Simple Random Sampling Estimation of the Number of Local Optima

We evaluate the performance of estimating the number of local optima by estimating their proportion in the search space using simple random sampling (SRS). The performance of this method is compared against that of the jackknife method. The methods are used to estimate the number of optima in two landscapes of random instances of some combinatorial optimisation problems. SRS provides a cheap, unbiased and accurate estimate when the proportion is not exceedingly small. We discuss choices of confidence interval in the case of extremely small proportion. In such cases, the method more likely provides an upper bound to the number of optima and can be combined with other methods to obtain a better lower bound. We suggest that SRS should be the first choice for estimating the number of optima when no prior information is available about the landscape under study.