Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences
Abstract
This paper continues our work on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences. We extend the method developed for the Schmüdgen-type hierarchy to the Putinar-type, Krivine--Stengle-type, extended-Handelman-type, and Hol-Scherer-type hierarchies. The main idea is to lift a pseudo-moment sequence to a simple set, approximate it by a moment sequence, and project the atoms of a representing measure onto the original feasible set. The Łojasiewicz inequality then converts the error in the defining constraints into an error in the moments. With the Łojasiewicz exponent 0<L≤ 1, we obtain the convergence rates O((log2 r)3L/2/rL) for the Putinar-type hierarchy and O(1/rL/2) for the normalized Krivine--Stengle-type and extended-Handelman-type hierarchies. In the matrix setting, we utilize a Chebyshev-type kernel on [−1,1]n to derive the rate O((log2 r)3L/2/rL) for the Hol-Scherer-type hierarchy. Together with our preceding work, the results provide a universal method for studying convergence rates of different types of moment-SOS hierarchies.
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BibTeX
@misc{tran2026extension,
title = {{Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences}},
author = {Hoang Anh Tran and Kim-Chuan Toh},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37286},
eprint = {2609.37286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37286},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences
AU - Hoang Anh Tran
AU - Kim-Chuan Toh
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37286
UR - https://arxiv.org/abs/2609.37286
AB - This paper continues our work on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences. We extend the method developed for the Schmüdgen-type hierarchy to the Putinar-type, Krivine--Stengle-type, extended-Handelman-type, and Hol-Scherer-type hierarchies. The main idea is to lift a pseudo-moment sequence to a simple set, approximate it by a moment sequence, and project the atoms of a representing measure onto the original feasible set. The Łojasiewicz inequality then converts the error in the defining constraints into an error in the moments. With the Łojasiewicz exponent 0<L≤ 1, we obtain the convergence rates O((log_2 r)^3L/2/r^L) for the Putinar-type hierarchy and O(1/r^L/2) for the normalized Krivine--Stengle-type and extended-Handelman-type hierarchies. In the matrix setting, we utilize a Chebyshev-type kernel on [−1,1]^n to derive the rate O((log_2 r)^3L/2/r^L) for the Hol-Scherer-type hierarchy. Together with our preceding work, the results provide a universal method for studying convergence rates of different types of moment-SOS hierarchies.
ER -