Skip to main content
SpringerLink
Log in

Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMI's). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.

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. Bellare, M. and Rogaway, P. (1995), The Complexity of Approximating a Nonlinear Program, Mathematical Programming 69, 429–441.

    Google Scholar 

  2. Berkelaar, A.B., Jansen, B., Roos, C. and Terlaky, T. (1996), Sensitivity Analysis in (Degenerate) Quadratic Programming. Technical Report TWI 96–26, Reports of the Faculty of Technical Mathematics and Informatics, Delft University of Technology.

  3. Bomze, I.M. (1998), On Standard Quadratic Optimization Problems, Journal of Global Optimization 13, 369–387.

    Google Scholar 

  4. Bomze, I.M., Dür, M., de Klerk, E., Roos, C., Quist, A. and Terlaky, T. (2000), On Copositive Programming and Standard Quadratic Optimization Problems, Journal of Global Optimization 18, 301–320.

    Google Scholar 

  5. de Klerk, E. and Pasechnik, D.V. (2002), Approximation of the Stability Number of a Graph via Copositive Programming, SIAM Journal of Optimization 12(4), 875–892.

    Google Scholar 

  6. Goemans, M.X. and Williamson, D.P. (1995), Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM 42, 1115–1145.

    Google Scholar 

  7. Hastad, J. (1999), Clique is Hard to Approximate Within uV u, Acta Mathematica 182, 105–142.

    Google Scholar 

  8. Motzkin, T.S. and Straus, E.G. (1965), Maxima for Graphs and a New Proof of a Theorem of Túran, Canadian J. Math. 17, 533–540.

    Google Scholar 

  9. Murty, K.G. and Kabadi, S.N. (1987), Some NP-Complete Problems in Quadratic and Linear Programming, Mathematical Programming 39, 117–129.

    Google Scholar 

  10. Nesterov, Y.E. (1999), Global Quadratic Optimization on the Sets with Simplex Structure. Discussion paper 9915, CORE, Katholic University of Louvain, Belgium.

    Google Scholar 

  11. Nesterov, Y.E. and Nemirovskii, A.S. (1994), Interior Point Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia, PA.

    Google Scholar 

  12. Nesterov, Y.E., Wolkowicz, H. and Ye, Y. (2000), Nonconvex Quadratic Optimization, in Wolkowicz, H., Saigal, R. and Vandenberghe, L. (eds.), Handbook of Semidefinite Programming, pp. 361–416. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  13. Papadimitriou, C.H. and Steiglitz, K. (1982), Combinatorial Optimization. Algorithms and Complexity. Prentice Hall, Inc., Englewood Cliffs, N.J.

    Google Scholar 

  14. Parrilo, P.A. (2000), Structured Semidefinite Programs and Semi-algebraic Geometry Methods in Robustness and Optimization. PhD thesis, California Institute of Technology, Pasadena, California, USA. Available at: http://www.cds.caltech.edu/pablo/.

    Google Scholar 

  15. Parrilo, P.A. and Sturmfels, B. (2001), Minimizing polynomial functions, Tech. Rep. math. OC/0103170, DIMACS, Rutgers University, March.

    Google Scholar 

  16. Pólya, G. (1928), Uber positive Darstellung von Polynomen. Vierteljschr. Naturforsch. Ges. ¨ Zurich, 73, 141–145 (also Collected Papers 2, 309–313, MIT Press, Cambridge, MA, London, 1974).

    Google Scholar 

  17. Powers, V. and Reznick, B. (2001), A New Bound for Pólya's Theorem with Applications to Polynomials Positive on Polyhedra, J. Pure Appl. Alg. 164, 221–229.

    Google Scholar 

  18. J. Renegar (2001), A Mathematical View of Interior-Point Methods in Convex Optimization. Forthcoming, SIAM, Philadelphia, PA.

  19. Quist, A.J., de Klerk, E., Roos, C. and Terlaky, T. (1998), Copositive Relaxation for General Quadratic Programming, Optimization Methods and Software 9, 185–209.

    Google Scholar 

  20. Sturm, J.F. (1999), Using SeDuMi 1.02, a MATLAB Toolbox for Optimization Over Symmetric Cones, Optimization Methods and Software 11–12, 625–653.

    Google Scholar 

  21. Vickers, G.T. and Cannings, C. (1988), Patterns of ESS's I, J. Theor. Biol. 132, 387–408.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ISDS, Universität Wien, Vienna, Austria

    Immanuel M. Bomze

  2. Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Etienne De Klerk

Authors
  1. Immanuel M. Bomze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Etienne De Klerk
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bomze, I.M., De Klerk, E. Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming. Journal of Global Optimization 24, 163–185 (2002). https://doi.org/10.1023/A:1020209017701

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020209017701

Search

Navigation