Mathematical Programming

, Volume 40, Issue 1–3, pp 247–263 | Cite as

Approximate solution of the trust region problem by minimization over two-dimensional subspaces

  • Richard H. Byrd
  • Robert B. Schnabel
  • Gerald A. Shultz


The trust region problem, minimization of a quadratic function subject to a spherical trust region constraint, occurs in many optimization algorithms. In a previous paper, the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two-dimensional trust region problem. They showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem. This paper reports computational results showing that the two-dimensional minimization approach gives nearly optimal reductions in then-dimension quadratic model over a wide range of test cases. We also show that there is very little difference, in efficiency and reliability, between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization. These results may encourage the application of similar approximate trust region techniques in other contexts.

Key words

Unconstrained optimization trust region 


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  1. M.R. Celis, J.E. Dennis 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 1984 (SIAM, Philadelphia, 1984) pp. 71–82.Google Scholar
  2. J.E. Dennis, Jr. and H.H.W. Mei, “Two new unconstrained optimization algorithms which use function and gradient values,”Journal of Optimization Theory and its Applications 28 (1979) 453–482.Google Scholar
  3. D.M. Gay, “Computing optimal locally constrained steps,”SIAM Journal on Scientific and Statistical Computing 2 (1981) 186–197.Google Scholar
  4. M.D. Hebden, “An algorithm for minimization using exact second derivatives,” Rept. TP515, A.E.R.E. (Harwell, England, 1973).Google Scholar
  5. J.J. Moré, “The Levenberg-Marquardt algorithm: implementation and theory,” in: G.A. Watson, ed.,Numerical Analysis Dundee 1977, Lecture Notes in Mathematics 630 (Springer-Verlag, Berlin, 1977) pp. 105–116.Google Scholar
  6. J.J. Moré, B.S. Garbow and K.E. Hillstrom, “Testing unconstrained optimization software,”ACM Transactions on Mathematical Software 7 (1981) 17–41.Google Scholar
  7. J.J. Moré and D.C. Sorensen, “Computing a trust region step,”SIAM Journal on Scientific and Statistical Computing 4 (1983) 553–572.Google Scholar
  8. B.N. Parlett,The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).Google Scholar
  9. M.J.D. Powell, “A new algorithm for unconstrained optimization,” in: J.B. Rosen, O. L. Mangasarian and K. Ritter, eds.,Nonlinear Programming (Academic Press, New York, 1970) pp. 31–65.Google Scholar
  10. R.B. Schnabel and P. Frank, “Tensor methods for nonlinear equations,”SIAM Journal on Numerical Analysis 21 (1984) 815–843.Google Scholar
  11. R.B. Schnabel and P. Frank, “Solving systems of nonlinear equations by tensor methods,” Technical Report CU-CS-334-86, Department of Computer Science, University of Colorado at Boulder (Boulder, CO, 1986).Google Scholar
  12. R.B. Schnabel, B.E. Weiss and J.E. Koontz, “A modular system of algorithms for unconstrained minimization,”ACM Transactions on Mathematical Software 11 (1985) 419–440.Google Scholar
  13. L. Schrage, “A more portable Fortran random number generator,”ACM Transactions on Mathematical Software 5 (1979) 132–138.Google Scholar
  14. G. A. Shultz, R. B. Schnabel and R.H. Byrd, “A family of trust-region-based algorithms for unconstrained minimization with strong global convergence properties“,SIAM Journal on Numerical Analysis 22 (1985) 47–67.Google Scholar
  15. D.C. Sorensen, “Newton's method with a model trust region modification,”SIAM Journal on Numerical Analysis 19 (1982) 409–426.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1988

Authors and Affiliations

  • Richard H. Byrd
    • 1
  • Robert B. Schnabel
    • 1
  • Gerald A. Shultz
    • 1
    • 2
  1. 1.Department of Computer ScienceUniversity of ColoradoBoulderUSA
  2. 2.Department of Mathematical SciencesMetropolitan State CollegeDenverUSA

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