Local Search Works Well in Practice

The figure below is an abstract, stylized depiction of local search. The solutions crowd the unshaded area, and cost decreases when we move downward. Starting from an initial solution, the algorithm moves downhill until a local optimum is reached. In general, the search space might be riddled with local optima, and some of them may be of very poor quality. The hope is that with a judicious choice of neighborhood structure, most local optima will be reasonable. Whether this is the reality or merely misplaced faith, it is an empirical fact that local search algorithms are the top performers on a broad range of optimization problems. Let's look at another such example.