In this paper, we show that the direct semidefinite programming (SDP) bound for thenonconvex quadratic optimization problem over ℓ1 unit ball(QPL1) is equivalent to the optimal d.c. (difference between convex) bound for thestandard quadratic programming reformulation of QPL1. Then we disprove a conjecture aboutthe tightness of the direct SDP bound. Finally, as an extension of QPL1, we study therelaxation problem of the sparse principal component analysis, denoted by QPL2L1. We showthat the existing direct SDP bound for QPL2L1 is equivalent to the doubly nonnegativerelaxation for variable-splitting reformulation of QPL2L1.
