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Global Convergence Analysis of a New Nonmonotone BFGS Algorithm on Convex Objective Functions

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Abstract

In this paper, a new nonmonotone BFGS algorithmfor unconstrained optimization is introduced. Under mild conditions,the global convergence of this new algorithm on convex functions isproved. Some numerical experiments show that this new nonmonotoneBFGS algorithm is competitive to the BFGS algorithm.

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References

  1. R.H. Byrd, "If BFGS converges, it converges fast," manuscript, 1989.

  2. R.H. Byrd and J. Nocedal, "A tool for the analysis of quasi-Newton methods with application to unconstrained minimization," SIAM J. Numer. Anal., vol. 26, pp. 727–739, 1989.

    Google Scholar 

  3. R.H. Byrd, J. Nocedal, and Y. Yuan, "Global convergence of a class of quasi-Newton methods on convex problems," SIAM J. Numer. Anal., vol. 24, pp. 1171–1189, 1987.

    Google Scholar 

  4. J.C. Gilbert and J. Nocedal, "Global convergence properties of conjugate gradient methods for optimization," SIAM J. Optimization, vol. 2, no. 1, pp. 21–42, 1992.

    Google Scholar 

  5. L. Grippo, F. Lampariello, and S. Lucidi, "A nonmonotone linesearch technique for Newton's methods," SIAM J. Numer. Anal., vol. 23, pp. 707–716, 1986.

    Google Scholar 

  6. Han Jiye and Liu Guanghui, "General form of stepsize selection rule of linesearch and relevant analysis of global convergence of BFGS algorithm," to appear in Acta Mathematicae Applicatae Sinica, 1992.

  7. Han Jiye and Liu Guanghui, "Notes on the general form of stepsize selection," OR and Decision Making, vol. I, pp. 619-624, 1992.

  8. Liu Guanghui, Han Jiye, and Sun Defeng, "Global convergence of the BFGS algorithm with nonmonotone linesearch," Optimization, vol. 34, pp. 147–159, 1995.

    Google Scholar 

  9. Liu Guanghui and Han Jiye, "Global convergence of the variable metric algorithms with a generalized Wolfe linesearch," Technical Report, Institute of Applied Mathematics, Academia Sinica, Beijing, China, no. 029, 1993 (to be published in J. of Mathematical Research and Exposition).

    Google Scholar 

  10. Liu Guanghui and Peng Jimen, "A unified model of a variety of nonmonotonic linesearches," OR and Decision Making, vol. I, pp. 616–618.

  11. Liu Guanghui and Peng Jimen, "The convergence properties of a sort of nonmonotonic algorithm," J. of Computation Mathematics, vol. 1, pp. 65–71, 1992.

    Google Scholar 

  12. J. Nocedal, "Theory of algorithms for unconstrained optimization," Acta Numerica, pp. 1–36, 1991.

  13. J. Nocedal and C. Zhu, "Test driver for BFGS," Optimization Technology Center, 1995.

  14. J.D. Pearson, "Variable metric methods of minimization," The Computer J., vol. 12, pp. 171–178, 1969.

    Google Scholar 

  15. M.J.D. Powell, "On the Convergence of the variable metric algorithm," J. of the Institute of Mathematics and its Applications, vol. 7, pp. 21–36, 1971.

    Google Scholar 

  16. M.J.D. Powell, "Some properties of the variable metric algorithm," in F.A. Lootsma, (Ed.), Numerical Methods for Nonlinear Optimization, Academia Press: London, 1972.

    Google Scholar 

  17. M.J.D. Powell, "Some global convergence properties of a variable metric algorithm for minimization without exact linesearches," in Nonlinear Programming, SIAM-AMS Proceedings, R.W. Cottle and C.E. Lemke (Eds.), vol. IX., American Mathematical Society, Providence, RI, 1976.

  18. J. Werner, "Uber die Globale Konvergenze von variable-metric verfahren mit nichtexakter schrittweitenbestimmung," Numer. Math., vol. 31, pp. 321–334, 1978.

    Google Scholar 

  19. J. Werner, "Global convergence of quasi-Newton methods with practical linesearches," Technical Report, NAM-Bericht Nr. 67, Marz, 1989.

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Han, J., Liu, G. Global Convergence Analysis of a New Nonmonotone BFGS Algorithm on Convex Objective Functions. Computational Optimization and Applications 7, 277–289 (1997). https://doi.org/10.1023/A:1008656711925

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