Computational Optimization and Applications

, Volume 53, Issue 3, pp 823–844 | Cite as

Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization

  • Hayato Waki
  • Maho Nakata
  • Masakazu Muramatsu


We observe that in a simple one-dimensional polynomial optimization problem (POP), the ‘optimal’ values of semidefinite programming (SDP) relaxation problems reported by the standard SDP solvers converge to the optimal value of the POP, while the true optimal values of SDP relaxation problems are strictly and significantly less than that value. Some pieces of circumstantial evidences for the strange behaviors of the SDP solvers are given. This result gives a warning to users of the SDP relaxation method for POPs to be careful in believing the results of the SDP solvers. We also demonstrate how SDPA-GMP, a multiple precision SDP solver developed by one of the authors, can deal with this situation correctly.


Polynomial optimization Semidefinite programming Numerical stability 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceThe University of Electro-CommunicationsTokyoJapan
  2. 2.Advanced Center for Computing and CommunicationRIKENSaitamaJapan

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