Journal of Global Optimization

, Volume 24, Issue 2, pp 163–185

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

  • Immanuel M. Bomze
  • Etienne De Klerk

DOI: 10.1023/A:1020209017701

Cite this article as:
Bomze, I.M. & De Klerk, E. Journal of Global Optimization (2002) 24: 163. doi:10.1023/A:1020209017701


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.

Approximation algorithmsStability numberSemidefinite programmingCopositive coneStandard quadratic optimizationLinear matrix inequalities

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  • Etienne De Klerk
    • 2
  1. 1.ISDSUniversität WienViennaAustria
  2. 2.Faculty of Information Technology and SystemsDelft University of TechnologyDelftThe Netherlands