NisB=NisW -Towards a fair benchmark for iterative optimisers

In a previous paper [1] we have defined the Nearer is Better degree of a function, defined in [0, 1]. We use it here as a measure of how difficult it is to find the position of the minimum of a function by using an iterative optimiser. The function is said to be deceptive when the degree is greater than 0.5, neutral for value 0.5, and nice for smaller values. We will show, either formally (mainly for dimension one) or experimentally, that the cardinality of the set of neutral functions is negligible and that there are as many deceptive functions as nice ones. We also show that deceptive functions are not necessarily "monstrous". We define a taxonomy of all possible functions, according to four criteria: degree of difficulty, presence of plateaus, number of global minima, and modality (unimodal or not). It appears that the structures of two classical benchmarks (CEC 2005 and CEC 2011) are far from this taxonomy. In particular they contain almost no deceptive functions. As all iterative optimisers assume, either implicitly or explicitly, that the functions they work on are nice, it means their efficiency on such benchmarks is not a good estimate of their performance on other functions