Mathematical Programming

, Volume 86, Issue 3, pp 463–473

On maximization of quadratic form over intersection of ellipsoids with common center

  • A. Nemirovski
  • C. Roos
  • T. Terlaky

DOI: 10.1007/s101070050100

Cite this article as:
Nemirovski, A., Roos, C. & Terlaky, T. Math. Program. (1999) 86: 463. doi:10.1007/s101070050100

Abstract.

We demonstrate that if A1,...,Am are symmetric positive semidefinite n×n matrices with positive definite sum and A is an arbitrary symmetric n×n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxation \(\max_X\{\Tr(AX)\mid\, \Tr(A_iX)\le1,\,\,i=1,...,m;\,X\succeq0\} \eqno{\hbox{(SDP)}}\) of the optimization program \(x^TAx\to\max\mid\, x^TA_ix\le 1,\,\,i=1,...,m \eqno{\hbox{(P)}}\) is not worse than \(1-\frac{1}{{2\ln(2m^2)}}\). It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a~feasible solution x to (P) with \(x^TAx\ge \frac{{\Opt(\hbox{{\rm SDP}})}}{{2\ln(2m^2)}} \eqno{(*)}\) can be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to (*) is established for the case when all Ai are of rank 1.

Key words: semidefinite relaxations – quadratic programming Mathematics Subject Classification (1991): 20E28, 20G40, 20C20

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • A. Nemirovski
    • 1
  • C. Roos
    • 2
  • T. Terlaky
    • 3
  1. 1.Faculty of Industrial Engineering and Management, Technion – Israel Institute of Technology, e-mail: nemirovs@ie.technion.ac.ilIL
  2. 2.Faculty of Technical Mathematics and Informatics, Technical University of Delft, The Netherlands, e-mail: C.Roos@twi.tudelft.nlNL
  3. 3.Faculty of Technical Mathematics and Informatics, Technical University of Delft, The Netherlands, e-mail: T.Terlaky@twi.tudelft.nlNL