Journal of Global Optimization

, Volume 18, Issue 4, pp 301–320

On Copositive Programming and Standard Quadratic Optimization Problems

  • Immanuel M. Bomze
  • Mirjam Dür
  • Etienne de Klerk
  • Cornelis Roos
  • Arie J. Quist
  • Tamás Terlaky
Article

DOI: 10.1023/A:1026583532263

Cite this article as:
Bomze, I.M., Dür, M., de Klerk, E. et al. Journal of Global Optimization (2000) 18: 301. doi:10.1023/A:1026583532263

Abstract

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

Copositive programmingGlobal maximizationPositive semidefinite matricesStandard quadratic optimization

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Immanuel M. Bomze
    • 1
  • Mirjam Dür
    • 2
  • Etienne de Klerk
    • 3
  • Cornelis Roos
    • 3
  • Arie J. Quist
    • 3
  • Tamás Terlaky
    • 4
  1. 1.ISDSUniversität WienAustria
  2. 2.Department of StatisticsVienna University of EconomicsAustria
  3. 3.Faculty ITS/TWI/SSORDelft University of TechnologyThe Netherlands
  4. 4.Department of Computing and SoftwareMcMaster University HamiltonCanada