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New results on semidefinite bounds for ℓ1-constrained nonconvex quadratic optimization∗

Published online by Cambridge University Press:  26 August 2013

Yong Xia
Yong Xia*
Affiliation:
State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing 100191, P.R. China
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Abstract

In this paper, we show that the direct semidefinite programming (SDP) bound for the nonconvex quadratic optimization problem over ℓ1 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.

Keywords

Quadratic programmingsemidefinite programmingℓ1unit ballsparse principal component analysis
Type
Research Article
Information
RAIRO - Operations Research , Volume 47 , Issue 3 , July 2013 , pp. 285 - 297
Copyright
© EDP Sciences, ROADEF, SMAI, 2013

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