A note on semidefinite programming relaxations for polynomial optimization over a single sphere

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 objectiv...

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Bibliographic Details
Published in:	Science China. Mathematics 2016-08, Vol.59 (8), p.1543-1560
Main Authors:	Hu, Jiang, Jiang, Bo, Liu, Xin, Wen, ZaiWen
Format:	Article
Language:	eng
Subjects:	Applications of Mathematics Approximation BEC Bose-Einstein condensates Mathematical analysis Mathematical models Mathematics Mathematics and Statistics NP难问题 Optimization Polynomials Rounding Tensors 优化问题半定规划松弛多项式注记玻色爱因斯坦凝聚球体
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Summary:	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 rela~xations are illustrated on a few preliminary numerical experiments.
ISSN:	1674-7283 1869-1862