Local search techniques like simulated annealing and tabu search are based on a neighborhood structure defined on the set of feasible solutions of a discrete optimization problem. For the scheduling problems Pm ∥ Cmax, 1 | prec | ∑ Ui, and a large class of sequencing problems with precedence constraints having local interchange properties we replace a simple neighborhood by a neighborhood on the set of all locally optimal solutions. This allows local search on the set of solutions that are locally optimal. Computational results are presented.