Abstract
In this article, we show that the \(\ell _1\)-constrained nonconvex quadratic optimization problem (QPL1) is NP-hard, even when the off-diagonal elements are all nonnegative. Then we give an answer to Pinar and Teboulle’s open problem on the nonlinear semidefinite programming relaxation of (QPL1). The analytical approach is extended to \(\ell _p\)-constrained quadratic programs with \(1<p<2\).
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Notes
The conjecture was raised in the personal communication with Professor Shu-Cherng Fang on December 2012.
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Acknowledgments
This research was supported by National Natural Science Foundation of China under Grants 11001006 and 91130019/A011702, and by the fund of State Key Laboratory of Software Development Environment under Grant SKLSDE-2013ZX-13. The author is grateful to the two anonymous referees for their valuable comments and suggestions that have greatly helped the author improve the paper.
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Hsia, Y. Complexity and nonlinear semidefinite programming reformulation of \(\ell _1\)-constrained nonconvex quadratic optimization. Optim Lett 8, 1433–1442 (2014). https://doi.org/10.1007/s11590-013-0670-1
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DOI: https://doi.org/10.1007/s11590-013-0670-1
Keywords
- Complexity
- Quadratic optimization
- \(\ell _{1}\) Unit ball
- Standard quadratic program
- Semidefinite programming