Length Scale for Characterising Continuous Optimization Problems

In metaheuristic optimization, understanding the relationship between problems and algorithms is important but non-trivial. There has been a growing interest in the literature on techniques for analysing problems, however previous work has mainly been developed for discrete problems. In this paper, we develop a novel framework for characterising continuous optimization problems based on the concept of length scale. We argue that length scale is an important property for the characterisation of continuous problems that is not captured by existing techniques. Intuitively, length scale measures the ratio of changes in the objective function value to steps between points in the search space. The concept is simple, makes few assumptions and can be calculated or estimated based only on the information available in black-box optimization (objective function values and search points). Some fundamental properties of length scale and its distribution are described. Experimental results show the potential use of length scale and directions to develop the framework further are discussed.