Improved SDP bounds for minimizing quadratic functions over the \ell^{1}-ball

Nesterov suggested an SDP-based bound for the problem to minimize a quadratic form over the -ball. In this note, we introduce a tighter SDP-based bound, the so-called copositive bound, and illustrate the improvement by simulation results. The copositive bound has the additional advantage that it can be easily extended to the inhomogeneous case of quadratic objectives including a linear term. We also indicate some improvements of the eigenvalue bound for the quadratic optimization over the -ball with 1 < p < 2, at least for p close to one.