  For safety-critical black-box optimization tasks, observations of the
constraints and the objective are often noisy and available only for the
feasible points. We propose an approach based on log barriers to find a local
solution of a non-convex non-smooth black-box optimization problem $\min
f^0(x)$ subject to $f^i(x)\leq 0,~ i = 1,\ldots, m$, at the same time,
guaranteeing constraint satisfaction while learning an optimal solution with
high probability. Our proposed algorithm exploits noisy observations to
iteratively improve on an initial safe point until convergence. We derive the
convergence rate and prove safety of our algorithm. We demonstrate its
performance in an application to an iterative control design problem.
