A new semidefinite relaxation for $\ell_{1}$-constrained quadratic   optimization and extensions
  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. 
Yong Xia, Yu-Jun Gong, Sheng-Nan Han
math.OC