Confidence Intervals for the Minimum of a Function Using Extreme Value Statistics

Stochastic search algorithms are becoming an increasingly popular tool in the optimisation community. The random structure of these methods allows us to sample from the range of a function and to obtain estimates of its global minimum. However, a major advantage of stochastic search algorithms over deterministic algorithms, which is frequently unexplored, is that they also allow us to obtain interval estimates. In this paper, we put forward such advantage by providing guidance on how to combine stochastic search and optimisation methods with extreme value theory. To illustrate this approach we use several well-known objective functions. The obtained results are encouraging, suggesting that the interval estimates yield by this approach can be helpful for supplementing point estimates produced by other sophisticated optimisation methods.