Advertisement

The Sherman-Morrison Formula for the Determinant and its Application for Optimizing Quadratic Functions on Condition Sets Given by Extreme Generators

  • Gerzson Kéri
Part of the Applied Optimization book series (APOP, volume 59)

Abstract

First a short survey is made of formulas, which deal with either the inverse, or the determinant of perturbed matrices, when a given matrix is modified with a scalar multiple of a dyad or a finite sum of dyads. By applying these formulas, an algorithmic solution will be developed for optimizing general (i. e. nonconcave, nonconvex) quadratic functions on condition sets given by extreme generators. (In other words: the condition set is given by its internal representation.) The main idea of our algorithm is testing copositivity of parametral matrices.

Keywords

copositive matrix determinant calculus extreme generator matrix inversion matrix perturbation optimization parametral matrix quadratic programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Bodewig, Matrix calculus, North-Holland Publishing Co., Amsterdam, 1956.Google Scholar
  2. 2.
    I. M. Bomze, On standard quadratic optimization problems, J. Global Optimiz., 13:369–387, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    I. M. Bomze, and G. Danninger, A finite algorithm for solving general quadratic problems, J. Global Optimiz.,4:1–16,1994.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    R. W. Cottle, G. J. Habetler, and C. E. Lemke, On classes of copositive matrices, Linear Algebra and its Applications, 3:295–310,1970.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    R. W. Cottle, J. S. Pang, and R. E. Stone, The linear complementarity problem, Academic Press, 1992.zbMATHGoogle Scholar
  6. 6.
    G. Danninger, Role of copositivity in optimality criteria for nonconvex optimization problems, J. Optim. Theo. Appl.,75:535–558,1992.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    G. Danninger, and I. M. Bomze, Using copositivity for global optimality criteria in concave quadratic programming problems, Math. Programming, 62:575–580, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    W. J. Duncan, Some devices for the solution of large sets of simultaneous linear equations, Phil. Mag., 35:660–670,1944.Google Scholar
  9. 9.
    K. P. Hadeler, On Copositive Matrices, Linear Algebra and its Applications, 49:79–89,1983.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    A. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co., Boston, Massachusets, 1964.zbMATHGoogle Scholar
  11. 11.
    G. Kéri, On a class of quadratic forms, in: A. Prékopa, ed., Survey of mathematical programming, 231–247, Akadémiai Kiadó, Budapest, 1979.Google Scholar
  12. 12.
    K. G. Murty, and S. N. Kabadi, Some NP-complete problems in quadratic and linear programming, Math. Programming, 39:117–129,1987.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, McGraw-Hill, New York, 1970.zbMATHGoogle Scholar
  14. 14.
    J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix, Ann. Math. Statist., 20:621,1949.Google Scholar
  15. 15.
    H. Valiaho, Criteria for Copositive Matrices, Linear Algebra and its Applications, 81:19–34,1986.MathSciNetCrossRefGoogle Scholar
  16. 16.
    H Valiaho, Almost Copositive Matrices, Linear Algebra and its Applications, 116:121–134,1989.MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. A. Woodbury, Inverting modified matrices, Statist. Res. Group, Mem. Rep., No. 42., Princeton Univ., Princeton, N. J., 1950.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Gerzson Kéri
    • 1
  1. 1.Computer and Automation Research InstituteHungarian Academy of SciencesHungary

Personalised recommendations