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Uniform sample generation in semialgebraic sets

We propose efficient techniques for generating independent identically distributed uniform random samples inside semialgebraic sets. The proposed algorithm leverages recent results on the approximation of indicator functions by polynomials %\cite{DabHen:13} to develop acceptance/rejection based sample generation algorithms with guaranteed performance in terms of rejection rate (the number of samples that should be generated in order to obtain an accepted sample). Moreover, the {acceptance} rate is shown to be is asymptotically optimal, in the sense that it tends to one (all samples accepted) as the degree of the polynomial approximation increases. The performance of the proposed method is illustrated by a numerical example.