The case for strategic oscillation
Résumé
We study a “hard” optimization problem for metaheuristic search, where a natural neighborhood (that consists of moves for flipping the values of zero-one variables) confronts two local optima, separated by a maximum possible number of moves in the feasible space. Once a descent method reaches the first local optimum, all sequences of feasible moves to reach the second, which is the global optimum, must ultimately pass through solutions that are progressively worse until reaching the worst solution of all, which is adjacent to the global optimum. We show how certain alternative neighborhoods can locate the global more readily, but disclose that each of these approaches encounters serious difficulties by 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.