Stochastic local search (SLS) algorithms have proven to be very competitive in solving hard computational problems. This paper investigates the foundations of SLS algorithms. We develop a simple SLS algorithm, MarkovSLS, with three search operators: greedy, noise, and restart. The search operators are controlled by probability parameters, leading to soft (probabilistic) rather than hard (deterministic) restarts. We consider two special cases of the MarkovSLS algorithm: SoftSLS and AdaptiveSLS. In SoftSLS, the probability parameters are fixed, enabling analysis using standard homogeneous Markov chains. We study the interaction between the restart and noise parameters in SoftSLS, and optimize them analytically in addition to the traditional empirical approach. Experimentally, we investigate the dependency of SoftSLS’s performance on its noise and restart parameters, validating the analytical results. AdaptiveSLS dynamically adjusts its noise and restart parameters during search. Experimentally, on synthetic and feature selection problems, we compare AdaptiveSLS with other algorithms including an analytically optimized version of SoftSLS, and find that it performs well while not requiring prior knowledge of the search space.