Formal equivalence between global optimization consistency and random search

We formalize a proof that any stochastic and iterative global optimization algorithm is consistent over Lipschitz continuous functions if and only if it samples the whole search space. To achieve this, we use the L∃∀N theorem prover and the Mathlib library. The major challenge of this formalization, apart from the technical aspects of the proof itself, is to converge to a definition of a stochastic and iterative global optimization algorithm that is both general enough to encompass all algorithms of this type and specific enough to be used in a formal proof. We define such an algorithm as a pair of an initial probability measure and a sequence of Markov kernels that describe the distribution of the next point sampled by the algorithm given the previous points and their evaluations. We then construct a probability measure on finite and infinite sequences of iterations of the algorithm using the Ionescu-Tulcea theorem.