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

Abstract.

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 A_i are of rank 1.