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On Producing Multiple Solutions Using Repeated Trials

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Abstract

The number of trials that is required by an algorithm to produce a given fraction of the problem solutions with a specified level of confidence is analyzed. The analysis indicates that the number of trials required to find a large fraction of the solutions rapidly decreases as the number of solutions obtained on each trial by an algorithm increases. In applications where multiple solutions are sought, this decrease in the number of trials could potentially offset the additional computational cost of algorithms that produce multiple solutions on a single trial. The analysis framework presented is used to compare the efficiency of a homotopy algorithm to that of a Newton method by measuring both the number of trials and the number of calculations required to obtain a specified fraction of the solutions.

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References

  • Alexander, J.C. (1978), The topological theory of an embedding method, in H. Wacker (ed.), Continuation Methods (pp. 37–68), Academic Press.

  • Coetzee, F.M. (1995), Homotopy approaches for the analysis and solution of neural network and other nonlinear systems of equations, Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA.

    Google Scholar 

  • Coetzee, F.M. and Stonick, V.L. (1995), Sequential homotopy-based computation of multiple solutions to nonlinear equations, Proceedings ICASSP 1995, IEEE.

  • Diener, I. (1987), On the global convergence of path following methods to determine all solutions to a system of nonlinear equations, Mathematical Programming 39: 181–188.

    Google Scholar 

  • Diener, I. and Schaback, R. (1990), An extended continuous Newton method, Journal of Optimization Theory and Applications 67(1): 57–77.

    Google Scholar 

  • Dixon, L.C.W. and Szegö, G.P. (1978), The global optimization problem: An introduction, in L.C.W. Dixon and G.P. Szegö (eds), Towards Global Optimization 2 (pp. 1–15), North-Holland, Amsterdam.

    Google Scholar 

  • Drexler, F.J. (1978), A homotopy-method for the calculation of all zero-dimensional polynomial ideals, in H. Wacker (ed.), Continuation Methods (pp. 69–93), Academic Press.

  • Garcia, C.B. and Zangwill, W.I. (1979), Determining all solutions to certain systems of nonlinear equations, Mathematics of Operations Research 4(1): 1–14.

    Google Scholar 

  • Morgan, A. and Sommese, A. (1987), Computing all solutions to polynomial systems using homotopy continuation, Applied Mathematics and Computation 24: 115–138.

    Google Scholar 

  • Trajković, L., Melville, R.C., Fang, S.-C. and Watson, L.T. (1993), Artificial parameter homotopy methods for the d.c. operating point problem, IEEE Trans. on CAD 12: 861–877.

    Google Scholar 

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Authors and Affiliations

  1. Siemens Corporate Research, Princeton, NJ, 08540, USA

    Frans M. Coetzee

  2. ECE Department, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Virginia L. Stonick

Authors
  1. Frans M. Coetzee
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  2. Virginia L. Stonick
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Coetzee, F.M., Stonick, V.L. On Producing Multiple Solutions Using Repeated Trials. Journal of Global Optimization 13, 241–254 (1998). https://doi.org/10.1023/A:1008253325174

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  • DOI: https://doi.org/10.1023/A:1008253325174

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