New results on semidefinite bounds for l1‐constrained nonconvex quadratic optimization

In this paper, we show that the direct semidefinite programming (SDP) bound for the nonconvex quadratic optimization problem over 𝓁 1 unit ball (QPL1) is equivalent to the optimal d.c. (difference between convex) bound for the standard quadratic programming reformulation of QPL1. Then we disprove a conjecture about the tightness of the direct SDP bound. Finally, as an extension of QPL1, we study the relaxation problem of the sparse principal component analysis, denoted by QPL2L1. We show that the existing direct SDP bound for QPL2L1 is equivalent to the doubly nonnegative relaxation for variable‐splitting reformulation of QPL2L1.