On the reduced-set pareto-lipschitzian optimization

A well-known example of global optimization that provides solutions within fixed error limits is optimization of functions with a known Lipschitz constant. In many real-life problems this constant is unknown. To address that a method called Pareto-Lipschitzian Optimization (PLO) was described that provides solutions within fixed error limits for functions with unknown Lipschitz constants. In this approach, a set of all unknown Lipschitz constants is regarded as multiple criteria using the concept of Pareto Optimality (PO). In this paper, a new version of the Pareto-Lipschitzian Optimization method (PLOR) is proposed where a set of unknown Lipschitzian constants is reduced just to the minimal and maximal ones. In the both methods, partition patterns are similar to those of DIRECT. The difference is in the rules of sequential partitions defining non-dominated sets. In PLO, it includes all Pareto-Optimal sets defined by all Lipschitz constants. In PLOR, it considers just two elements corresponding to the maximal and minimal Lipschitz constant. in DIRECT, it selects a part of the Pareto-Optimal set which is determined by some heuristic parameter .