# American Institute of Mathematical Sciences

2015, 5(2): 185-195. doi: 10.3934/naco.2015.5.185

## A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions

 1 State Key Laboratory of Software Development Environment, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China 2 LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China, China

Received  September 2014 Revised  April 2015 Published  June 2015

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.
Citation: Yong Xia, Yu-Jun Gong, Sheng-Nan Han. A new semidefinite relaxation for $L_{1}$-constrained quadratic optimization and extensions. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 185-195. doi: 10.3934/naco.2015.5.185
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