Hill-Climbing Algorithm: Let's Go for a Walk Before Finding the Optimum

Local search algorithms are one of the most developed metaheuristics to solve combinatorial optimisation problems. Particularly, hill-climbing algorithms are simple but effective techniques that have been extensively used to deal with this kind of problems. These algorithms draw paths through the search space, choosing at each step a better solution than the current solution. As already known, they stop when a local optimum is reached. It is commonly believed that the closer the solution to the local optimum, the better its fitness. This premiss has a main implication: under this intuition, it is assumed that at each step of a hill-climbing algorithm, the new solution reduces the distance to the local optima. In this paper, we prove that this statement is not necessarily true. In fact, for some permutation-based combinatorial optimisation problems, such as the Permutation Flowshop Scheduling Problem, the Linear Ordering Problem and the Quadratic Assignment Problem, when considering the 2-exchange and the insert neighbourhoods, we find that, in most of the cases, the paths defined by a hill-climbing algorithm do not monotonically reduce the distance to the local optimum. Moreover, this distance remains constant for several steps, or it even increases for some steps. We provide an analysis of the solutions found in the attraction basins according to the distance to the local optimum and to the number of steps of the algorithm. We also give some visual examples of the paths followed by the algorithm.