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arXiv · 2610.17371

Global Extendability of Flat Cliques in Sparse Moment Relaxations

Haoran Sun
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Abstract

Moment relaxations provide lower bounds for polynomial optimization problems, while the moment matrix carries more information than the bound itself. When the moment matrix satisfies the flat truncation condition, the bound is tight and global minimizers can be extracted from the matrix. In correlatively sparse relaxations, it is common that only some blocks are flat. We study whether such partial flatness still reveals part of a global minimizer. On a unicycle trajectory optimization problem, repeatedly extracting the atoms of flat cliques and fixing them recovers feasible, near-optimal trajectories. We then construct a counterexample whose relaxation is tight and has a rank-one flat clique, yet the unique extracted atom cannot be extended to a global minimizer. Finally, we give sufficient conditions for global extendability that involve only the boundary of a clique. Every extracted atom minimizes its clique objective given its boundary value, and when the bound is tight and the boundary moment block has rank at most one, the extracted atom of the clique is globally extendable.

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Haoran Sun. 2026-10-12. Global Extendability of Flat Cliques in Sparse Moment Relaxations. https://arxiv.org/abs/2610.17371

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