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On semidefinite bounds for maximizationof a non-convex quadraticobjective over the l1 unit ball

Published online by Cambridge University Press:  08 November 2006

Mustafa Ç. Pinar  and
Marc Teboulle
Mustafa Ç. Pinar
Affiliation:
Department of Industrial Engineering,Bilkent University, 06533 Ankara, Turkey; [email protected]
Marc Teboulle
Affiliation:
School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel; [email protected]
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Abstract

We consider the non-convex quadratic maximization problem subjectto the l1 unit ball constraint. The nature of the l 1 normstructure makes this problem extremely hard to analyze, and as aconsequence, the same difficulties are encountered when trying tobuild suitable approximations for this problem by some tractableconvex counterpart formulations. We explore some properties ofthis problem, derive SDP-like relaxations and raise openquestions.

Keywords

Non-convex quadratic optimizationL1-norm constraintsemidefinite programming relaxationduality.
Type
Research Article
Information
RAIRO - Operations Research , Volume 40 , Issue 3 , July 2006 , pp. 253 - 265
Copyright
© EDP Sciences, 2006

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References

Goemans, M.X. and Williamson, D.P., Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems using Semidefinite Programming. J. ACM 42 (1995) 1115–1145. CrossRef
J.-B. Hiriart-Urruty, Conditions for Global Optimality, in Handbook for Global Optimization. Kluwer Academic Publishers, Dordrecht, Holland (1999) 1–26.
Hiriart-Urruty, J.-B., Conditions for Global Optimality 2. J. Global Optim. 13 (1998) 349–367. CrossRef
Hiriart-Urruty, J.-B., Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints. J. Global Optim. 21 (2001) 445–455. CrossRef
R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press, Cambridge (1985).
V. Jeyakumar, A.M. Rubinov and Z.Y. Wu, Non-convex Quadratic Minimization Problems with Quadratic Constraints: Global Optimality Conditions. Technical Report AMR05/19, University of South Wales, School of Mathematics (2005).
V. Jeyakumar, A.M. Rubinov and Z.Y. Wu, Sufficient Global Optimality Conditions for Non-convex Quadratic Minimization Problems with Box Constraints. Technical Report AMR05/20, University of South Wales, School of Mathematics (2005).
Lasserre, J.-B., Global Optimization with Polynomials and the problem of moments. SIAM J. Optim. 11 (2001) 796–817. CrossRef
Lasserre, J.-B., GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi. ACM Trans. Math. Software 29 (2003) 165–194.
Laurent, M., A comparison of the Sherali-Adams, Lovasz-Schrijver and Lasserre relaxations for 0-1 programming. Math. Oper. Res. 28 (2003) 470–496. CrossRef
Lovasz, L. and Schrijver, A., Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1 (1991) 166–190. CrossRef
Nesterov, Y., Semidefinite Relaxation and Non-convex Quadratic Optimization. Optim. Methods Softw. 12 (1997) 1–20.
Y. Nesterov, Global Quadratic Optimization via Conic Relaxation, in Handbook of Semidefinite Programming, edited by H. Wolkowicz, R. Saigal and L. Vandenberghe. Kluwer Academic Publishers, Boston (2000) 363–384.
R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, NJ (1970).
Shor, N.Z., On a bounding method for quadratic extremal problems with 0-1 variables. Kibernetika 2 (1985) 48–50.
H. Wolkowicz, R. Saigal and L. Vandenberghe (Editors), Handbook of semidefinite programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers, Boston, MA (2000).
Approximating Quadratic Programming, Y. Ye with Bound and Quadratic Constraints. Math. Program. 84 (1999) 219–226.
Zhang, S., Quadratic Maximization and Semidefinite Relaxation. Math. Program. 87 (2000) 453–465. CrossRef