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Exactness and Effective Degree Bound of Lasserre's Relaxation for Polynomial Optimization over Finite Variety

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arXiv:2110.13766v5 Announce Type: replace
Abstract: We consider the effective degree bound problem for Lasserre's hierarchy of sum of squares (SOS) relaxations for polynomial optimization with $n$ variables. Under the assumption that the first $n$ equality constraint defining polynomials $g_1,\ldots,g_n$ have no nontrivial common complex zero locus at infinity, and a nonsingularity condition, we establish an effective degree bound for the exactness of Lasserre's hierarchy. Our assumption holds on a Zariski open set in the space of polynomials of fixed degrees, which is much weaker than the grid condition under which the same effective degree bound was previously known. As a direct application we obtain the first explicit degree bound for gradient type SOS relaxation under a generic condition.

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