In this article we present a robustness analysis of the extraction of optimizers in polynomial optimization. Optimizers can be extracted by solving moment problems using flatness and the Gelfand-Naimark-Segal (GNS) construction. Here a modification of the GNS construction is presented that applies even to non-flat data, and then its sensitivity under perturbations is studied. The focus is on eigenvalue optimization for noncommutative polynomials, but it is also explained how the main results pertain to commutative and tracial optimization.
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