An Interior-Point Algorithm for Nonconvex Nonlinear Programming

  • Robert J. Vanderbei
  • David F. Shanno
Article

Abstract

The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.

nonlinear programming interior-point methods nonconvex optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M.P. Bendsøe, A. Ben-Tal and J. Zowe, “Optimization methods for truss geometry and topology design,” Structural Optimization, vol. 7, pp. 141-159, 1994.Google Scholar
  2. 2.
    A. Brooke, D. Kendrick and A. Meeraus, GAMS: A User's Guide, Scientific Press, 1988.Google Scholar
  3. 3.
    R.H. Byrd, M.E. Hribar and J. Nocedal, “An interior point algorithm for large scale nonlinear programming,” Technical Report OTC 97/05, Optimization Technology Center, Northwestern University, 1997.Google Scholar
  4. 4.
    A.R. Conn, N. Gould and Ph.L. Toint, “A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds,” Math. of Computation, vol. 66, pp. 261-288, 1997.Google Scholar
  5. 5.
    A.R. Conn, N. Gould and Ph.L. Toint, “A primal-dual algorithm for minimizing a non-convex function subject to bound and linear equality constraints,” Technical Report Report 96/9, Dept of Mathematics, FUNDP, Namur (B), 1997.Google Scholar
  6. 6.
    A.R. Conn, N.I.M. Gould and Ph.L. Toint, LANCELOT: a Fortran Package for Large-Scale Nonlinear Optimization (Release A). Springer Verlag, Heidelberg, New York, 1992.Google Scholar
  7. 7.
    A. El-Bakry, R. Tapia, T. Tsuchiya and Y. Zhang, “On the formulation and theory of the Newton interior-point method for nonlinear programming,” J. of Optimization Theory and Appl., vol. 89, pp. 507-541, 1996.Google Scholar
  8. 8.
    A.V. Fiacco and G.P. McCormick, “Nonlinear Programming: Sequential Unconstrainted Minimization Techniques,” Research Analysis Corporation, McLean Virginia, 1968 (Republished in 1990 by SIAM, Philadelphia).Google Scholar
  9. 9.
    R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function,” Technical Report NA/171, University of Dundee, Dept. of Mathematics, Dundee, Scotland, 1997.Google Scholar
  10. 10.
    A. Forsgren and P.E. Gill, “Primal-dual interior methods for nonconvex nonlinear programming,” Technical Report NA 96-3, Department of Mathematics, University of California, San Diego, 1996.Google Scholar
  11. 11.
    R. Fourer, D.M. Gay and B.W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, Scientific Press, 1993.Google Scholar
  12. 12.
    R.S. Gajulapalli and L.S. Lasdon, “Computational experience with a safeguarded barrier algorithm for sparse nonlinear programming,” Technical report, University Texas at Austin, 1997.Google Scholar
  13. 13.
    D.M. Gay, M. L. Overton and M.H. Wright, “A primal-dual interior method for nonconvex nonlinear programming,” in Proceedings of the 1996 International Conference on Nonlinear Programming, Kluwer: Boston, 1998, to appear.Google Scholar
  14. 14.
    P.E. Gill, W. Murray, D.B. Ponceleón and M.A. Saunders. “Solving reduced KKT systems in barrier methods for linear and quadratic programming,” Technical Report SOL 91-7, Systems Optimization Laboratory, Stanford University, Stanford, CA, 1991.Google Scholar
  15. 15.
    R.L. Graham, “The largest small hexagon,” Journal of Combinatorial Theory, vol. 18, pp. 165-170, 1975.Google Scholar
  16. 16.
    W. Hock and K. Schittkowski, “Test examples for nonlinear programming codes,” volume 187 of Lecture Notes in Economics and Mathematical Systems, Springer Verlag: Heidelberg, 1981.Google Scholar
  17. 17.
    H. Lebret and S. Boyd, “Antenna array pattern synthesis via convex optimization,” IEEE Transactions on Signal Processing, vol. 45, pp. 526-532, 1997.Google Scholar
  18. 18.
    I.J. Lustig, R.E. Marsten and D.F. Shanno, “Interior point methods for linear programming: computational state of the art,” ORSA J. on Computing, vol. 6, pp. 1-14, 1994.Google Scholar
  19. 19.
    H. Mittelmann, Benchmarks for optimization software, http://plato.la.asu.edu/bench.html.Google Scholar
  20. 20.
    B.A. Murtagh and M.A. Saunders, “MINOS 5.4 user's guide,” Technical Report SOL 83-20R, Systems Optimization Laboratory, Stanford University, 1983 (Revised February, 1995).Google Scholar
  21. 21.
    D.F. Shanno and E.M. Simantiraki, “Interior-point methods for linear and nonlinear programming,” in The State of the Art in Numerical Analysis, Oxford University Press: New York, pp. 339-362, 1997.Google Scholar
  22. 22.
    R.J. Vanderbei, Large-scale nonlinear AMPL models, http://www.sor.princeton.edu/~rvdb/ampl/nlmodels/.Google Scholar
  23. 23.
    R.J. Vanderbei, “LOQO: An interior point code for quadratic programming,” Technical Report SOR 94-15, Princeton University, 1994.Google Scholar
  24. 24.
    R.J. Vanderbei, “Symmetric quasi-definite matrices,” SIAM Journal on Optimization, vol. 5,no. 1, pp. 100-113, 1995.Google Scholar
  25. 25.
    R.J. Vanderbei, Linear Programming: Foundations and Extensions, Kluwer Academic Publishers: Boston, MA, 1996.Google Scholar
  26. 26.
    T. Wang, R.D.C. Monteiro and J.S. Pang, “An interior-point potential reduction method for constrained equations,” Mathematical Programming, vol. 74, pp. 159-195, 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  • David F. Shanno
    • 1
  1. 1.Princeton UniversityUSA

Personalised recommendations