Mathematical Programming

, Volume 45, Issue 1–3, pp 373–406 | Cite as

A direct active set algorithm for large sparse quadratic programs with simple bounds

  • Thomas F. Coleman
  • Laurie A. Hulbert
Article

Abstract

We show how a direct active set method for solving definite and indefinite quadratic programs with simple bounds can be efficiently implemented for large sparse problems. All of the necessary factorizations can be carried out in a static data structure that is set up before the numeric computation begins. The space required for these factorizations is no larger than that required for a single sparse Cholesky factorization of the Hessian of the quadratic. We propose several improvements to this basic algorithm: a new way to find a search direction in the indefinite case that allows us to free more than one variable at a time and a new heuristic method for finding a starting point. These ideas are motivated by the two-norm trust region problem. Additionally, we also show how projection techniques can be used to add several constraints to the active set at each iteration. Our experimental results show that an algorithm with these improvements runs much faster than the basic algorithm for positive definite problems and finds local minima with lower function values for indefinite problems.

AMS/MOS Subject Classifications

65K05 90C20 65K10 65F30 

Key words

Quadratic programming large sparse minimization active set methods trust region methods sparse Cholesky factorization updates simple bounds box constraints 
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References

  1. [1]
    J.L. Barlow, “The use of three different rank detection methods in the solution of sparse weighted and equality constrained least squares problems,” Technical Report CS-88-18, The Pennsylvania State University (University Park, PA, 1988).Google Scholar
  2. [2]
    D.P. Bertsekas, “Projected Newton methods for optimization problems with simple constraints,”SIAM Journal on Control and Optimization 20 (1982) 221–246.Google Scholar
  3. [3]
    A. Björck, “A direct method for sparse least squares problems with lower and upper bounds,”Numerische Mathematik 54 (1988) 19–32.Google Scholar
  4. [4]
    P.H. Calamai and J.J. Moré, “Projected gradient methods for linearly constrained problems,”Mathematical Programming 39 (1987) 93–116.Google Scholar
  5. [5]
    T.F. Coleman, “A chordal preconditioner for large scale optimization,”Mathematical Programming 40 (1988) 265–287.Google Scholar
  6. [6]
    A.R. Conn, N.I.M. Gould and Ph.L. Toint, “Global convergence of a class of trust region algorithms for optimization with simple bounds,”SIAM Journal on Numerical Analysis 25 (1988) 433–460.Google Scholar
  7. [7]
    A.R. Conn, N.I.M. Gould and Ph.L. Toint, “Testing a class of methods for solving minimization problems with simple bounds on the variables,”Mathematics of Computation 50 (1988) 399–430.Google Scholar
  8. [8]
    R.S. Dembo and U. Tulowitzki, “On the minimization of quadratic functions subject to box constraints,” Technical Report B 71, Yale University (New Haven, CT, 1983).Google Scholar
  9. [9]
    J.J. Dongarra, J.R. Bunch, C.B. Moler and G.W. Stewart,LINPACK Users' Guide (SIAM, Philadelphia, PA, 1979).Google Scholar
  10. [10]
    G.C. Everstine, “A comparison of three resequencing algorithms for the reduction of matrix profile and wave front“,International Journal for Numerical Methods in Engineering 14 (1979) 837–853.Google Scholar
  11. [11]
    R. Fletcher, “A general quadratic programming algorithm,”Journal of the Institute of Mathematics and its Applications 7 (1971) 76–91.Google Scholar
  12. [12]
    R. Fletcher and M.P. Jackson, “Minimization of a quadratic function of many variables subject only to lower and upper bounds,”Journal of the Institute of Mathematics and Its Applications 14 (1974) 159–174.Google Scholar
  13. [13]
    A. George and J.W.H. Liu,Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NY, 1981).Google Scholar
  14. [14]
    J.R. Gilbert and T. Peierls, “Sparse partial pivoting in time proportional to arithmetic operations,”SIAM Journal on Scientific and Statistical Computing 9 (1988) 862–874.Google Scholar
  15. [15]
    P.E. Gill and W. Murray, “Numerically stable methods for quadratic programming,Mathematical Programming 14 (1978) 349–372.Google Scholar
  16. [16]
    P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, New York, 1981).Google Scholar
  17. [17]
    G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, MD, 1983).Google Scholar
  18. [18]
    N.J. Higham, “Fortran codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation,” Technical Report Numerical Analysis Report No. 135, University of Manchester (Manchester, UK, 1987).Google Scholar
  19. [19]
    J.J. Moré, “Numerical solution of bound constrained problems,” Technical Report 96, Argonne National Laboratory (Argonne, IL, 1987).Google Scholar
  20. [20]
    J.J. Moré and D.C. Sorensen, “Computing a trust region step,”SIAM Journal on Scientific and Statistical Computing (1983) 553–572.Google Scholar
  21. [21]
    J.J. Moré and G. Toraldo, “Algorithms for bound constrained quadratic programming problems,” Technical Report 117, Argonne National Laboratory (Argonne, IL, 1988).Google Scholar
  22. [22]
    K.G. Murty and S.N. Kabadi, “Some NP-complete problems in quadratic and nonlinear programming,”Mathematical Programming 39 (1987) 117–129.Google Scholar
  23. [23]
    S. Parter, “The use of linear graphs in Gauss elimination,”SIAM Review 3 (1961) 119–130.Google Scholar
  24. [24]
    D.J. Rose, R.E. Tarjan and G.S. Leuker, “Algorithmic aspects of vertex elimination on graphs,”SIAM Journal on Computing 5 (1976) 266–283.Google Scholar
  25. [25]
    M. Yannakakis, “Computing the minimum fill-in is NP-complete,”SIAM Journal on algebraic and Discrete Methods 2 (1981) 77–79.Google Scholar
  26. [26]
    Y. Ye and E. Tse, “A polynomial-time algorithm for convex quadratic programming,” Technical Report, Stanford University (Stanford, CA, 1986).Google Scholar

Copyright information

© North-Holland 1989

Authors and Affiliations

  • Thomas F. Coleman
    • 1
  • Laurie A. Hulbert
    • 1
  1. 1.Computer Science Department & Center for Applied MathematicsCornell UniversityIthacaUSA

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