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Using approximate secant equations in limited memory methods for multilevel unconstrained optimization

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Abstract

The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behavior of the objective function. Following earlier work by Gratton and Toint (2009) we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take advantage of this new second-order information. We then present numerical experiments with these methods and formulate recommendations for their practical use.

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References

  1. Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  2. Broyden, C.G.: The convergence of a class of double-rank minimization algorithms. J. Inst. Math. Appl. 6, 76–90 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dai, Y., Yuan, Y.: An efficient hybrid conjugate gradient method for unconstrained optimization. Ann. Oper. Res. 103, 33–47 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dai, Y.H., Yuan, Y.: A nonlinear conjugate gradient method with a strong global convergence property. SIAM J. Optim. 10(1), 177–182 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs (1983). Reprinted as Classics in Applied Mathematics, vol. 16, SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  6. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fisher, M.: Minimization algorithms for variational data assimilation. In: Recent Developments in Numerical Methods for Atmospheric Modelling, pp. 364–385 (1998). ECMWF

    Google Scholar 

  8. Fletcher, R.: A new approach to variable metric algorithms. Comput. J. 13, 317–322 (1970)

    Article  MATH  Google Scholar 

  9. Fletcher, R., Reeves, C.M.: Function minimization by conjugate gradients. Comput. J. 7, 149–154 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  10. Goldfarb, D.: A family of variable metric methods derived by variational means. Math. Comput. 24, 23–26 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gratton, S., Malmedy, V., Toint, Ph.L.: Using approximate secant equations in limited memory methods for multilevel unconstrained optimization. Tech. rep. 2009/18, Department of Mathematics, University of Namur, Namur, Belgium (2009)

  12. Gratton, S., Mouffe, M., Sartenaer, A., Toint, Ph.L., Tomanos, D.: Numerical experience with a recursive trust-region method for multilevel nonlinear optimization. Optim. Methods Softw. 25(3), 359–386 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gratton, S., Mouffe, M., Toint, Ph.L., Weber-Mendonça, M.: A recursive trust-region method in infinity norm for bound-constrained nonlinear optimization. IMA J. Numer. Anal. 28(4), 827–861 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gratton, S., Sartenaer, A., Toint, Ph.L.: Recursive trust-region methods for multiscale nonlinear optimization. SIAM J. Optim. 19(1), 414–444 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gratton, S., Toint, Ph.L.: Approximate invariant subspaces and quasi-Newton optimization methods. Optim. Methods Softw. 25(4), 507–529 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16(1), 170–192 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hager, W.W., Zhang, H.: Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent. ACM Trans. Math. Softw. 32(1), 113–137 (2006)

    Article  MathSciNet  Google Scholar 

  18. Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2, 35–58 (2006)

    MathSciNet  MATH  Google Scholar 

  19. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49(6), 409–436 (1952)

    MathSciNet  MATH  Google Scholar 

  20. Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. Ser. B 45(1), 503–528 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nash, S.G.: A multigrid approach to discretized optimization problems. Optim. Methods Softw. 14, 99–116 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  23. Oh, S., Milstein, A., Bouman, C., Webb, K.: Multigrid algorithms for optimization and inverse problems. In: Bouman, C., Stevenson, R. (eds.) Computational Imaging. Proceedings of the SPIE, vol. 5016, pp. 59–70 (2003). DDM

    Google Scholar 

  24. Shanno, D.F.: Conditioning of quasi-Newton methods for function minimization. Math. Comput. 24, 647–657 (1970)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. ENSEEIHT-IRIT, 2, rue Camichel, 31000, Toulouse, France

    Serge Gratton

  2. Fund for Scientific Research (FNRS), 5 Rue d’Egmont, 1000, Brussels, Belgium

    Vincent Malmedy

  3. Department of Mathematics, University of Namur (FUNDP), 61, rue de Bruxelles, 5000, Namur, Belgium

    Vincent Malmedy

  4. Namur Center for Complex Systems (NAXYS), University of Namur (FUNDP), 61, rue de Bruxelles, 5000, Namur, Belgium

    Philippe L. Toint

Authors
  1. Serge Gratton
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  2. Vincent Malmedy
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  3. Philippe L. Toint
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Correspondence to Philippe L. Toint.

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Gratton, S., Malmedy, V. & Toint, P.L. Using approximate secant equations in limited memory methods for multilevel unconstrained optimization. Comput Optim Appl 51, 967–979 (2012). https://doi.org/10.1007/s10589-011-9393-3

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  • DOI: https://doi.org/10.1007/s10589-011-9393-3

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