The convergence rate of the moment-SOS hierarchy on a compact semi-algebraic set is O(1/r^L), where L is the Lojasiewicz exponent of the set's defining polynomials.
Square distance functions are Polyak-{\L}ojasiewicz and vice-versa
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abstract
This short note gathers known results to state that the squared distance function to a (nonconvex) closed set of an Euclidean space is Polyak-{\L}ojasiewicz. As a fuzzy reciprocate, we also recall that every Polyak-{\L}ojasiewicz function can be bounded from below by the squared distance function to its set of minimizers.
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On the convergence rates of moment-SOS hierarchies approximation of truncated moment sequences
The convergence rate of the moment-SOS hierarchy on a compact semi-algebraic set is O(1/r^L), where L is the Lojasiewicz exponent of the set's defining polynomials.