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.
Mots-clés : quadratic programming, semidefinite programming, $\ell _1$-unit ball, sparse principal component analysis
@article{RO_2013__47_3_285_0,
author = {Xia, Yong},
title = {New results on semidefinite bounds for $\ell _1$-constrained nonconvex quadratic optimization},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {285--297},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/ro/2013039},
mrnumber = {3143753},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ro/2013039/}
}
TY - JOUR AU - Xia, Yong TI - New results on semidefinite bounds for $\ell _1$-constrained nonconvex quadratic optimization JO - RAIRO - Operations Research - Recherche Opérationnelle PY - 2013 SP - 285 EP - 297 VL - 47 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ro/2013039/ DO - 10.1051/ro/2013039 LA - en ID - RO_2013__47_3_285_0 ER -
%0 Journal Article %A Xia, Yong %T New results on semidefinite bounds for $\ell _1$-constrained nonconvex quadratic optimization %J RAIRO - Operations Research - Recherche Opérationnelle %D 2013 %P 285-297 %V 47 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ro/2013039/ %R 10.1051/ro/2013039 %G en %F RO_2013__47_3_285_0
Xia, Yong. New results on semidefinite bounds for $\ell _1$-constrained nonconvex quadratic optimization. RAIRO - Operations Research - Recherche Opérationnelle, Tome 47 (2013) no. 3, pp. 285-297. doi : 10.1051/ro/2013039. http://www.numdam.org/articles/10.1051/ro/2013039/
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