Some of the most successful algorithms for satisfiability, such as Walksat, are based on random walks. Similarly, local search algorithms for solving constraint optimization problems benefit significantly from randomization. However, well-known algorithms such as stochastic search or simulated annealing perform a less directed random walk than used in satisfiability. By making a closer analogy to the technique used in Walksat, we obtain a different kind of randomization called random subset optimization. Experiments on both structured and random problems show strong performance compared with other local search algorithms.