BIT Numerical Mathematics

, Volume 34, Issue 3, pp 372–387 | Cite as

Corrected sequential linear programming for sparse minimax optimization

  • Kristján Jónasson
  • Kaj Madsen


We present a new algorithm for nonlinear minimax optimization which is well suited for large and sparse problems. The method is based on trust regions and sequential linear programming. On each iteration a linear minimax problem is solved for a basic step. If necessary, this is followed by the determination of a minimum norm corrective step based on a first-order Taylor approximation. No Hessian information needs to be stored. Global convergence is proved. This new method has been extensively tested and compared with other methods, including two well known codes for nonlinear programming. The numerical tests indicate that in many cases the new method can find the solution in just as few iterations as methods based on approximate second-order information. The tests also show that for some problems the corrective steps give much faster convergence than for similar methods which do not employ such steps.

AMS subject classification


Key words

Nonlinear minimax optimization sequential linear programming sparse nonlinear programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Arioli, I. S. Duff, and P. P. M. de Rijk,On the augmented system approach to sparse least-squares problems, Numer. Math., 55 (1989), pp. 667–684.CrossRefGoogle Scholar
  2. 2.
    J. W. Bandler and S. H. Chen,Circuit optimization: the state of the art, IEEE Trans. Microwave Theory Tech., 36 (1988), pp. 424–443.CrossRefGoogle Scholar
  3. 3.
    A. Ben-Tal, and M. P. Bendsøe,A new method for optimal truss topology design, MAT-report no. 1991–08, Mathematical Institute, The Technical University of Denmark, 2800 Lyngby, Denmark, 1991.Google Scholar
  4. 4.
    I. Bongartz, A. R. Conn, N. Gould, and Ph. L. Toint,CUTE: Constrained and unconstrained testing environment, Tech. report TR/PA/93/10, CERFACS, 42 Avenue Gustave Coriolis, 31057 Toulouse Cedex, France, 1993.Google Scholar
  5. 5.
    K. K. Chan and P. D. Patel,Optimization of contoured beams for satellite antennas, IEE Proc.-H, Microwaves, Antennas and Propagation, 132 (1985), pp. 400–406.Google Scholar
  6. 6.
    F. H. Clarke,Optimization and Nonsmooth Analysis, J. Wiley & Sons, New York, 1983.Google Scholar
  7. 7.
    A. R. Conn, N. I. M. Gould and Ph. L. Toint,LANCELOT: A Fortran Package for Large Scale Nonlinear Optimization (Release A), Springer Series in Computational Mathematics 17, Springer Verlag, Berlin, 1992.Google Scholar
  8. 8.
    A. R. Conn and Yuying Li,An efficient algorithm for nonlinear minimax problems, report CS-88-41, University of Waterloo, Ontario, Canada, 1989.Google Scholar
  9. 9.
    I. S. Duff and J. K. Reid,A comparison of some methods for the solution of sparse overdetermined systems of linear equation, J. Inst. Math. Applics., 17 (1976), pp. 267–280.Google Scholar
  10. 10.
    I. S. Duff and J. K. Reid,The multifrontal solution of indefinite sparse symmetric linear equations, ACM Trans. Math. Software, 9 (1983), pp. 302–323.CrossRefGoogle Scholar
  11. 11.
    R. Fletcher,A model algorithm for composite nondifferentiable optimization problems, Math. Programming Study, 17 (1982), pp. 67–76.Google Scholar
  12. 12.
    R. Fletcher,Practical Methods of Optimization (2nd ed, J. Wiley & Sons, New York, 1987.Google Scholar
  13. 13.
    R. Fletcher,A first derivative method for nonlinear programming based on successive l 1 LP, in New Methods in Optimization and their Industrial Uses. State of the Art, Perspectives (ed. J.-P. Penot), Int. Ser. of Num. Math. 87, Birkhäuser Verlag, Basel, 1989, pp. 43–56.Google Scholar
  14. 14.
    R. Fletcher,Resolving degeneracy in quadratic programming, Report NA/135, Dept. of Mathematical Sciences, University of Dundee, Dundee DD1 4HN, Scotland, 1991, Annals of Oper. Res., to appear.Google Scholar
  15. 15.
    R. Fletcher and E. Sainz de la Maza,Nonlinear programming and nonsmooth optimization by successive linear programming, Math. Programming, 43 (1989), pp. 235–256.CrossRefGoogle Scholar
  16. 16.
    A. Griewank and Ph. L. Toint,Partitioned variable metric updates for large structured optimization problems, Numer. Math., 39 (1982), pp. 119–137.CrossRefGoogle Scholar
  17. 17.
    J. Hald,MMLA1Q, a FORTRAN subroutine for linearly constrained minimax optimization, Report NI-81-01, Inst. for Num. Anal., Technical University of Denmark, 2800 Lyngby, Denmark, 1981.Google Scholar
  18. 18.
    J. Hald and K. Madsen,Combined LP and quasi-Newton methods for minimax optimization, Math. Programming, 20 (1981), pp. 49–62.CrossRefGoogle Scholar
  19. 19.
    S. P. Han,Variable metric methods for minimizing a class of nondifferentiable functions, Math Programming, 20 (1981), pp. 1–13.CrossRefGoogle Scholar
  20. 20.
    W. Hock and K. Schittkowski,Test Examples for Nonlinear Programming Codes, Lecture Notes in Econ. and Math. Syst. 187, Springer Verlag, Berlin, 1981.Google Scholar
  21. 21.
    K. Jónasson,Minimax optimization using sequential linear programming with minimum norm corrective steps, Report NI-92-01, Inst. for Num. Anal., Technical University of Denmark, 2800 Lyngby, Denmark, 1992.Google Scholar
  22. 22.
    K. Jónasson and K. Madsen,Corrected sequential linear programming for sparse minimax optimization, Report NI-92-06, Inst. for Num. Anal., Technical University of Denmark, 2800 Lyngby, Denmark, 1992.Google Scholar
  23. 23.
    K. Madsen,An algorithm for minimax solution of over-determined systems of nonlinear equations, J. Inst. Math. Applics., 16 (1975), pp. 321–328.Google Scholar
  24. 24.
    K. Madsen,Minimization of Non-linear Approximation Functions, Dr. Techn. thesis, Technical University of Denmark, 2800 Lyngby, Denmark, 1986.Google Scholar
  25. 25.
    K. Madsen, O. Tingleff, P. C. Hansen, and W. Owczarz,Robust subroutines for non-linear optimization, Rep. NI-90-06, Inst. for Num. Anal., Technical University of Denmark, 2800 Lyngby, Denmark, 1990.Google Scholar
  26. 26.
    K. Madsen and H. Schjær-Jacobsen,Linearly constrained mini-max optimization, Math. Programming, 14 (1978), pp. 208–223.CrossRefGoogle Scholar
  27. 27.
    D. Q. Mayne,On the use of exact penalty functions to determine step length in optimization algorithms, in Numerical Analysis, Dundee 1979 (ed. G. A. Watson), Lecture Notes in Mathematics 773, Springer Verlag, Berlin, 1980.Google Scholar
  28. 28.
    W. Murray and M. L. Overton,A projected Lagrangian algorithm for nonlinear minimax optimization, SIAM J. Sci. Statist. Comput., 1 (1980), pp. 345–370.CrossRefGoogle Scholar
  29. 29.
    R. H. Nickel and J. W. Tolle,A sparse sequential quadratic programming algorithm, J. Optim. Theory Appl., 60, no. 3 (1989), pp. 453–473.CrossRefGoogle Scholar
  30. 30.
    A. Potchinkov and R. Reemtsen,The design of FIR filters in the complex domain by a semiinfinite optimization technique, Reports no. 327/1992 and 328/1992, Technische Universität Berlin, Berlin, 1992.Google Scholar
  31. 31.
    J. M. Ortega and W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.Google Scholar
  32. 32.
    G. A. Watson,The minimax solution of an overdetermined system of non-linear equations, J. Inst. Math. Applics., 23 (1979), pp. 167–180.Google Scholar
  33. 33.
    J. Zhang, N. Kim, and L. Lasdon,An improved successive linear programming algorithm, Management Sci., 31 no. 10 (1985), pp. 1312–1331.Google Scholar

Copyright information

© BIT Foundation 1994

Authors and Affiliations

  • Kristján Jónasson
    • 1
  • Kaj Madsen
    • 1
  1. 1.Institute of Mathematical ModellingTechnical University of DenmarkLyngbyDenmark

Personalised recommendations