TY - JOUR
T1 - Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization
AU - Waki, Hayato
AU - Nakata, Maho
AU - Muramatsu, Masakazu
N1 - Funding Information:
Acknowledgements The first author thanks Professor Monique Laurent for hosting him during his stay in Centrum Wiskunde & Informatica (CWI). The first author was supported by Grant-in-Aid for JSPS Fellows 18005736 and 20003236. The second author was supported by The Special Postdoctoral Researchers’ Program of RIKEN. The third author was partially supported by Grant-in-Aid for Scientific Research (C) 19560063.
PY - 2012/12
Y1 - 2012/12
N2 - 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.
AB - 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.
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U2 - 10.1007/s10589-011-9437-8
DO - 10.1007/s10589-011-9437-8
M3 - Article
AN - SCOPUS:80053360075
SN - 0926-6003
VL - 53
SP - 823
EP - 844
JO - Computational Optimization and Applications
JF - Computational Optimization and Applications
IS - 3
ER -