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

Authors:
Hayato Waki;Maho Nakata;Masakazu Muramatsu
Affiliations:
Department of Computer Science, The University of Electro-Communications, Tokyo, Japan 182-8585;Advanced Center for Computing and Communication, RIKEN, Saitama, Japan 351-0198;Department of Computer Science, The University of Electro-Communications, Tokyo, Japan 182-8585
Venue:
Computational Optimization and Applications
Year:
2012

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Abstract

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.