Efficiency of Local Search with Multiple Local Optima

The first contribution of this paper is a theoretical investigation of combinatorial optimization problems. Their landscapes are specified by the set of neighborhoods of all points of the search space. The aim of the paper consists of the estimation of the number N of local optima and the distributions of the sizes (αj) of their attraction basins. For different types of landscapes we give precise estimates of the size of the random sample that ensures that at least one point lies in each attraction basin. A practical methodology is then proposed for identifying these quantities (N and (αj) distributions) for an unknown landscape, given a random sample of starting points and a local steepest ascent search. This methodology can be applied to any landscape specified with a modification operator and provides bounds on search complexity to detect all local optima. Experiments demonstrate the efficiency of this methodology for guiding the choice of modification operators, eventually leading to the design of problem-dependent optimization heuristics.