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 minf0(x)minf0(x)\min f^0(x) subject to fi(x)≤0,i=1,…,mfi(x)≤0,i=1,…,mf^i(x)\leq 0, i = 1,\ldots, m, at the same time, guaranteeing constraint satisfaction while learning 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.