  This paper addresses black-box smooth optimization problems, where the
objective and constraint functions are not explicitly known but can be queried.
The main goal of this work is to generate a sequence of feasible points
converging towards a KKT primal-dual pair. Assuming to have prior knowledge on
the smoothness of the unknown objective and constraints, we propose a novel
zeroth-order method that iteratively computes quadratic approximations of the
constraint functions, constructs local feasible sets and optimizes over them.
Under some mild assumptions, we prove that this method returns an $\eta$-KKT
pair (a property reflecting how close a primal-dual pair is to the exact KKT
condition) within $O({1}/{\eta^{2}})$ iterations. Moreover, we numerically show
that our method can achieve faster convergence compared with some
state-of-the-art zeroth-order approaches. The effectiveness of the proposed
approach is also illustrated by applying it to nonconvex optimization problems
in optimal control and power system operation.
