On the Pareto Optimality in the Context of Lipschitzian Optimization

Abstract

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, we propose a novel method called Pareto–Lipschitzian Optimization (PLO) that provides solutions within fixed error limits for functions with unknown Lipschitz constants. In the proposed approach, a set of all unknown Lipschitz constants is regarded as multiple criteria using the concept of Pareto Optimality (PO).We compare PLO to the existing algorithm DIRECT. We show that, in contrast to PLO, the DIRECT algorithm considers only a subset of PO decisions that are selected by a heuristic rule depending on an adjustable parameter. It means that some PO decisions are preferred to others. By contrast, PLO regards all PO decisions without preferences and is naturally suited to utilize highly parallel computing.