A numerically stable dual method for solving strictly convex quadratic programs


An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example.

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  1. [1]

    R.H. Bartels, G.H. Golub, and M.A. Saunders, “Numerical techniques in mathematical programming”. in: J.B. Rosen, O.I. Mangasarian and K. Ritter, eds.,Nonlinear programming (Academic Press, New York, 1970) pp. 123–176.

    Google Scholar 

  2. [2]

    E.M.L. Beale, “On minimizing a convex function subject to linear inequalities,”Journal of the Royal Statistical Society Series B 17 (1955) 173–184.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    E.M.L. Beale, “On quadratic programming,”Naval Research Logistics Quarterly 6 (1959) 227–243.

    MathSciNet  Google Scholar 

  4. [4]

    M.G. Biggs, “Constrained minimization using recursive quadratic programming: some alternative subproblem formulations,” in: L.C.W. Dixon and G.P. Szego, eds.,Towards global optimization (North-Holland, Amsterdam, 1975) pp. 341–349.

    Google Scholar 

  5. [5]

    J.W. Bunch and L. Kaufman, “Indefinite quadratic programming,” Computing Science Technical Report 61, Bell. Labs, Murray Hill. NJ (1977).

    Google Scholar 

  6. [6]

    A.R. Conn and J.W. Sinclair, “Quadratic programming via a nondifferentiable penalty function,” Department of Combinatorics and Optimization Research Report CORR 75-15, University of Waterloo, Waterloo, Ont. (1975).

    Google Scholar 

  7. [7]

    R.W. Cottle and G.B. Dantzig, “Complementary pivot theory of mathematical programming,” in: G.B. Dantzig and A.F. Veinott, eds.Lectures in applied mathematics II, Mathematics of the decision sciences, Part 1 (American Mathematical Society, Providence, RI, 1968) pp 115–136.

    Google Scholar 

  8. [8]

    J.W. Daniel, W.B. Graggs, L. Kaufman and G.W. Stewart, “Reorthogonalization and stable algorithms for updating the Gram-Schmidt QR factorizations,”Mathematics of Computation 30 (1976) 772–795.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    A. Dax “The gradient projection method for quadratic programming,” Institute of Mathematics Report, The Hebrew University of Jerusalem (Jerusalem, 1978).

    Google Scholar 

  10. [10]

    G.B. Dantzig,Linear programming and extensions (Princeton University Press, Princeton, NJ, 1963) Chapter 24, Section 4.

    Google Scholar 

  11. [11]

    R. Fletcher, “The calculation of feasible points for linearly constrained optimization problems”.

  12. [12]

    R. Fletcher, “A FORTRAN subroutine for quadratic programming”, UKAEA Research Group Report. AERE R6370 (1970).

  13. [13]

    R. Fletcher, “A general quadratic programming algorithm”,Journal of the Institute of Mathematics and Its Applications (1971) 76–91.

  14. [14]

    P.E. Gill, G.H. Golub, W. Murray and M.A. Saunders, “Methods for modifying matrix factorizations,”Mathematics of Computation 28 (1974) 505–535.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    P.E. Gill and W. Murray, “Numerically stable methods for quadratic programming,”Mathematical programming 14 (1978) 349–372.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    D. Goldfarb, “Extension of Newton's method and simplex methods for solving quadratic programs,” in: F.A. Lootsma, ed.,Numerical methods for nonlinear optimization (Academic Press, London, 1972) pp. 239–254.

    Google Scholar 

  17. [17]

    D. Goldfarb, “Matrix factorizations in optimization of nonlinear functions subject to linear constraints,”Mathematical Programming 10 (1975) 1–31.

    Article  MathSciNet  Google Scholar 

  18. [18]

    D. Goldfarb and A. Idnani, “Dual and primal-dual methods for solving strictly convex quadratic programs,” in: J.P. Hennart, ed.,Numerical Analysis, Proceedings Cocoyoc, Mexico 1981, Lecture Notes in Mathematics 909 (Springer-Verlag, Berlin, 1982) pp. 226–239.

    Google Scholar 

  19. [19]

    A.S. Goncalves, “A primal-dual method for quadratic programming with bounded variables,” in F.A. Lootsma, ed.,Numerical methods for nonlinear optimization (Academic Press, London, 1972) pp. 255–263.

    Google Scholar 

  20. [20]

    M.D. Grigoriadis and K. Ritter, “A parametric method for semidefinite quadratic programs,”SIAM Journal of Control 7 (1969) 559–577.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    S-P. Han, “Superlinearly convergent variable metric algorithms for general nonlinear programming problems,”Mathematical Programming 11 (1976) 263–282.

    Article  MathSciNet  Google Scholar 

  22. [22]

    S-P. Han, “Solving quadratic programs by an exact penalty function,” MRC Technical Summary Report No. 2180, M.R.C., University of Wisconsin (Madison, WI, 1981).

    Google Scholar 

  23. [23]

    A.U. Idnani, “Extension of Newton's method for solving positive definite quadratic programs— A computational experience,” Master's Thesis, City College of New York, Department of Computer Science (New York, 1973).

    Google Scholar 

  24. [24]

    A.U. Idnani, “Numerically stable dual projection methods for solving positive definite quadratic programs,” Ph.D. Thesis. City College of New York. Department of Computer Science, (New York, 1980).

    Google Scholar 

  25. [25]

    C.L. Lawson and R.J. Hanson,Solving least squares problems (Prentice-Hall, Engelwood Cliffs, N.J., 1974).

    Google Scholar 

  26. [26]

    C.E. Lemke, “A method of solution for quadratic programs,”Management Science 8 (1962) 442–453.

    MATH  MathSciNet  Google Scholar 

  27. [27]

    R. Mifflin, “A stable method for solving certain constrained least squares problems,”Mathematical Programming 16 (1979) 141–158.

    MATH  Article  MathSciNet  Google Scholar 

  28. [28]

    W. Murray “An algorithm for finding a local minimum of an indefinite quadratic program”, NPL NAC Report No. 1 (1971).

  29. [29]

    B.A. Murtagh and M.A. Saunders, “Large-scale linearly constrained optimization,”Mathematical Programming 14 (1978) 41–72.

    MATH  Article  MathSciNet  Google Scholar 

  30. [30]

    M.J.D. Powell, “A fast algorithm for nonlinearly constrained optimization calculations,” in:Numerical analysis, Dundee, 1977, Lecture Notes in Mathematics 630 (Springer Verlag, Berlin, 1978) pp. 144–157.

    Google Scholar 

  31. [31]

    M.J.D. Powell, “An example of cycling in a feasible point algorithm”,Mathematical Programming 20 (1981) 353–357.

    MATH  Article  MathSciNet  Google Scholar 

  32. [32]

    K. Ritter, “Ein Verfahren zur Lösung parameter-abhängiger, nichtlinearer Maximum-Probleme”,Unternehmensforschung 6 (1962) 149–166: English transl.,Naval Research Logistics Quarterly 14 (1967) 147–162.

    MATH  Article  Google Scholar 

  33. [33]

    J.B. Rosen, “The gradient projection method for nonlinear programming. Part 1. Linear constraints,”SIAM Journal of Applied Mathematics 8 (1960) 181–217.

    MATH  Article  Google Scholar 

  34. [34]

    J.B. Rosen and S. Suzuki, “Construction of nonlinear programming test problems”,Communications of the ACM (1965) 113.

  35. [35]

    K. Schittkowski,Nonlinear programming codes—Information, tests, performance. Lecture Notes in Economics and Mathematical Systems, No. 183 (Springer-Verlag, Berlin, 1980).

    Google Scholar 

  36. [36]

    K. Schittkowski and J. Stoer, “A factorization method for the solution of constrained linear least squares problems allowing subsequent data changes,”Numerische Mathematik 31 (1979) 431–463.

    MATH  Article  MathSciNet  Google Scholar 

  37. [37]

    J. Stoer, “On the numerical solution of constrained least squares problems”,SIAM Journal on Numerical Analysis 8 (1971) 382–411.

    MATH  Article  MathSciNet  Google Scholar 

  38. [38]

    H. Theil and C. Van De Panne, “Quadratic programming as an extension of conventional quadratic maximization,”Management Science 7 (1960) 1–20.

    MATH  MathSciNet  Article  Google Scholar 

  39. [39]

    C. Van de Panne and A. Whinston, “The simplex and the dual method for quadratic programming,”Operations Research Quarterly 15 (1964) 355–389.

    Article  Google Scholar 

  40. [40]

    R.B. Wilson, “A simplicial algorithm for concave programming,” Dissertation, Garduate School of Business Administration, Harvard University (Boston, MA, 1963).

    Google Scholar 

  41. [41]

    P. Wolfe, “The simplex method for quadratic programming,”Econometrica 27 (1959) 382–398.

    MATH  Article  MathSciNet  Google Scholar 

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Additional information

This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.

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Goldfarb, D., Idnani, A. A numerically stable dual method for solving strictly convex quadratic programs. Mathematical Programming 27, 1–33 (1983). https://doi.org/10.1007/BF02591962

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Key words

  • Positive Definite Quadratic Programming
  • Matrix Factorizations
  • Dual Algorithms
  • Successive Quadratic Programming Methods