Mathematical Programming

, Volume 64, Issue 1–3, pp 249–276 | Cite as

On the solution of a two ball trust region subproblem



In this paper we investigate the structure of a two ball trust region subproblem arising frequently in nonlinear parameter identification problems and propose a method for its solution. The method decomposes the subproblem and allows the application of efficient, well studied methods for the solution of trust region subproblems arising in unconstrained optimization. In the discussion of the structure we focus on the case where both constraints are active and on the treatment of the unconstrained problem.

AMS Subject Classifications



Trust region methods Quadratic programming Gauss—Newton method Nonlinear least squares 


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  1. [1]
    H.T. Banks and K. Kunisch,Estimation Techniques for Distributed Parameter Systems (Birkhäuser-Verlag, Boston, Basel, Berlin, 1989).Google Scholar
  2. [2]
    M.R. Celis and J.E. Dennis, Jr. and R.A. Tapia “A trust region strategy for nonlinear equality constrained optimization,” in: P.T. Boggs, R.H. Byrd and R.B. Schnabel, eds.,Numerical Optimization (SIAM, Philadelphia, 1984) pp. 71–82.Google Scholar
  3. [3]
    J.E. Dennis, Jr. and R.B. Schnabel,Numerical Methods for Nonlinear Equations and Unconstrained Optimization (Prentice-Hall, Englewood Cliffs, N.J., 1983).Google Scholar
  4. [4]
    P.C. Hansen, “Truncated SVD solutions to discrete ill-posed problems with ill-determined numerical rank,”SIAM Journal on Scientific and Statistical Computations 11 (1990) 503–518.Google Scholar
  5. [5]
    J.J. Dongarra, J.R. Bunch, C.B. Moler and G.W. Stewart,LINPACK User's Guide (SIAM, Philadelphia, 1979).Google Scholar
  6. [6]
    R. Fletcher,Practical Methods of Optimization, 2nd edition (John Wiley & Sons, New York, 1987).Google Scholar
  7. [7]
    G.H. Golub and C.F. van Loan,Matrix Computations, 2nd edition (The John Hopkins University Press Baltimore, 1989).Google Scholar
  8. [8]
    M. Heinkenschloss, “Mesh independence for nonlinear least squares problems with norm constraints,”SIAM Journal on Optimization 3 (1993) 81–117.Google Scholar
  9. [9]
    M. Heinkenschloss, “Gauss—Newton methods for infinite dimensional least squares problems with norm constraints,” Ph.D. Thesis, Universität Trier (1991).Google Scholar
  10. [10]
    O.L. Mangasarian, “Solution of the symmetric linear complementarity problem by iterative methods,”Journal of Optimization Theory and Applications 22 (1977) 465–485.Google Scholar
  11. [11]
    J.M. Martinez, “Local minimizers of quadratic functions on Euclidean balls and spheres,”Siam Journal on Optimization 4 (1994) 159–176.Google Scholar
  12. [12]
    J.J. Moré, “The Levenberg—Marquardt algorithm: implementation and theory,” in: G.A. Watson, ed.,Numerical Analysis, Proceedings, Biennial Conference, Dundee 1977 (Springer Verlag, Berlin, 1977) pp. 105–116.Google Scholar
  13. [13]
    J.J. Moré, “Recent developments in algorithms and software for trust region methods,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming, The State of The Art (Springer Verlag, Berlin, 1983) pp. 258–287.Google Scholar
  14. [14]
    J.J. Moré and D. Sorensen, “Computing a trust region step,”SIAM Journal on Scientific and Statistical Computations 4 (1983) 553–572.Google Scholar
  15. [15]
    S. Omatu and J.H. Seinfeld,Distributed Parameter Systems. Theory and Applications (Oxford University Press, Oxford, 1989).Google Scholar
  16. [16]
    C.C. Paige, “Bidiagonalization of matrices and solution of linear equations,”SIAM Journal on Numerical Analysis 11 (1974) 197–209.Google Scholar
  17. [17]
    C.C. Paige and M.A. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,”ACM Transactions on Mathematical Software 8 (1982) 43–71.Google Scholar
  18. [18]
    J.-S. Pang, “Inexact Newton methods for the nonlinear complementarity problem,”Mathematical Programming (Series A) 36 (1986) 54–71.Google Scholar
  19. [19]
    J.-S. Pang, “More results on the convergence of iterative methods for the symmetric linear complementarity problem,”Journal of Optimization Theory and Applications 42 (1986) 107–134.Google Scholar
  20. [20]
    Ph.L. Toint, “Global convergence of a class of trust-region methods for nonconvex minimization in Hilbert space,”IMA Journal of Numerical Analysis 8 (1988) 231–252.Google Scholar
  21. [21]
    C.R. Vogel, “A constrained least squares regularization method for nonlinear ill-posed problem,”SIAM Journal on Control and Optimization 28 (1990) 34–49.Google Scholar
  22. [22]
    K. Williamson, “A robust trust region algorithm for nonlinear programming,” Report TR88-10, Dept. of Math. Sciences (Rice University, Houston, Texas, 1990, (revised May, 1991)).Google Scholar
  23. [23]
    Y. Yuan, “A dual algorithm for minimizing a quadratic function with two quadratic constraints,”Journal on Computational Mathematics 9 (1991) 348–359.Google Scholar
  24. [24]
    Y. Yuan, “On a subproblem of trust region algorithms for constrained optimization,”Mathematical Programming (Series A) 47 (1990) 53–63.Google Scholar
  25. [25]
    Y. Zhang, “Computing a Celis—Dennis—Tapia trust region step for equality constrained optimization,”Mathematical Programming (Series A) 55 (1992) 109–124.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc 1994

Authors and Affiliations

  1. 1.FB IV-MathematikUniversität TrierTrierGermany
  2. 2.Department of MathematicsVirginia Polytechnic Institute and State UniversityBlacksburgUSA

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