, Volume 86, Issue 3, pp 463-473

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

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We demonstrate that if A 1,...,A m 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.

Received August 13, 1998 / Revised version received May 25, 1999¶ Published online September 15, 1999