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Local convergence of collinear scaling algorithms related to direct least‐change secant update methods for minimization

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Abstract

Collinear scaling algorithms related to direct fixed-scale and rescaled least-change secant update methods for unconstrained minimization are derived. Theorems on local and q-superlinear convergence of these algorithms are presented. These results are extensions of those of Dennis and Walker [14] for direct least-change secant update methods.

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References

  1. K.A. Ariyawansa, Deriving collinear scaling algorithms as extensions of quasi-Newton methods and the local convergence of DFP-and BFGS-related collinear scaling algorithms, Math. Programming 49 (1990) 23–48.

    Article  MATH  MathSciNet  Google Scholar 

  2. K.A. Ariyawansa, Line search termination criteria for collinear scaling algorithms for minimizing a class of convex functions, Numer. Math. 80 (1998) 363–376.

    Article  MATH  MathSciNet  Google Scholar 

  3. K.A. Ariyawansa, W.C. Davidon and K.D. McKennon, A characterization of convexity-preserving maps on open, connected subsets of affine spaces (1997), submitted for publication.

  4. K.A. Ariyawansa, W.C. Davidon and K.D. McKennon, On Davidon’s collinear scalings and Karmarkar’s projective transformations (1997), submitted for publication.

  5. K.A. Ariyawansa and D.T.M. Lau, Local and q-superlinear convergence of a class of collinear scaling algorithms that extends quasi-Newton methods with Broyden’s bounded φ-class of updates, Optimization 23 (1992) 323–339.

    MATH  MathSciNet  Google Scholar 

  6. K.A. Ariyawansa and D.T.M. Lau, On the updating scheme in a class of collinear scaling algorithms for sparse minimization, J. Optim. Theory Appl. 75 (1992) 183–193.

    Article  MATH  MathSciNet  Google Scholar 

  7. C.G. Broyden, J.E. Dennis Jr. and J.J. Moré, On the local and superlinear convergence of quasi-Newton methods, J. Inst. Math. Appl. 12 (1973) 223–246.

    MATH  MathSciNet  Google Scholar 

  8. W.C. Davidon, Optimization by nonlinear scaling, in: Numerical Software – Needs and Availability, ed. D. Jacobs (Academic Press, London, 1977) pp. 377–383.

    Google Scholar 

  9. W.C. Davidon, Conic approximations and collinear scalings for optimizers, SIAM J. Numer. Anal. 17 (1980) 268–281.

    Article  MATH  MathSciNet  Google Scholar 

  10. J.E. Dennis Jr., D.M. Gay and R.E. Welsch, An adaptive nonlinear least-squares algorithm, ACM Trans. Math. Software 7 (1981) 348–368.

    Article  MATH  Google Scholar 

  11. J.E. Dennis Jr. and J.J. Moré, A characterization of superlinear convergence and its application to quasi-Newton methods, Math. Comp. 28 (1974) 549–560.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.E. Dennis Jr. and J.J. Moré, Quasi-Newton methods, motivation and theory, SIAM Rev. 19 (1977) 46–89.

    Article  MATH  MathSciNet  Google Scholar 

  13. J.E. Dennis Jr. and R.B. Schnabel, Least-change secant updates for quasi-Newton methods, SIAM Rev. 21 (1979) 443–459.

    Article  MATH  MathSciNet  Google Scholar 

  14. J.E. Dennis Jr. and H.F. Walker, Convergence theorems for least-change secant update methods, SIAM J. Numer. Anal. 18 (1981) 949–987; 19 (1981) 443.

    Article  MATH  MathSciNet  Google Scholar 

  15. J.E. Dennis Jr. and H.F. Walker, Inaccuracy in quasi-Newton methods: local improvement theorems, Math. Programming Study 22 (1984) 70–85.

    MATH  MathSciNet  Google Scholar 

  16. J.E. Dennis Jr. and H.F. Walker, Least-change sparse secant update methods with inaccurate secant conditions, SIAM J. Numer. Anal. 22 (1985) 760–778.

    Article  MATH  MathSciNet  Google Scholar 

  17. D.M. Gay, On solving robust and generalized linear regression problems, in: Ottimizazione non Lineare e Applicazioni, eds. S. Incerti and G. Treccani, Quaderni dell’Unione Matematica Italiana 17 (Pitagora Editrice, Bologna, Italy, 1980) pp. 55–83.

    Google Scholar 

  18. P.E. Gill and W. Murray, Newton-type methods for unconstrained and linearly constrained optimization, Math. Programming 7 (1974) 311–350.

    Article  MATH  MathSciNet  Google Scholar 

  19. L. Grandinetti, Some investigations in a new algorithm for nonlinear optimization based on conic models of the objective function, J. Optim. Theory Appl. 43 (1984) 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  20. N. Karmarkar, A new polynomial time algorithm for linear programming, Combinatorica 4 (1984) 373–395.

    MATH  MathSciNet  Google Scholar 

  21. J.E. Marsden and M.J. Hoffman, Elementary Classical Analysis (Freeman, New York, 1993).

    MATH  Google Scholar 

  22. E.S. Marwill, Exploiting sparsity in Newton-like methods, Technical Report TR 78-335, Department of Computer Science, Cornell University, Ithaca, New York (1978).

    Google Scholar 

  23. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).

    MATH  Google Scholar 

  24. D.C. Sorensen, The q-superlinear convergence of a collinear scaling algorithm for unconstrained optimization, SIAM J. Numer. Anal. 17 (1980) 84–114.

    Article  MATH  MathSciNet  Google Scholar 

  25. P.L. Toint, On sparse and symmetric updating subject to a linear equation, Math. Comp. 31 (1977) 954–961.

    Article  MATH  MathSciNet  Google Scholar 

  26. J.H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University Press, London, 1965).

    MATH  Google Scholar 

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  1. K.A. Ariyawansa
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Ariyawansa, K. Local convergence of collinear scaling algorithms related to direct least‐change secant update methods for minimization. Numerical Algorithms 18, 293–320 (1998). https://doi.org/10.1023/A:1019181701715

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