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New results on semidefinite bounds forℓ1-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 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.

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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References

Anstreicher, K., Burer, S. and Versus, D.C., Copositive Bounds for Standard QP. J. Global Optim. 33 (2005) 299–312. Google Scholar
Anstreicher, K.M. and Burer, S., Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program. 124 (2010) 33–43. Google Scholar
Bomze, I.M., Dür, M., De Klerk, E., Roos, C., Quist, A.J. and Terlaky, T., On copositive programming and standard quadratic optimization problems. J. Global Optim. 18 (2000) 301–320. Google Scholar
Bomze, I.M., Locatelli, M. and Tardella, F., New and old bounds for standard quadratic optimization: dominance, equivalence and incomparability. Math. Program. Ser. A 115 (2008) 31–64. Google Scholar
d’Aspremont, A., El Ghaoui, L., Jordan, M.I. and Lanckriet, G.R.G., A direct formulation for sparse PCA using semidefinite programming. SIAM Rev. 48 (2007) 434–448. Google Scholar
Lovasz, L. and Schrijver, A., Cones of matrices and set-functions and 0-1 optimization, SIAM. J. Optim. 1 (1991) 166–190. Google Scholar
Luss, R. and Teboulle, M., Convex Approximations to Sparse PCA via Lagrangian Duality, Oper. Res. Lett. 39 (2011) 57–61. Google Scholar
Y. Nesterov, Global Quadratic Optimization via Conic Relaxation, in Handbook of Semidefinite Programming, H. Wolkowicz, R. Saigal and L. Vandenberghe, eds., Kluwer Academic Publishers, Boston (2000) 363–384.
Pinar, M.Ç. and Teboulle, M., On semidefinite bounds for maximization of a non-convex quadratic objective over the ℓ 1 unit ball. RAIRO-Oper. Res. 40 (2006) 253–265. Google Scholar
Shor, N.Z., Quadratic optimization problems. Soviet Journal of Computer and Systems Sciences 25 (1987) 1–11. Google Scholar
H. Wolkowicz, R. Saigal and L. Vandenberghe eds., Handbook of semidefinite programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers, Boston, MA (2000).
Xia, Y., Sheu, R.L., Sun, X.L. and Li, D., Tightening a copositive relaxation for standard quadratic optimization problems. Comput. Optim. Appl. 55 (2013) 379–398. Google Scholar