Benchmark Functions for the CEC'2010 Special Session and Competition on Large-Scale

In the past decades, different kinds of metaheuristic optimization algorithms [1, 2] have been developed; Simulated Annealing (SA) [3, 4], Evolutionary Algorithms (EAs) [5–7], Differential Evolution (DE) [8, 9], Particle Swarm Optimization (PSO) [10, 11], Ant Colony Optimization (ACO) [12, 13], and Estimation of Distribution Algorithms (EDAs) [14, 15] are just a few of them. These algorithms have shown excellent search abilities but often lose their efficacy when applied to large and complex problems, e.g., problem instances with high dimensions, such as those with more than one hundred decision variables. Many optimization methods suffer from the “curse of dimensionality” [16, 17], which implies that their performance deteriorates quickly as the dimensionality of the search space increases. The reasons for this phenomenon appear to be two-fold. First, the solution space of a problem often increases exponentially with the problem dimension [16, 17] and more efficient search strategies are required to explore all promising regions within a given time budget. Second, also the characteristics of a problem may change with the scale. Rosenbrock’s function [18] (see also Section 2.6), for instance, is unimodal for two dimension but becomes multimodal for higher ones [19]. Because of such a worsening of the features of an optimization problem resulting from an increase in scale, a previously successful search strategy may no longer be capable of finding the optimal solution. Historically, scaling EAs to large-scale problems has attracted much interest, including both theoretical and practical studies. The earliest practical approach might be parallelizing an existing EA [20–22]. Later, cooperative coevolution appeared as another promising method [23, 24]. However, existing works on this topic are often limited to test problems used in individual studies and a systematic evaluation platform is still not available in literature for comparing the scalability of different EAs. This report aims to contribute to solving this problem. In particular, we provide a suite of benchmark functions for large-scale numerical optimization. Although the difficulty of a problem generally increases with its dimensionality, it is natural that some highdimensional problems are easier than others. For example, if the decision variables involved in a problem are independent of each other, the problem can be easily solved by decomposing it into a number of sub-problems, each of which involving only one decision variable while treating all others as constants. This way, even a line search or greedy method can solve the problem efficiently [25]. This class of problem is known as separable problems, and has been formally defined in [26] as follows: