  We introduce the class of subdifferentially polynomially bounded (SPB)
functions, which is a rich class of locally Lipschitz functions that
encompasses all Lipschitz functions, all gradient- or Hessian-Lipschitz
functions, and even some non-smooth locally Lipschitz functions. We show that
SPB functions are compatible with Gaussian smoothing (GS), in the sense that
the GS of any SPB function is well-defined and satisfies a descent lemma akin
to gradient-Lipschitz functions, with the Lipschitz constant replaced by a
polynomial function. Leveraging this descent lemma, we propose GS-based
zeroth-order optimization algorithms with an adaptive stepsize strategy for
constrained minimization of SPB functions, and analyze their iteration
complexity. An important instrument in our analysis, which could be of
independent interest, is the quantification of Goldstein stationarity via the
GS gradient.
