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A modified nearly exact method for solving low-rank trust region subproblem

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Abstract

In this paper, we first discuss how the nearly exact (NE) method proposed by Moré and Sorensen [14] for solving trust region (TR) subproblems can be modified to solve large-scale “low-rank” TR subproblems efficiently. Our modified algorithm completely avoids computation of Cholesky factorizations by instead relying primarily on the Sherman–Morrison–Woodbury formula for computing inverses of “diagonal plus low-rank” type matrices. We also implement a specific version of the modified log-barrier (MLB) algorithm proposed by Polyak [17] where the generated log-barrier subproblems are solved by a trust region method. The corresponding direction finding TR subproblems are of the low-rank type and are then solved by our modified NE method. We finally discuss the computational results of our implementation of the MLB method and its comparison with a version of LANCELOT [5] based on a collection extracted from CUTEr [12] of nonlinear programming problems with simple bound constraints.

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References

  1. Breitfeld M.G., Shanno D.F. (1994) Preliminary computational experience with modified log-barrier functions for large-scale nonlinear programming. In: Hager W., Hearn D., Pardalos P. (eds) Large Scale Optimization: State of the Art. Kluwer, Dordrecht, pp. 45–67

    Google Scholar 

  2. Breitfeld M.G., Shanno D.F. (1996) Computational experience with penalty-barrier methods for nonlinear programming. Ann. Oper. Res. 62, 439–463

    Article  MathSciNet  Google Scholar 

  3. Bunch J.R., Parlett B.N. (1971) Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639–655

    Article  MathSciNet  Google Scholar 

  4. Cline A.K., Moler C.B., Stewart G.W., Wilkinson J.H. (1979) An estimate for the condition number of a matrix. SIAM J. Numer. Anal. 16, 368–375

    Article  MathSciNet  Google Scholar 

  5. Conn A.R., Gould N.I.M., Toint P.L. (1992) LANCELOT: A Fortran package for large-scale nonlinear optimization. Springer Series in Computational Mathematics 17. Springer, Berlin New York

    Google Scholar 

  6. Conn A.R., Gould N.I.M., Toint P.L. (2000) Trust-region methods. SIAM Publications, Philadelphia

    MATH  Google Scholar 

  7. Dennis J.E., Mei H.W. (1979) Two new unconstrained optimization algorithms which use function and gradient values. J. Optim. Theor. Appl. 28, 453–482

    Article  MathSciNet  Google Scholar 

  8. Fletcher R. (1980) Practical Methods of Optimization. Unconstrained Optimization 1. John Wiley, New York

    Google Scholar 

  9. Fortin C., Wolkowicz H. (2004) The trust region subproblem and semidefinite programming. Optim. Methods Softw. 19(1): 41–67

    Article  MathSciNet  Google Scholar 

  10. Gay D.M. (1981) Computing optimal locally constrained steps. SIAM J. Sci. Stat. Comput. 4(2): 186–197

    Article  MathSciNet  Google Scholar 

  11. Gould N.I.M., Lucidi S., Roma M., Toint P.L. (1999) Solving the trust-region subproblem using the Lanczos method. SIAM J. Optimization 9(2): 504–525

    Article  MathSciNet  Google Scholar 

  12. Gould, N.I.M., Orban, D., Toint, P.L.: General CUTEr documentation. Technical Report TR/PA/02/13, CERFACS, Toulouse, France (2003)

  13. Hebden, M.D.: An algorithm for minimization using exact second derivatives. Atomic Energy Research Establishment, Report T.P.515, Harwell, England (1973)

  14. Moré J.J., Sorensen D.C. (1983) Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3): 553–572

    Article  Google Scholar 

  15. Moré J.J. (1983) Recent developments in algorithms and software for trust region methods. In: Bachem A., Grotschel M., Korte B. (eds) Mathematical Programming: the State of the Art, University of Bonn. Springer, Berlin Heidelberg New York, pp. 258–287

    Google Scholar 

  16. Nocedal J., Wright S.J. (1999) Numerical optimization. Springer, Berlin Heidelberg New York

    Book  MATH  Google Scholar 

  17. Polyak R. (1992) Modified barrier functions (theory and methods). Math. Program. 54, 177–222

    Article  MathSciNet  Google Scholar 

  18. Powell M.J.D. (1970) A hybrid method for nonlinear equations. In: Rabinowitz P. (ed) Numerical Methods for Nonlinear Algebraic Equations. Gordon and Breach, New York

    Google Scholar 

  19. Reinsch C.H. (1971) Smoothing by spline functions II. Numer. Math.16, 451–454

    Article  MathSciNet  Google Scholar 

  20. Shultz G.A., Schnabel R.B., Byrd R.H. (1985) A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties. SIAM J. Numer. Anal. 22, 47–67

    Article  MathSciNet  Google Scholar 

  21. Sorensen D.C. (1982) Newton’s method with a model trust region modification. SIAM J. Numer. Anal. 19, 404–426

    Article  MathSciNet  Google Scholar 

  22. Steihaug T. (1983) The conjugate gradient method and trust regions in large-scale optimization. SIAM J. Numer. Anal. 20, 626–637

    Article  MathSciNet  Google Scholar 

  23. Toint, P.L.: Towards an efficient sparsity exploiting newton method for minimization. In: Duff, I.S. (ed.) Sparse Matrices and Their Uses, Institute of Mathematics and its Applications Conference Series, xii+387 Academic Press, Inc. London (1981)

  24. Wang Z.H., Wen Z.W., Yuan Y. (2004) A subspace trust region method for large scale unconstrained optimization. In: Yuan Y. (ed) Numerical Linear Algebra and Optimization, Proceeding of the 2003 International Conference on Numerical Optimization and Linear Algebra. Science Press, Beijing, pp. 265–274

    Google Scholar 

  25. Yuan, Y.: A Subspace Trust Region Algorithm. In: Presented at the Conference of Multiscale Optimization Methods and Applications, Center for Applied Optimization, University of Florida, USA, 26–28, February 2004

  26. Zhang J., Xu C. (1999) A class of indefinite dogleg path methods for unconstrained minimization. SIAM J. Optim. 9(3): 646–667

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 156, CANADA

    Zhaosong Lu

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Renato D. C. Monteiro

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  1. Zhaosong Lu
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  2. Renato D. C. Monteiro
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Correspondence to Zhaosong Lu.

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Lu, Z., Monteiro, R.D.C. A modified nearly exact method for solving low-rank trust region subproblem. Math. Program. 109, 385–411 (2007). https://doi.org/10.1007/s10107-006-0025-0

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