New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation
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We consider a quadratic program with a few negative eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified \(\epsilon \)-tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within \(\epsilon \)-tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when \(r\le 10\), and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739.
KeywordsQuadratic programming Alternative direction method Convex relaxation Branch-and-bound Line search Computational complexity
Mathematics Subject Classification90C20 90C22 90C26
We would like to thank all the anonymous reviewers and the associate editor for their useful suggestions that has helped to substantially improve the presentation of this work.
- 1.Anjos, M .F., Lasserre, J .B.: Handbook of Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications. International Series in Operational Research and Management Science, p. 166. Springer, Berlin (2001)Google Scholar
- 10.Floudas, C .A., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 217–270. Kluwer Academic Publishers, Boston (1994)Google Scholar
- 12.Gould, N .I .M., Toint, P .L.: Numerical methods for large-scale non-convex quadratic programming. In: Siddiqi, A.H., Kocvara, M. (eds.) Trends in Industrial and Applied Mathematics. Applied Optimization, vol. 72, pp. 149–179. Springer, Boston (2002)Google Scholar
- 15.IBM ILOG CPLEX. IBM ILOG CPLEX 12.3 User’s Manual for CPLEX, 89 (2011)Google Scholar
- 35.Pham Dinh, T., Le Thi, H. A.: Recent advances in DC programming and DCA. In: Transactions on Computational Intelligence XIII, pp. 1–37. Springer, Berlin (2014)Google Scholar
- 48.Zhang, Y.J., So, A.M.-C.: Optimal spectrum sharing in MIMO cognitive radio networks via semidefinite programming. IEEE J. Sel. Areas Commun. 29, 362–373 (2011)Google Scholar