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

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. Keywordspolynomial optimization over a single sphere, semidefinite programming, best rank-1 tensor approximation, Bose-Einstein condensates MSC(2010) 65K05, 90C22, 90C26 Citation: Hu J, Jiang B, Liu X, et al. A note on semidefinite programming relaxations for polynomial optimization over a single sphere.

Greedy Approaches to Symmetric Orthogonal Tensor Decomposition

“…Though general-purpose polynomial solvers based on the sum-of-squares framework [28,29,30,31,32,27,36] can be utilized (as we did in the subsection 4.2), more efficient and scalable methods (e.g. projected gradient method [26], semidefinite programming relaxations [20,21,23]), specifically tailored to the structure of (1.3), may be anticipated. This is definitely a promising future research direction.…”

“…Review on SROAwRD. Algorithm 1 is intensively studied in the tensor community, though most papers [14,8,15,16,17,18,12,13,19,20,21,22,7,23,24] focus on the numerical aspects of how to solve the best tensor rank-one approximation (1.2). Regarding theoretical guarantees for the symmetric and orthogonal decomposition, Zhang and Golub [8] first prove that SROAwRD outputs the exact symmetric and orthogonal decomposition if the input tensor is symmetric and orthogonally decomposable:…”

On The Geometric Analysis of A Quartic-quadratic Optimization Problem under A Spherical Constraint

“…Despite recent progress on the geometric properties of nonconvex minimization problems and due to the complex interaction of the quadratic and quartic terms, the landscape of (1.1) is still elusive. We further note that in [40], Hu et al have shown that the minimization problem (1.1) can be interpreted as a special instance of the partition problem and thus, it is generally NP-hard to solve (1.1).…”

A Brief Introduction to Manifold Optimization

“…The problem above can be also regarded as the best rank-1 tensor approximation of a fourth-order tensor F [39], with…”