Abstract
We study two instances of polynomial optimization problem over a single sphere. The first problem is to compute the best rank-1 tensor approximation. We show the equivalence between two recent semidefinite relaxations methods. The other one arises from Bose-Einstein condensates (BEC), whose objective function is a summation of a probably nonconvex quadratic function and a quartic term. These two polynomial optimization problems are closely connected since the BEC problem can be viewed as a structured fourth-order best rank-1 tensor approximation. We show that the BEC problem is NP-hard and propose a semidefinite relaxation with both deterministic and randomized rounding procedures. Explicit approximation ratios for these rounding procedures are presented. The performance of these semidefinite relaxations are illustrated on a few preliminary numerical experiments.
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Hu, J., Jiang, B., Liu, X. et al. A note on semidefinite programming relaxations for polynomial optimization over a single sphere. Sci. China Math. 59, 1543–1560 (2016). https://doi.org/10.1007/s11425-016-0301-5
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DOI: https://doi.org/10.1007/s11425-016-0301-5
Keywords
- polynomial optimization over a single sphere
- semidefinite programming
- best rank-1 tensor approximation
- Bose-Einstein condensates