Advertisement

Mathematical Programming

, Volume 49, Issue 1–3, pp 189–211 | Cite as

A trust region algorithm for equality constrained optimization

  • M. J. D. Powell
  • Y. Yuan
Article

Abstract

A trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function. Under certain conditions global convergence and local superlinear convergence results are proved.

Key words

Equality constrained optimization exact penalty functions nonlinear programming superlinear convergence trust regions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.P. Bertsekas,Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).Google Scholar
  2. M.C. Biggs, “On the convergence of some constrained minimization algorithms based on recursive quadratic programming,”Journal of the Institute of Mathematics and its Applications 21 (1978) 67–82.Google Scholar
  3. M.C. Biggs, “A recursive quadratic programming algorithm based on the augmented Lagrangian function,” Technical Report 139, Numerical Optimisation Centre, Hatfield Polytechnic (Hatfield, England, 1983).Google Scholar
  4. P.T. Boggs, J.W. Tolle and P. Wang, “On the local convergence of quasi-Newton methods for constrained optimization,”SIAM Journal on Control and Optimization 20 (1982) 161–171.Google Scholar
  5. R.H. Byrd, R.B. Schnabel and G.A. Shultz, “A trust region algorithm for nonlinearly constrained optimization,” Technical Report CU-CS-313-85, University of Colorado (Boulder, CO, 1985).Google Scholar
  6. 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, 1985) pp. 71–82.Google Scholar
  7. R. Fletcher,Practical Methods of Optimization (John Wiley and Sons, Chichester, 1987, 2nd ed.).Google Scholar
  8. D.M. Gay, “Computing optimal locally constrained steps,”SIAM Journal on Scientific and Statistical Computing 2 (1981) 186–197.Google Scholar
  9. 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
  10. M.J.D. Powell, “Convergence properties of a class of minimization algorithms,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming, Vol. 2 (Academic Press, New York, 1975) pp. 1–27.Google Scholar
  11. M.J.D. Powell, “The convergence of variable metric methods for nonlinearly constrained optimization,” in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,Nonlinear Programming, Vol. 3 (Academic Press, New York, 1978) pp. 27–63.Google Scholar
  12. M.J.D. Powell, “Variable metric methods for constrained optimization,” in: A. Bachem, M. Grötschel and B. Korte, eds.,Mathematical Programming, The State of the Art (Springer-Verlag, Berlin, 1983) pp. 288–311.Google Scholar
  13. M.J.D. Powell, “The performance of two subroutines for constrained optimization on some difficult test problems,” in: P.T. Boggs, R.H. Byrd and R.B. Schnabel, eds.,Numerical Optimization 1984 (SIAM, Philadelphia, 1985) pp. 160–177.Google Scholar
  14. M.J.D. Powell and Y. Yuan, “A recursive quadratic programming algorithm for equality constrained optimization,”Mathematical Programming 35 (1986a) 265–278.Google Scholar
  15. M.J.D. Powell and Y. Yuan, “A trust region algorithm for equality constrained optimization,” Report DAMTP 1986/NA2, University of Cambridge, 1986b.Google Scholar
  16. K. Schittkowski, “The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function, Part I: convergence analysis,”Numerische Mathematik 38 (1981) 83–114.Google Scholar
  17. K. Schittkowski, “On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function,”Mathematische Operationsforschung und Statistik, Series Optimization 14 (1983) 197–216.Google Scholar
  18. D.C. Sorensen, “An example concerning quasi-Newton estimates of a sparse Hessian,”SIGNUM Newsletter 16, No. 2 (1981) 8–10.Google Scholar
  19. D.C. Sorensen, “Trust region methods for unconstrained optimization,” in: M.J.D. Powell, ed.,Nonlinear Optimization 1981 (Academic Press, London, 1982) pp. 29–38.Google Scholar
  20. A. Vardi, “A trust region algorithm for equality constrained minimization: convergence properties and implementation,”SIAM Journal on Numerical Analysis 22 (1985) 575–591.Google Scholar
  21. Y. Yuan, “Conditions for convergence of trust region algorithms for nonsmooth optimization,”Mathematical Programming 31 (1985) 220–228.Google Scholar
  22. Y. Yuan, “A dual algorithm for minimizing a quadratic function with two quadratic constraints,” Report DAMTP 1988/NA3, University of Cambridge (Cambridge, 1988).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1990

Authors and Affiliations

  • M. J. D. Powell
    • 1
  • Y. Yuan
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeEngland

Personalised recommendations