CrossRefGoogle Scholar
  18. 18.
    Ordonez, C.A., Pacheco, J.L., Weathers, D.L.: Spatially periodic electromagnetic force field for plasma confinement and control. Open Plasma Phys. J. 5, 1–10 (2012)CrossRefGoogle Scholar
  19. 19.
    Kolokathis, P.D., Theodorou, D.N.: On solving the master equation in spatially periodic systems. J. Chem. Phys. 137, 034112 (2012)CrossRefGoogle Scholar
  20. 20.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1997)zbMATHGoogle Scholar
  21. 21.
    Hartman, A.: Software and hardware testing using combinatorial covering suites. In: Graph Theory, Combinatorics and Algorithms, pp. 237–266. Springer, New York (2005)Google Scholar
  22. 22.
    Greene, C.: Sperner families and partitions of a partially ordered set. In: Hall, M., Jr., van Lint, J. (eds.) Combinatorics, pp. 277–290. Dordrecht, Holland (1975)Google Scholar
  23. 23.
    Crescenzi, P., Silvestri, R.: Sperner’s lemma and robust machines. In: Proceedings of the IEEE on Eighth Annual Structure in Complexity Theory Conference, pp. 194–199 (1993)Google Scholar
  24. 24.
    Chiang, H.D., Wang, B., Jiang, Q.Y.: Applications of TRUST-TECH methodology in optimal power flow of power systems. In: Optimization in the Energy Industry, pp. 297–318. Springer, Berlin (2009)Google Scholar
  25. 25.
    Atkins, P., De Paula, J.: Physical Chemistry, 8th edn. Oxford University Press, Oxford (2006)Google Scholar
  26. 26.
    Tessier, S.R., Brennecke, J.F., Stadtherr, M.A.: Reliable phase stability analysis for excess Gibbs energy models. Chem. Eng. Sci. 55(10), 1785–1796 (2000)CrossRefGoogle Scholar
  27. 27.
    Onuchic, J.N., Luthey-Schulten, Z., Wolynes, P.G.: Theory of protein folding: the energy landscape perspective. Ann. Rev. Phys. Chem. 48(1), 545–600 (1997)CrossRefGoogle Scholar
  28. 28.
    Engelking, R.: Dimension Theory. PWN Polish Scientific Publishers, Warszawa (1978)zbMATHGoogle Scholar
  29. 29.
    Conway, J.B.: Functions of One complex Variable II. Springer, New York (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringCornell UniversityIthacaUSA

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