Failure Modes of Interior-Point SDP Solvers on Moment--SOS Hierarchies
Abstract
Standard semidefinite programming (SDP) solvers working in double precision can become unreliable on difficult problems, even when their output appears plausible. We study this issue for SDP problems arising from high-order moment-SOS hierarchies. Three polynomial optimization problems are analysed by comparing standard double-precision computations with reference solutions obtained using the extended-precision solver \mbox{Loraine\,64x4}. We observe two distinct patterns of numerical failure. In \emph{false convergence}, the computed bounds form a smooth sequence but approach the limit at the wrong rate. In \emph{stagnation}, the sequence develops a plateau or oscillations and no longer reflects the convergence of the underlying hierarchy. We introduce an empirical effective condition number based on the positive spectral components of the primal and dual solution matrices. It serves as a warning indicator rather than an error bound, and its growth accompanies numerical deterioration in our examples. We also examine the imbalance between primal and dual spectral conditioning and a Cholesky inverse residuum as potential diagnostic indicators. The results show that solver termination messages and apparently regular convergence curves may be insufficient to establish the reliability of high-order moment-SOS bounds.
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BibTeX
@misc{kocvara2026failure,
title = {{Failure Modes of Interior-Point SDP Solvers on Moment--SOS Hierarchies}},
author = {Michal Kocvara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17261},
eprint = {2610.17261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17261},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Failure Modes of Interior-Point SDP Solvers on Moment--SOS Hierarchies
AU - Michal Kocvara
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17261
UR - https://arxiv.org/abs/2610.17261
AB - Standard semidefinite programming (SDP) solvers working in double precision can become unreliable on difficult problems, even when their output appears plausible. We study this issue for SDP problems arising from high-order moment-SOS hierarchies. Three polynomial optimization problems are analysed by comparing standard double-precision computations with reference solutions obtained using the extended-precision solver \mbox{Loraine\,64x4}. We observe two distinct patterns of numerical failure. In \emph{false convergence}, the computed bounds form a smooth sequence but approach the limit at the wrong rate. In \emph{stagnation}, the sequence develops a plateau or oscillations and no longer reflects the convergence of the underlying hierarchy. We introduce an empirical effective condition number based on the positive spectral components of the primal and dual solution matrices. It serves as a warning indicator rather than an error bound, and its growth accompanies numerical deterioration in our examples. We also examine the imbalance between primal and dual spectral conditioning and a Cholesky inverse residuum as potential diagnostic indicators. The results show that solver termination messages and apparently regular convergence curves may be insufficient to establish the reliability of high-order moment-SOS bounds.
ER -