Skip to main content
SpringerLink
Log in

Minimal representation of quadratically constrained convex feasible regions

  • Published:
Mathematical Programming Submit manuscript

Abstract

In this paper we are concerned with characterizing minimal representations of feasible regions defined by both linear and convex quadratic constraints. We say that representation is minimal if every other representation has either more quadratic constraints, or has the same number of quadratic constraints and at least as many linear constraints. We will prove that a representation is minimal if and only if it contains no redundant constraints, no pseudo-quadratic constraints and no implicit equality constraints. We define a pseudo-quadratic constraint as a quadratic constraint that can be replaced by a finite number of linear constraints. In order to prove the minimal representation theorem, we also prove that if the surfaces of two quadratic constraints match on a ball, then they match everywhere.

In this paper we also provide algorithms that can be used to detect implicit equalities and pseudoquadratic constraints. The redundant constraints can be identified using the hypersphere directions (HD) method.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D.P. Baron, “Quadratic programming with quadratic constraints,”Naval Research Logistics Quarterly 19 (1972) 105–119.

    Google Scholar 

  2. A. Boneh and A. Golan, “Constraint redundancy and feasible region boundedness by a random feasible point generator (RFPG),” contributed paper, EURO III (Amsterdam, April 1979).

    Google Scholar 

  3. R.J. Caron, M. Hlynka and J.F. McDonald, “On the best case performance of hit and run methods for detecting necessary constraints,”Mathematical Programming 54 (1992) 233–249.

    Google Scholar 

  4. R.J. Caron, J.F. McDonald and C.M. Ponic, “A degenerate extreme point strategy for the classification of linear constraints as redundant or necessary,”Journal of Optimization Theory and Applications 62 (1989) 225–237.

    Google Scholar 

  5. R.J. Caron and W.T. Obuchowska, “Quadratically constrained convex quadratic programmes: Faulty feasible regions,” Technical Report WMSR 92-05, Department of Mathematics and Statistics, University of Windsor (Windsor, Ont., 1992).

    Google Scholar 

  6. F. Cole, J.G. Ecker and W. Gochet, “A reduced gradient method for quadratic programs with quadratic constraints andl p -constrainedl p -approximation problems,”European Journal of Operational Research 9 (1982) 194–203.

    Google Scholar 

  7. J.G. Ecker and R.D. Niemi, “A dual method for quadratic programs with quadratic constraints,”SIAM Journal of Applied Mathematics 28 (1974) 568–576.

    Google Scholar 

  8. U. Eckhardt, “Representation of Convex Sets,” Proceedings of International Symposium on External Methods and Systems Analysis (Austin, Texas, 1979).

  9. S.C. Fang and J.R. Rajasekera, “Controlled perturbations for quadratically constrained quadratic programs,”Mathematical Programming 36 (1986) 276–289.

    Google Scholar 

  10. D. Goldfarb, S. Liu and S. Wang, “A logarithmic barrier function algorithm for quadratically constrained convex quadratic programming,”SIAM Journal on Optimization 1 (1991) 252–267.

    Google Scholar 

  11. W. Hock and K. Schittkowski,Test Examples for Nonlinear Programming Codes (Springer, Berlin, 1980).

    Google Scholar 

  12. F. Jarre, “On the convergence of the method of analytic centers when applied to convex quadratic programs,”Mathematical Programming 49 (1990/91) 341–358.

    Google Scholar 

  13. S. Mehrotra and J. Sun, “A method of analytic centres for quadratically constrained convex quadratic programmes,”SIAM Journal on Numerical Analysis 28 (1991) 529–544.

    Google Scholar 

  14. E. Phan-huy-Hao, “Quadratically constrained programming: some applications and a method for solution,”Zeitschrift für Operations Research 8 (1982) 105–119.

    Google Scholar 

  15. A. Schrijver,Theory of Linear and Integer Programming (J. Wiley, New York, 1986).

    Google Scholar 

  16. J. Telgen, “Minimal representation of convex polyhedral sets,”Journal of Optimization Theory and Applications 38 (1982) 1–24.

    Google Scholar 

  17. J. Telgen, “Identifying redundant constraints and implicit equalities in systems of linear constraints,”Management Science 29 (1983) 1209–1222.

    Google Scholar 

  18. C.R. Vogel, “A constrained least squares method for nonlinear ill-posed problems,”SIAM Journal on Control and Optimization 28 (1990) 34–49.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Windsor, N9B 3P4, Windsor, Ont., Canada

    Wiesława T. Obuchowska & Richard J. Caron

Authors
  1. Wiesława T. Obuchowska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Richard J. Caron
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Corresponding author.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Obuchowska, W.T., Caron, R.J. Minimal representation of quadratically constrained convex feasible regions. Mathematical Programming 68, 169–186 (1995). https://doi.org/10.1007/BF01585763

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01585763

Keywords

Search

Navigation