  In this paper, by improving the variable-splitting approach, we propose a new
semidefinite programming (SDP) relaxation for the nonconvex quadratic
optimization problem over the $\ell_1$ unit ball (QPL1). It dominates the
state-of-the-art SDP-based bound for (QPL1). As extensions, we apply the new
approach to the relaxation problem of the sparse principal component analysis
and the nonconvex quadratic optimization problem over the $\ell_p$ ($1< p<2$)
unit ball and then show the dominance of the new relaxation.
