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In this note, we prove that for homogeneous polynomial optimization on thesphere, if the objective $f$ is generic in the input space, all feasible pointssatisfying the first order and second order necessary optimality conditions arelocal minimizers, which addresses an issue raised in the recent work byLasserre (Optimization Letters, 2021). As a corollary, this implies thatLasserre's hierarchy has finite convergence when $f$ is generic.

Optimality conditions for homogeneous polynomial optimization on the unit sphere