This letter considers the iterative numerical optimization of time-varying cost functions where no gradient information is available at each iteration. In this case, the proposed algorithm estimates a directional derivative by finite differences. The main contributions are the derivation of error bounds for such algorithms and proposal of optimal algorithm parameter values, e.g., step-sizes, for strongly convex cost functions. The algorithm is applied to tackle a source localization problem using a sensing agent where the source actively evades the agent. Numerical examples are provided to illustrate the theoretical results.