Abstract
In this paper, we propose a shifted power method for a type of polynomial optimization problem over unit spheres. The global convergence of the proposed method is established and an easily implemented scope of the shifted parameter is provided.
Highlights
We propose a shifted power method for a type of polynomial optimization problem over unit spheres
It is well known that tensor is a useful tool in polynomial optimization as a polynomial, especially a homogenous polynomial, has a very simple expression with the aid of tensor, and the optimal condition of a homogenous polynomial optimization with a special structured feasible region can be deeply characterized (Qi 2005, Qi 2009)
We extend the shifted power method to a more general type of polynomial optimization problem (1.1) and establish the convergence of the method
Summary
The polynomial optimization problem in which the objective function and constraints are polynomial functions received much attention recently due to their wide applications in such as signal processing (Ghosh 2008, Qi 2003), biomedical engineering (Kofidis 2002, Lasserre 2001), material science (Soare 2008), quantum mechanics (Dahl et al 2008, Wang et al 2009), and numerical linear algebra (Hof 2009, Qi 2005), see (Klerk 2008) for a survey on the various classes of polynomial optimization with simplex, hypercube, or sphere constraints. Where A is a partially symmetric tensor w.r.t. index blocks (i1i2 · · · id1 ), (id1+1id1+2, · · · id1+d2 ), · · · , (id1+d2+···+ds−1+1 · · · id) This problem contains the problem of finding best rank-1 approximation or computing the largest eigenvalue in magnitude of a super-symmetric tensor (Kofidis 2002, Kolda 2011, Zhang 2012, Qi 2009) as special cases. We extend the shifted power method to a more general type of polynomial optimization problem (1.1) and establish the convergence of the method. (1) We apply the shifted power method to a more general type of polynomial optimization problem defined on the unit spheres and establish its convergence.
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