Redefining the solution space is a way to make local search heuristics perform better. This is useful for tightly constrained problems where moving from one feasible solution to another is difficult.
Reformulating the problem, in particular redefining the set of decision variables, is a way to proceed. Such reformulations are well known in integer programming. They are called "extended formulations", thus leading to a considerable (possibly exponential) increase in the number of variables. For example, have a look at the paper by Sadykov and Vanderbeck for an introduction to this method.
For example, consider a machine scheduling problem where many complex constraints are defined to rule the way tasks can be performed together on each machine. In such a context, basic local search approaches perform poorly. Indeed, moving from one feasible schedule to another feasible schedule by exploring basic, local neighborhoods (that is, using basic moves and exchanges of tasks) is very hard. In that case, an extended reformulation consists of generating sets of feasible schedules for each machine (possibly heuristically) and then deciding the schedule for each machine (possibly heuristically). The second step can be viewed as solving a set partitioning problem, which can be done by using integer linear programming techniques, but also by using local search methods (moving and exchanging schedules, instead of tasks).