Problems with numerical noise form the key domain of Derivative-Free Optimization (DFO) algorithms and response surface methodology. We propose a general DFO algorithm for noisy problems. This algorithm utilizes regularized regression models that leverage between the model complexity and the level of noise in a function. For problems with controllable noise, an accuracy adjustment procedure is employed to dynamically reduce the noise level. Numerical studies on a protein alignment problem and a series of test problems illustrate the effectiveness of the proposed techniques.