Analysing and characterising optimization problems using length scale

Analysis of optimization problem landscapes is fundamental in the understanding and characterisation of problems and the subsequent practical performance of algorithms. In this paper, a general framework is developed for characterising black-box optimization problems based on length scale, which measures the change in objective function with respect to the distance between candidate solution pairs. Both discrete and continuous problems can be analysed using the framework, however, in this paper, we focus on continuous optimization. Length scale analysis aims to efficiently and effectively utilise the information available in black-box optimization. Analytical properties regarding length scale are discussed and illustrated using simple example problems. A rigorous sampling methodology is developed and demonstrated to improve upon current practice. The framework is applied to the black-box optimization benchmarking problem set, and shows greater ability to discriminate between the problems in comparison to seven well-known landscape analysis techniques. Dimensionality reduction and clustering techniques are applied comparatively to an ensemble of the seven techniques and the length scale information to gain insight into the relationships between problems. A fundamental summary of length scale information is an estimate of its probability density function, which is shown to capture salient structural characteristics of the benchmark problems.