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

Published online by Cambridge University Press:  08 November 2006

Mustafa Ç. Pinar
Department of Industrial Engineering,Bilkent University, 06533 Ankara, Turkey;
Marc Teboulle
School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv 69978, Israel;
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We consider the non-convex quadratic maximization problem subject to the l1 unit ball constraint. The nature of the l1 norm structure makes this problem extremely hard to analyze, and as a consequence, the same difficulties are encountered when trying to build suitable approximations for this problem by some tractable convex counterpart formulations. We explore some properties of this problem, derive SDP-like relaxations and raise open questions.

Research Article
© EDP Sciences, 2006

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Goemans, M.X. and Williamson, D.P., Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems using Semidefinite Programming. J. ACM 42 (1995) 11151145. 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) 349367. CrossRef
Hiriart-Urruty, J.-B., Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints. J. Global Optim. 21 (2001) 445455. 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) 796817. CrossRef
Lasserre, J.-B., GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi. ACM Trans. Math. Software 29 (2003) 165194.
Laurent, M., A comparison of the Sherali-Adams, Lovasz-Schrijver and Lasserre relaxations for 0-1 programming. Math. Oper. Res. 28 (2003) 470496. CrossRef
Lovasz, L. and Schrijver, A., Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1 (1991) 166190. CrossRef
Nesterov, Y., Semidefinite Relaxation and Non-convex Quadratic Optimization. Optim. Methods Softw. 12 (1997) 120.
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) 4850.
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) 219226.
Zhang, S., Quadratic Maximization and Semidefinite Relaxation. Math. Program. 87 (2000) 453465. CrossRef