Convergent regions of the Newton homotopy method for nonlinear systems: theory and computational applications

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications

Volumn 48, Issue 1, 2001, Pages 51-66

  Lee, Jaewook ^a,b   Chiang, Hsiao Dong ^a,c  

^a Department of Obstetrics Gynecology   (United States)

^b Korea University   (South Korea)

^c Cornell University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; DIFFERENTIAL EQUATIONS; EIGENVALUES AND EIGENFUNCTIONS; GLOBAL OPTIMIZATION; NONLINEAR EQUATIONS; NUMERICAL METHODS; SET THEORY;

CONVERGENT REGIONS; NEWTON HOMOTOPY METHOD;

NONLINEAR NETWORK ANALYSIS;

EID: 0035088090     PISSN: 10577122     EISSN: None     Source Type: Journal    
DOI: 10.1109/81.903187     Document Type: Article

References (47)

1. E.L. Allgower and K. Georg, Introduction to Numerical Continuation Methods. Berlin, Germany: Springer-Verlag, 1990.
2. J.M. Anglada and J. M. Bofill, "On the restricted step method coupled with the augmented hessian for the search of stationary points of any continuous function," Int. J. Quantum Chem., vol. 62, pp. 153-165, 1997.
3. V.I. Arnold, Ordinary Differential Equations. Cambridge, MA: MIT Press, 1973.
4. _, Geometrical Methods in the Theory of Ordinary Differential Equations. Berlin, Germany: Springer-Verlag, 1977.
5. F. H. Branin, "A widely convergent method for finding multiple solutions of nonlinear systems of equations," IBM J. Res. Develop., vol. 16, pp. 504-522, 1972.
6. C.J. Cerjan and W. H. Miller, "On finding transition states," J. Chem. Phys., vol. 75, pp. 2800-2805, 1981.
7. H.D. Chiang, M. W. Hirsch, and F. F. Wu, "Stability region of nonlinear autonomous dynamical systems," IEEE Trans. Automat. Contr., vol. 33, pp. 16-27, Jan., 1988.
8. H.D. Chiang, C.-C. Chu, and G. Cauley, "Direct stability analysis of electric power systems using energy functions: Theory, applications, and perspective," Proc. IEEE, vol. 83, pp. 1497-1529, Nov., 1995.
9. H.D. Chiang and C.-C. Chu, "A systematic search method for obtaining multiple local optimal solutions of nonlinear programming problems," IEEE Trans. Circuits Syst., vol. 43, pp. 99-109, 1996.
10. L.O. Chua and A. Ushida, "A switching-parameter algorithm for finding multiple solutions of nonlinear resistive circuits," Int. J. Circuit Theory Appl., vol. 4, pp. 215-239, 1976.
11. J.E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Philadelphia, PA: SIAM, 1996.
12. I. Diener, "On the global convergence of path-following methods to determine all solutions to a system of nonlinear equations," Math. Progr., vol. 39, pp. 181-188, 1987.
13. _, "An extended continuous Newton method,"J. Optim. Theory Appl., vol. 67, pp. 57-77, 1990.
14. _, "Trajectory methods in global optimization," in Hand book of Global Optimization,R. Horst and P. M. Pardalos, Eds. Norwell, MA: Kluwer, 1995, pp. 649-668.
15. H.D. Chiang and L. Fekih-Ahmed, "Quasistability regions of nonlinear dynamical systems: Theory," IEEE Trans. Circuits Syst., vol. 43, pp. 627-635, Aug., 1996.
16. P. Fletcher, Practical Methods of Optimization. New York: Wiley, 1987.
17. C. A. Floudas and P. M. Pardalos, Recent Advances in Global Optimization, ser. Princeton Series in Computer Science, NJ: Princeton, 1992.
18. W. Forster, "Homotopy methods," in Hand book of Global Optimization,R. Horst and P. M. Pardalos, Eds. Norwell, MA: Kluwer, 1995, pp. 669-750.
19. C.B. Garcia and F. J. Gould, "A theorem on homotopy paths," Math. Oper. Res., vol. 3, pp. 282-289, 1978.
20. _, "Relations between several path following algorithms and local and global Newton methods," SIAM Rev., vol. 22, pp. 263-274, 1980.
21. R. Genesio, M. Tartaglia, and A. Vicino, "On the estimation of asymptotic stability regions: State of the art and new proposals," IEEE Trans. Circuits Syst., vol. CAS-31, pp. 574-581, Aug. 1984.
22. J. Guckenheimer and P. Homes, "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields," Applied Math. Sci., vol. 42, 1986.
23. J. Guddat,F. Guerra Vazquez, and H. T. Jongen, Parametric Optimization: Singularities, Pathfollowing and Jumps. New York: Wiley, 1990.
24. V. Guillemin and A. Pollack, Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974.
25. M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra. New York, NY: Academic, 1974.
26. M.W. Hirsch, Differential Topology. New York, Heidelberg, Berlin: Springer-Verlag, 1976.
27. R. Horst and P. M. Pardalos, Hand book of Global Optimization. Norwell, MA: Kluwer, 1994.
28. W. Hurewicz and H. Wallman, Dimension Theory. Princeton, NJ: Princeton Univ. Press, 1948.
29. H.T. Jongen and J. Sprekels, "The index-fc stabilizing differential equation," OR Spectrum, vol. 2, pp. 223-225, 1981.
30. H.T. Jongen, P. Jonker, and F. Twilt, Nonlinear Optimization in R". Frankfurt, Germany: Peter Lang Verlag, 1983.
31. H.K. Khalil, Nonlinear Systems. New York, NY: Macmillan, 1992.
32. M. Kuno and J. D. Seader, "Computing all real solutions to systems of nonlinear equations with global fixed-point homotopy," Ind. Eng. Chem., vol. 27, no. 7, pp. 1320-1329, 1988.
33. J. Lee, "Trajectory-based methods for global optimization," Ph.D. dissertation, Dep. Appl. Math., Cornell Univ., Ithaca, NY, 1999.
34. D. A. Liotard, "Algorithmic tools in the study of semi empirical potential surfaces," Int. J. Quantum Chem., vol. 44, pp. 723-741, 1992.
35. N. Mousseau and G. T. Barkerma, "Traveling through potential energy land scapes of disordered materials: The activation-relaxation technique," Phys. Rev. E, vol. 57, no. 2, pp. 2419-2424, 1998.
36. J. Palis and W. deMelo, Geometric Theory of Dynamical Systems: An Introduction. , Berlin, Germany: Springer-Verlag, 1982.
37. L. Perko, Differential Equations and Dynamical Systems. New York: Springer-Verlag, 1991.
38. F. Salam, A. Arapostathis, and P. Varaiya, "Analytical expression of the unstable manifold at equilibrium points in dynamical systems of differential equations," in Proc. 22nd CDC Conf., Dec. 1983.
39. J.E. Sinclair and R. Fletcher, "A new method of saddle-point location for the calculation of defect migration energies," J. Phy., Solid State Phys., vol. 7, pp. 864-870, 1974.
40. S. Smale, "Differential dynamical systems," Bull. Amer. Math. Soc., vol. 73, pp. 747-817, 1967.
41. _, "Convergent process of price adjustment and global Newton methods,"J. Math. Econom., vol. 3, pp. 107-120, 1976.
42. C.M. Smith, "How to find a saddle point," Int. J. Quantum Chem., vol. 37, pp. 773-783, 1990.
43. S. Ushiki, "Analytical expression of unstable manifolds," in Proc. Jpn. Acad., vol. 56A, 1980,p. 259.
44. L.T. Watson, "Globally convergent homotopy methods: A tutorial," Appl. Math. Comp., vol. 31, pp. 369-396, 1989.
45. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer, 1990.
46. J. Zaborsky, G. Huang, B. Zheng, and T. C. Leung, "On the phase portrait of a class of large nonlinear dynamic systems such as the power system," IEEE Trans. Automat. Contr., vol. 33, pp. 4-15, Jan., 1988.
47. J. Zaborsky,G. Huang, and B. Zheng, "A counterexample on a theorem by Tsolas et al., and an independent result by Zaborsky et al." IEEE Trans. Automat. Contr.. vol. 33, pp. 316-317, Dec. 1988.