The case for strategic oscillation

Hard problems for metaheuristic search can be a source of insight for developing better methods. We examine a challenging instance of such a problem that has exactly two local optima, with the novel property that these two solutions are separated by a maximum possible number of moves in the feasible space. Once a descent method based on flipping 0-1 variables reaches the first local optimum, all sequences of feasible flip moves to reach the second, which is the global optimum, must ultimately pass through solutions that are progressively worse. The global optimum can only be reached by visiting these progressively worse solutions until reaching the worst solution of all, which is adjacent to the global optimum. This raises the question of what kind of method can prove successful in solving this problem efficiently. Flipping two or three variables at a time is likewise stymied by the problem structure, offering negligible improvement in the number of steps required, and entailing greatly increased effort. We show how certain alternative neighborhoods can locate the global more readily, but disclose that each of these approaches encounters serious difficulties upon slightly changing the problem formulation. We also identify other possible approaches that seem at first to be promising but turn out to have deficiencies. Finally, we observe that a strategic oscillation approach for transitioning between feasible and infeasible space overcomes these difficulties, reinforcing recent published observations about the utility of solution trajectories that alternate between feasibility and infeasibility. We also sketch features of such an approach that have implications for future research.