Maximum block improvement and polynomial optimization

Bilian Chen, Simai He, Zhening Li, Shuzhong Zhang

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In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we accept only a block update that achieves the maximum improvement, hence the name of our new search method: maximum block improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proved. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.
Original languageEnglish
Pages (from-to)87-107
JournalSIAM Journal on Optimization
Issue number1
Early online date24 Jan 2012
Publication statusPublished - Mar 2012


  • block coordinate descent
  • polynomial optimization problem
  • tensor form


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