Skip to main content
Log in

Linear inequalities, mathematical programming and matrix theory

  • Published:
Mathematical Programming Submit manuscript

Abstract

A survey is made of solvability theory for systems of complex linear inequalities.

This theory is applied to complex mathematical programming and stability and inertia theorems in matrix theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R.A. Abrams, “Nonlinear programming in complex space: Sufficient conditions and duality”, (to appear).

  2. R.A. Abrams, and A. Ben-Israel, “A duality theorem for complex quadratic programming”,Journal of Optimization Theory and Applications 4 (1969) 244–252.

    Google Scholar 

  3. R.A. Abrams, and A. Ben-Israel, “Nonlinear programming in complex space: Necessary conditions” Report No. 70-3, Series in Applied Mathematics (Northwestern University).

  4. R. Bellman,Introduction to matrix analysis (McGraw-Hill, New York, 1960).

    Google Scholar 

  5. A. Ben-Israel, “Linear equations and inequalities on finite dimensional, real or complex, vector spaces: A unified theory”,Journal of Mathematical Analysis and Applications 27 (1969) 367–389.

    Google Scholar 

  6. A. Berman, and A. Ben-Israel, “Linear equations over cones with interior: A solvability theorem with applications to matrix theory”,Linear Algebra and its Applications (Forthcoming).

  7. A. Berman, and A. Ben-Israel, “More on linear inequalities with applications to matrix theory”,Journal of Mathematical Analysis and Applications 33 (1971) 482–496.

    Google Scholar 

  8. A. Berman, and A. Ben-Israel, “A note on pencils of Hermitian or symmetric matrices”,SIAM Journal of Applied Mathematics 21 (1971) 51–54.

    Google Scholar 

  9. A. Berman, and P. Gaiha, “A generalization of irreducible monotonicity”,Linear Algebra and its Applications (Forthcoming).

  10. N. Bourbaki,Espaces vectoriels topologiques (Hermann & Cie, Paris, 1953) Chs. 1 et 2.

    Google Scholar 

  11. J. Farkas, “Über die Theorie des einfachen Ungleichungen”,Journal fur die reine und angewandte Mathematik 124 (1902) 1–24.

    Google Scholar 

  12. P. Gordan, “Über die Auflösungen linearer Gleichungen mit reelen Ceoffizienten”,Mathematische Annalen 6 (1873) 23–28.

    Google Scholar 

  13. R.D. Hill, “Inertia theorems for simultaneously triangulable complex matrices”,Linear Algebra and its Applications 2 (1969) 131–142.

    Google Scholar 

  14. N. Levinson, “Linear programming in complex space”,Journal of Mathematical Analysis and Applications 14 (1966) 44–62.

    Google Scholar 

  15. A. Lyapunov, “Problème général de stabilité du mouvement”,Annals of Mathematics Studies 17 (Princeton University Press, Princeton, N.J., 1947).

    Google Scholar 

  16. M. Marcus and M. Minc,A survey of matrix theory and matrix inequalities (Allyn and Bacon, Boston, Mass., 1964).

    Google Scholar 

  17. H. Schneider, “Positive operators and an inertia theorem”,Numerische Mathematik 7 (1965) 11–17.

    Google Scholar 

  18. P. Stein, “Some general theorems on iterants”,Journal of Research of the National Bureau of Standards 48 (1952) 11–17.

    Google Scholar 

  19. E. Stiemke, “Über positive Lösungen homogener linearer Gleichungen”,Mathematische Annalen 76 (1915) 340–342.

    Google Scholar 

  20. O. Taussky, “MatricesC withC n → 0”,Journal of Algebra 1 (1964) 5–10.

    Google Scholar 

  21. R. Varga,Matrix iterative analysis (Prentice Hall, Englewood Cliffs, N.J., 1962).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berman, A., Ben-Israel, A. Linear inequalities, mathematical programming and matrix theory. Mathematical Programming 1, 291–300 (1971). https://doi.org/10.1007/BF01584093

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01584093

Keywords

Navigation