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On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

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Abstract

We present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined. Different sets of conditions allow this examination to take place when null-space, range-space or Lagrangian methods are used to find stationary points for the quadratic program.

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References

  • S. Afriat, “The quadratic form positive definite on a linear manifold”,Proceedings of the Cambridge Philosophical Society 47 (1951) 1–6.

    Google Scholar 

  • J.R. Bunch and L.C. Kaufman, “Some stable methods for calculating inertia and solving symmetric linear equations”,Mathematics of Computation 31 (1977) 163–179.

    Google Scholar 

  • J.R. Bunch and L.C. Kaufman, “A computational method for the indefinite quadratic programming problem”,Linear Algebra and its Applications 34 (1980) 341–370.

    Google Scholar 

  • J.R. Bunch and B.N. Parlett, “Direct methods for solving symmetric indefinite systems of linear equations”,SIAM Journal of Numerical Analysis 8 (1971) 639–655.

    Google Scholar 

  • A.R. Conn and N.I.M. Gould, “On the location of directions of infinite descent for non-linear programming algorithms”, Technical Report CS-83-21 (1983), University of Waterloo (To appear SIAM Journal on Numerical Analysis).

  • R.W. Cottle, “Manifestations of the Schur complement”,Linear Algebra and its Applications 8 (1974) 189–211.

    Google Scholar 

  • A. Dax, Partial pivoting strategies for symmetric Gaussian elimination”,Mathematical Programming 22 (1982) 288–303.

    Google Scholar 

  • G. Debreu, “Definite and semidefinite quadratic forms”,Econometrica 20 (1952) 295–300.

    Google Scholar 

  • R.S. Dembo, contribution 3.5 in: M.J.D. Powell, ed.,Nonlinear optimization 1981 (Academic Press, London and New York, 1982) pp. 161–162.

    Google Scholar 

  • D. Djang, “Algorithmic equivalence in quadratic programming”, Ph.D. Thesis, Stanford University (Stanford, CA, 1979).

    Google Scholar 

  • I.S. Duff and J.K. Reid, “The multi-frontal solution of indefinite sparse symmetric linear systems”, Report CSS122 (1982) A.E.R.E., Harwell, England.

    Google Scholar 

  • I.S. Duff, J.K. Reid, N. Munksgaard and H.B. Nielsen, “Direct solutions of linear equations whose matrix is sparse, symmetric and indefinite”,Journal of the Institute of Mathematics and its Applications 23 (1979) 235–250.

    Google Scholar 

  • R. Fletcher, “Factorizing symmetric indefinite matrices”,Linear Algebra and its Applications 14 (1956) 257–272.

    Google Scholar 

  • R. Fletcher,Practical methods of optimization, vol. 2: Constrained optimization (John Wiley & Sons, New York, 1981).

    Google Scholar 

  • R. Fletcher and T.L. Freeman, “A modified Newton method for minimization”,Journal of Optimization Theory and Applications 23 (1977) 357–372.

    Google Scholar 

  • P.E. Gill, N.I.M. Gould, W. Murray, M.A. Saunders and M.H. Wright, “Range-space methods for convex quadratic programming”, SOL Report SOL82-14 Department of Operations Research, Stanford University (Stanford, CA, 1982).

    Google Scholar 

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

    Google Scholar 

  • P.E. Gill, W. Murray and M.H. Wright,Practical optimization (Academic Press, London and New York, 1981).

    Google Scholar 

  • D. Goldfarb, “Curvilinear path steplength algorithms for minimization which use directions of negative curvature”,Mathematical Programming 18 (1980) 31–40.

    Google Scholar 

  • G.H. Golub and C.F. Van Loan,Matrix computations (Johns Hopkins University Press, Baltimore, 1983).

    Google Scholar 

  • N.I.M. Gould, “The existence and uniqueness of solutions to general equality quadratic programming problems by range-space methods”, Research Report CORR 83-1 University of Waterloo (Waterloo, Ontario, 1983).

  • E.V. Haynsworth, “Determination of the inertia of a partitioned Hermitian matrix”,Linear Algebra and its Applications 1 (1968) 73–81.

    Google Scholar 

  • H.W. Kuhn and A.W. Tucker, “Nonlinear programming”, in J. Neyman, ed.,Proceedings of the second Berkeley symposium on mathematical statistics and probability (University of California Press, Berkeley and Los Angeles, 1951) pp. 481–492.

    Google Scholar 

  • H.B. Mann, “Quadratic forms with linear constraints”,American Mathematical Monthly 50 (1943) 430–433.

    Google Scholar 

  • J.J. More and D.C. Sorensen, “On the use of directions of negative curvature in a modified Newton method”,Mathematical Programming 16 (1979) 1–20.

    Google Scholar 

  • J. Orden, “Stationary points of quadratic functions under linear constraints”,Computer Journal 7 (1974) 238–242.

    Google Scholar 

  • D.C. Sorensen, “Updating the symmetric indefinite factorization with applications in a modified Newton”, Report ANL 77-49 Argonne National Laboratory (Argonne, IL, 1977).

  • H. Väliaho, “On the definity of quadratic forms, subject to linear constraints”,Journal of Optimization Theory and Applications 38 (1982) 143–145.

    Google Scholar 

  • J. H. Wilkinson,The algebraic eigenvalue problem (Clarendon Press, Oxford, 1965).

    Google Scholar 

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This research supported in part by the Natural Sciences and Engineering Research Council, Canada.

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Gould, N.I.M. On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem. Mathematical Programming 32, 90–99 (1985). https://doi.org/10.1007/BF01585660

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  • DOI: https://doi.org/10.1007/BF01585660

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