On Maximization of Quadratic formover Intersection of Ellipsoids

We demonstrate that if A1; :::; Am are symmetric positive semide nite n n matrices with positive de nite sum and A is an arbitrary symmetric n n matrix, then the quality of the semide nite relaxation max X fTr(AX) j Tr(AiX) 1; i = 1; :::;m; X 0g (SDP) of the optimization program xTAx! max j xTAix 1; i = 1; :::;m (P) is not worse than 1 2 ln(2m2) . 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 xTAx Opt(SDP) 2 ln(2m2) ( ) can be found e ciently. This somehow improves one of the results of Nesterov (1998) where bound similar to (*) is established for the case when all Ai are of rank 1.