We consider the global optimization problem for d-variate Lipschitz functions which, in a certain sense, do not increase too slowly in a neighborhood of the global minimizer(s). On these functions, we apply optimization algorithms which use only function values. We propose two adaptive deterministic methods. The first one applies in a situation when the Lipschitz constant L is known. The second one applies if L is unknown. We show that for an optimal method, adaptiveness is necessary and that randomization (Monte-Carlo) yields no further advantage. Both algorithms presented have the optimal rate of convergence.
Horn REF has developed an optimization algorithm for Lipschitz continuous functions that does not require knowledge of the Lipschitz constant.
Optimal algorithms for global optimization in case of unknown Lipschitz constant
Algorithms and Complexity for Continuous Problems
Computer Science Mathematics