Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences

Hoang Anh Tran; Kim-Chuan Toh

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arXiv · 2609.37286

Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences

Hoang Anh Tran·Kim-Chuan Toh

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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 $0<L\leq 1$, we obtain the convergence rates $\mathrm{O}((\log_2 r)^{3L/2}/r^{L})$ for the Putinar-type hierarchy and $\mathrm{O}(1/r^{L/2})$ 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 $\mathrm{O}((\log_2 r)^{3L/2}/r^{L})$ 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.

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BibTeX RIS

Hoang Anh Tran, Kim-Chuan Toh. 2026-09-29. Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences. https://arxiv.org/abs/2609.37286

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