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

DOI: 10.1007/s101070050100

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

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## 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.